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Portfolio Selection with Multiple Criteria

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Portfolio Selection with Multiple Criteria Bernhard Pfaff [email protected] Invesco Asset Management GmbH Frankfurt am Main R in Finance, Chicago IL, 20 and 21 May 2016 Pfaff (Invesco) MCO RFinance 2016 1 / 24
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Portfolio Selection with Multiple Criteria

Bernhard [email protected]

Invesco Asset Management GmbHFrankfurt am Main

R in Finance, Chicago IL, 20 and 21 May 2016

Pfaff (Invesco) MCO RFinance 2016 1 / 24

Contents

1 Overview

2 Multi Criteria Optimization

3 Example: Multi-asset class portfolio

4 Summary

5 Bibliography

Pfaff (Invesco) MCO RFinance 2016 2 / 24

Overview

Overview

Definition of multiple (conflicting) criteria objectives with respect toportfolio optimization problems.

Concept of non-dominated solutions (Pareto efficiency).

Solutions/points can determined by:1 GA/EMO: elitist non-dominated sorting algorithms (e.g. NSGA-II)2 MCDM: (classical) optimization (e.g. weighted-sum method)

Applied to multi-asset portfolio with objectives for:1 Mean return,2 Total Covariance Risk,3 Diversification with respect to assets’ risk contributions.

Pfaff (Invesco) MCO RFinance 2016 3 / 24

Multi Criteria Optimization

Multi Criteria OptimizationProblem formulation

minimize fm(ω), m = 1, 2, . . . ,M;

subject to gj(ω) ≥ 0, j = 1, 2, . . . , J;

hk(ω) = 0, k = 1, 2, . . . ,K ;

ω(L)i ≤ ωi ≤ ω

(U)i , i = 1, 2, . . . , n.

Problem: M (conflicting) objective functions and n (constrained)variables.

A solution ω ∈ Ω is efficient (Pareto optimal or non-dominated) ifthere is no ω ∈ Ω such that fk(ω) ≤ fk(ω) for k = 1, . . . , p andfi (ω) < fi (ω) for some i ∈ 1, . . . , k.

Pfaff (Invesco) MCO RFinance 2016 4 / 24

Multi Criteria Optimization

Multi Criteria OptimizationGA/EMO: Pareto efficient solutions

Goal: find solutions which lie on Pareto-efficient front and encompassits entire range.

Solutions can be determined by genetic NSGA-II algorithm.

However, no guarantee in finding optimial points on (close to)frontier.

NSGA-II consists of the following steps (a) create random population,(b) selection (non-dominant/constraint-dominant), (c) applyingvariation (crossover, mutation), (d) elitism (crowding distances).

An implementation is provided in the R package mco (see Mersmann,2014).

Pfaff (Invesco) MCO RFinance 2016 5 / 24

Multi Criteria Optimization

Multi Criteria OptimizationMCDM: Weighted-sum method

minimizeM∑

m=1

λmfm(ω), with λm ≥ 0;

subject to gj(ω) ≥ 0, j = 1, 2, . . . , J;

hk(ω) = 0, k = 1, 2, . . . ,K ;

ω(L)i ≤ ωi ≤ ω

(U)i , i = 1, 2, . . . , n.

Scaling/normalization of objective functions is important.

Goals and/or satisfycing levels can be included.

Possibility of a hybrid approach, where some objectives are specifiedas ε-constraints.

Pfaff (Invesco) MCO RFinance 2016 6 / 24

Multi Criteria Optimization

Multi Criteria OptimizationSynopsis of portfolio applications

Computation of non-dominated surface in tri-criterion problems:Hirschberger et al. (2013).

Bi-criterion EMO problem formulation with discontinuities in searchspace (risk & return with (i) either zero or within bounds allocationsand (ii) cardinality constraint on count of included assets): Deb et al.(2011).

Tri-criterion problems (quad-lin-lin):

risk, return and& transaction costs: Steuer et al. (2005), Steuer et al.(2013).risk, return and& ESG index: Utz et al. (2015).

Pfaff (Invesco) MCO RFinance 2016 7 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioSpecification

Data set MultiAsset contained in FRAPO: month-end data from11/2004 until 11/2011.

MCO conisting of three objectives:1 mean return,2 volatility, and3 dispersion of risk contributions.

Targeted return of 6% p.a.

Targeted volatility of 4% p.a.

Pfaff (Invesco) MCO RFinance 2016 8 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: Initializing

> library(FRAPO)

> library(mco)

> ## Loading of data

> data(MultiAsset)

> Prices <- timeSeries(MultiAsset,

+ charvec = rownames(MultiAsset))

> NAssets <- ncol(Prices)

> R <- returns(Prices, method = "discrete", percentage = TRUE)

> ## Defining parameters

> TargetRpa <- 6 ## percentage p.a.

> TargetR <- 100 * ((1 + TargetRpa / 100)^(1 / 12) - 1)

> TargetVol <- 4 ## percentage p.a.

> l <- rep(1, 3) ## goal weighting

> WeightedSum <- FALSE

> mu <- colMeans(R)

> S <- cov(R)

Pfaff (Invesco) MCO RFinance 2016 9 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: Multiple criteria objective and budget constraint

> f <- function(x)

+ y <- numeric(3)

+ y[1] <- -1.0 * l[1] * drop(crossprod(x, mu)) / TargetR

+ y[2] <- l[2] * drop(sqrt(t(x) %*% S %*% x)) *

+ sqrt(12) / TargetVol

+ y[3] <- l[3] * sum((mrc(x, S) / 100)^2)

+ if(WeightedSum)

+ return(sum(y))

+ else

+ return(y)

+

+

> g <- function(x)

+ c(1.01 - sum(x), sum(x) - 0.99)

+

Pfaff (Invesco) MCO RFinance 2016 10 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: Determining Pareto efficient solutions

> ans <- nsga2(f, NAssets, 3,

+ lower.bounds = rep(0, NAssets),

+ upper.bounds = rep(1, NAssets),

+ constraints = g, cdim = 2, popsize = 500)

> names(ans)

[1] "par" "value" "pareto.optimal"

> ## Preparing objective values for graphics

> mco <- data.frame(ans$value[ans$pareto.optimal, ])

> mco[, 1] <- ((1 + (-1.0 * mco[, 1] * TargetR) / 100)^12

+ - 1.0) * 100

> mco[, 2] <- mco[, 2] * TargetVol

> colnames(mco) <- c("Return", "Risk", "Diversification")

Pfaff (Invesco) MCO RFinance 2016 11 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: 3D scatterplot

> library(scatterplot3d)

> scatterplot3d(mco,

+ main = "Pareto Efficient Solutions",

+ sub = "Pareto Frontier (Surface)",

+ xlab = "Return Objective",

+ ylab = "Risk Objective",

+ zlab = "Dispersion of MRC",

+ angle = 15,

+ highlight.3d = FALSE,

+ box = TRUE,

+ color = "steelblue",

+ pch = 19, type = "p",

+ cex.symbols = 0.6)

Pareto Efficient Solutions

Pareto Frontier (Surface)

4 6 8 10 12 14 16 18 20

0.0

0.2

0.4

0.6

0.8

1.0

2 4

6 8

1012

1416

Return Objective

Ris

k O

bjec

tive

Dis

pers

ion

of M

RC

Pfaff (Invesco) MCO RFinance 2016 12 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: image plot

> library(akima)

> library(fields)

> s <- interp(mco[, 2], mco[, 1], mco[, 3],

+ xo = seq(min(mco[, 2]), max(mco[, 2]), length = 100),

+ yo = seq(min(mco[, 1]), max(mco[, 1]), length = 100),

+ duplicate = "mean"

+ )

> par(mar = c(5, 6, 5, 6))

> image.plot(s, nlevel = 50,

+ main = "Image plot of efficient set",

+ legend.lab = "Dispersion of MRC",

+ xlab = "Risk Objective",

+ ylab = "Return Objective",

+ legend.mar = 4,

+ horizontal = TRUE,

+ legend.shrink = 0.7,

+ col = topo.colors(50))

> contour(s, add = TRUE, nlevels = 20, labcex = 0.8)

> points(mco[, 2], mco[, 1], pch = 18, cex = 0.4, col = "orange")

4 6 8 10 12 14

68

1012

1416

18

Image plot of efficient set

Risk ObjectiveR

etur

n O

bjec

tive

0.2 0.4 0.6 0.8Dispersion of MRC

0.15

0.2 0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.8

Pfaff (Invesco) MCO RFinance 2016 13 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: weighting of objectives

grid <- expand.grid(x = seq(0.05, 0.95, by = 0.05),

y = seq(0.05, 0.95, by = 0.05))

grid <- grid[which(rowSums(grid) <= 1.0), ]

wobj <- as.matrix(cbind(grid, 1 - rowSums(grid)),

nrow = nrow(grid), ncol = 3)

W <- matrix(NA, nrow = nrow(wobj), ncol = NAssets)

WeightedSum <- TRUE

IneqA <- matrix(1, nrow = 1, ncol = NAssets)

ew <- rep(1 / NAssets, NAssets)

library(fPortfolio) ## for donlp2NLP

for(i in 1:nrow(wobj))

l <- c(wobj[i, ])

W[i, ] <- donlp2NLP(start = ew, objective = f,

par.lower = rep(0, NAssets),

ineqA = IneqA, ineqA.lower = 1.0,

ineqA.upper = 1.0)$solution

Pfaff (Invesco) MCO RFinance 2016 14 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: weighting of objectives & ternary plot

> library(PerformanceAnalytics)

> library(ggtern) ## Wahrschau! version < 2.0.1

> Es95Mod <- apply(W, 1, function(x)

+ r <- timeSeries(R %*% x / 100, time(R))

+ -100 * ES(r)

+ )

> terndat <- data.frame(cbind(wobj, Es95Mod))

> colnames(terndat) <- c("x", "y", "z", "value")

> ## Theme for ternary plot

> terntheme <- function()

+ list(theme_rgbg(),

+ theme(legend.position = c(0, 1),

+ legend.justification = c(0, 1),

+ plot.margin=unit(c(0, 2,0, 2), "cm"))

+ )

+

Pfaff (Invesco) MCO RFinance 2016 15 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: ternary plot, cont’d

> ## ternary plot

> ggtern(terndat, aes(x = x, y = y, z = z, value = value)) +

+ geom_interpolate_tern(aes(value = value, color = ..level..),

+ binwidth = 1.0) +

+ terntheme() +

+ theme_hidegrid_minor() +

+ theme_showgrid_major() +

+ Lline(0.2, color = "blue", linetype = 2) + ## x

+ Tline(0.3, colour = "red2", linetype = 2) + ## y

+ Rline(0.5, color = "brown", linetype = 2) + ## z

+ scale_color_gradient(low = "green", high = "red") +

+ labs(x = "Return", y = "Risk", z = "MRC",

+ title = "Ternary Plot with ES Contour Lines",

+ color = "Level")

100

80

60

40

20

100

80 60 40 20

100

80

60

40

20

Risk

Return MRC

Ris

k

Return

MR

C

0

4

8

12

16Level

Ternary Plot with ES Contour Lines

Pfaff (Invesco) MCO RFinance 2016 16 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: backtest, part I

> library(cccp) ## for ERC portolio

> ## backtest, extending window

> ep <- time(R)[-c(1:59)]

> bs <- length(ep)

> sp <- rep(start(R), bs)

> ## initialising object

> Wmco <- matrix(NA, nrow = bs, ncol = NAssets)

> Wmsr <- Wmdp <- Wgmv <- Werc <- Wmco

> l <- c(0.2, 0.1, 0.7) ## goal weighting

Pfaff (Invesco) MCO RFinance 2016 17 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: backtest, part II

> for(i in 1:bs)

+ rdat <- window(R, start = sp[i], end = ep[i])

+ mu <- colMeans(rdat)

+ S <- cov(rdat)

+ Wmco[i, ] <- donlp2NLP(start = ew, objective = f,

+ par.lower = rep(0, NAssets),

+ ineqA = IneqA, ineqA.lower = 1.0,

+ ineqA.upper = 1.0)$solution

+ ans <- tangencyPortfolio(rdat)

+ Wmsr[i, ] <- getWeights(ans)

+ ans <- PMD(rdat)

+ Wmdp[i, ] <- FRAPO::Weights(ans) / 100

+ ans <- PGMV(rdat)

+ Wgmv[i, ] <- FRAPO::Weights(ans) / 100

+ ans <- rp(ew, S, ew, optctrl = ctrl(trace = FALSE))

+ Werc[i, ] <- c(getx(ans))

+

Iteration: 0

pobj: 0.126801

dobj: -0.96263

pinf: 2.22045e-016

dinf: 3.83253

dgap: 1.08943

Iteration: 1

pobj: 0.124852

dobj: 0.00264633

pinf: 2.22045e-016

dinf: 0.427881

dgap: 0.122206

Iteration: 2

pobj: 0.118802

dobj: 0.0835599

pinf: 4.44089e-016

dinf: 0.111199

dgap: 0.0352419

Iteration: 3

pobj: 0.113548

dobj: 0.0993458

pinf: 7.77156e-016

dinf: 0.0372481

dgap: 0.014202

Iteration: 4

pobj: 0.107482

dobj: 0.104089

pinf: 1.55431e-015

dinf: 1.66533e-016

dgap: 0.00339286

Iteration: 5

pobj: 0.105591

dobj: 0.104892

pinf: 2.9976e-015

dinf: 2.23773e-016

dgap: 0.000698946

Iteration: 6

pobj: 0.105115

dobj: 0.105007

pinf: 5.77316e-015

dinf: 1.8619e-016

dgap: 0.000108626

Iteration: 7

pobj: 0.105047

dobj: 0.105027

pinf: 1.13243e-014

dinf: 1.00074e-016

dgap: 1.98598e-005

Iteration: 8

pobj: 0.105036

dobj: 0.105032

pinf: 2.23155e-014

dinf: 1.90283e-016

dgap: 4.3372e-006

Iteration: 9

pobj: 0.105034

dobj: 0.105033

pinf: 4.44089e-014

dinf: 1.24127e-016

dgap: 8.77542e-007

Iteration: 10

pobj: 0.105033

dobj: 0.105033

pinf: 8.82627e-014

dinf: 1.64204e-016

dgap: 1.12839e-007

Iteration: 11

pobj: 0.105033

dobj: 0.105033

pinf: 1.72196e-013

dinf: 1.94289e-016

dgap: 8.26087e-009

Optimal solution found.

Iteration: 0

pobj: 0.362971

dobj: -0.826572

pinf: 3.36575

dinf: 5.07688

dgap: 17.2127

Iteration: 1

pobj: 0.419846

dobj: -0.610122

pinf: 0.352087

dinf: 0.531087

dgap: 2.61477

Iteration: 2

pobj: 0.422811

dobj: -0.146063

pinf: 0.0836643

dinf: 0.126199

dgap: 0.888119

Iteration: 3

pobj: 0.381137

dobj: 0.149456

pinf: 0.0151466

dinf: 0.0228471

dgap: 0.271697

Iteration: 4

pobj: 0.332313

dobj: 0.250453

pinf: 0.00208538

dinf: 0.00314558

dgap: 0.0846849

Iteration: 5

pobj: 0.313591

dobj: 0.288708

pinf: 0.000394894

dinf: 0.000595657

dgap: 0.0251527

Iteration: 6

pobj: 0.303817

dobj: 0.2957

pinf: 6.39925e-005

dinf: 9.65262e-005

dgap: 0.00815032

Iteration: 7

pobj: 0.299024

dobj: 0.297668

pinf: 3.49767e-006

dinf: 5.27588e-006

dgap: 0.00135742

Iteration: 8

pobj: 0.297976

dobj: 0.297794

pinf: 3.25552e-007

dinf: 4.91062e-007

dgap: 0.000182128

Iteration: 9

pobj: 0.297814

dobj: 0.297802

pinf: 1.91646e-008

dinf: 2.89079e-008

dgap: 1.23223e-005

Iteration: 10

pobj: 0.297803

dobj: 0.297802

pinf: 9.70921e-010

dinf: 1.46454e-009

dgap: 6.32737e-007

Iteration: 11

pobj: 0.297802

dobj: 0.297802

pinf: 4.85793e-011

dinf: 7.32765e-011

dgap: 3.16812e-008

Optimal solution found.

Iteration: 0

pobj: 0.123721

dobj: -0.963402

pinf: 1.11022e-016

dinf: 3.82468

dgap: 1.08712

Iteration: 1

pobj: 0.121862

dobj: 0.00123752

pinf: 2.22045e-016

dinf: 0.422508

dgap: 0.120624

Iteration: 2

pobj: 0.115814

dobj: 0.0810899

pinf: 2.22045e-016

dinf: 0.109938

dgap: 0.0347246

Iteration: 3

pobj: 0.11028

dobj: 0.0972651

pinf: 4.44089e-016

dinf: 0.0338733

dgap: 0.0130149

Iteration: 4

pobj: 0.104565

dobj: 0.101339

pinf: 8.88178e-016

dinf: 0.000902096

dgap: 0.00322598

Iteration: 5

pobj: 0.102671

dobj: 0.102068

pinf: 1.77636e-015

dinf: 5.70661e-005

dgap: 0.000603327

Iteration: 6

pobj: 0.102259

dobj: 0.102169

pinf: 3.44169e-015

dinf: 2.69901e-006

dgap: 9.01822e-005

Iteration: 7

pobj: 0.102204

dobj: 0.102187

pinf: 6.77236e-015

dinf: 1.28266e-007

dgap: 1.73152e-005

Iteration: 8

pobj: 0.102194

dobj: 0.10219

pinf: 1.33227e-014

dinf: 2.63174e-009

dgap: 3.7038e-006

Iteration: 9

pobj: 0.102191

dobj: 0.102191

pinf: 2.65343e-014

dinf: 4.96533e-011

dgap: 6.94795e-007

Iteration: 10

pobj: 0.102191

dobj: 0.102191

pinf: 5.19584e-014

dinf: 1.92505e-012

dgap: 7.89682e-008

Optimal solution found.

Iteration: 0

pobj: 0.35774

dobj: -0.832046

pinf: 3.3757

dinf: 5.0986

dgap: 17.3336

Iteration: 1

pobj: 0.415478

dobj: -0.61496

pinf: 0.362862

dinf: 0.548061

dgap: 2.67242

Iteration: 2

pobj: 0.420214

dobj: -0.158925

pinf: 0.0885245

dinf: 0.133706

dgap: 0.921424

Iteration: 3

pobj: 0.379027

dobj: 0.142668

pinf: 0.0154915

dinf: 0.0233981

dgap: 0.278041

Iteration: 4

pobj: 0.328734

dobj: 0.245164

pinf: 0.00211642

dinf: 0.00319662

dgap: 0.0865185

Iteration: 5

pobj: 0.308997

dobj: 0.284874

pinf: 0.000357781

dinf: 0.000540387

dgap: 0.0243708

Iteration: 6

pobj: 0.299609

dobj: 0.291457

pinf: 6.58067e-005

dinf: 9.93934e-005

dgap: 0.00818694

Iteration: 7

pobj: 0.294797

dobj: 0.2934

pinf: 4.03528e-006

dinf: 6.09483e-006

dgap: 0.00139899

Iteration: 8

pobj: 0.293728

dobj: 0.29354

pinf: 3.8613e-007

dinf: 5.83205e-007

dgap: 0.00018854

Iteration: 9

pobj: 0.293562

dobj: 0.293549

pinf: 2.28877e-008

dinf: 3.45692e-008

dgap: 1.27549e-005

Iteration: 10

pobj: 0.293551

dobj: 0.29355

pinf: 1.16015e-009

dinf: 1.75227e-009

dgap: 6.54844e-007

Iteration: 11

pobj: 0.29355

dobj: 0.29355

pinf: 5.80487e-011

dinf: 8.76747e-011

dgap: 3.27877e-008

Optimal solution found.

Iteration: 0

pobj: 0.123384

dobj: -0.962439

pinf: 1.11022e-016

dinf: 3.81849

dgap: 1.08582

Iteration: 1

pobj: 0.121607

dobj: 0.00201636

pinf: 1.11022e-016

dinf: 0.418756

dgap: 0.11959

Iteration: 2

pobj: 0.115834

dobj: 0.081482

pinf: 2.22045e-016

dinf: 0.108843

dgap: 0.0343518

Iteration: 3

pobj: 0.110506

dobj: 0.0978079

pinf: 3.33067e-016

dinf: 0.0329326

dgap: 0.0126977

Iteration: 4

pobj: 0.10513

dobj: 0.10191

pinf: 6.66134e-016

dinf: 0.00104159

dgap: 0.0032198

Iteration: 5

pobj: 0.103326

dobj: 0.102697

pinf: 1.33227e-015

dinf: 6.32514e-005

dgap: 0.000629017

Iteration: 6

pobj: 0.10292

dobj: 0.102822

pinf: 2.66454e-015

dinf: 2.75908e-006

dgap: 9.83016e-005

Iteration: 7

pobj: 0.102866

dobj: 0.102848

pinf: 5.21805e-015

dinf: 1.30988e-007

dgap: 1.79689e-005

Iteration: 8

pobj: 0.102859

dobj: 0.102856

pinf: 1.03251e-014

dinf: 2.35916e-009

dgap: 3.40098e-006

Iteration: 9

pobj: 0.102859

dobj: 0.102858

pinf: 2.02061e-014

dinf: 7.24439e-011

dgap: 5.93075e-007

Iteration: 10

pobj: 0.102859

dobj: 0.102859

pinf: 3.9746e-014

dinf: 3.26919e-012

dgap: 5.96908e-008

Optimal solution found.

Iteration: 0

pobj: 0.354538

dobj: -0.835719

pinf: 3.37699

dinf: 5.08478

dgap: 17.2936

Iteration: 1

pobj: 0.410595

dobj: -0.619505

pinf: 0.363421

dinf: 0.547207

dgap: 2.66966

Iteration: 2

pobj: 0.415523

dobj: -0.163452

pinf: 0.0886577

dinf: 0.133493

dgap: 0.920655

Iteration: 3

pobj: 0.37489

dobj: 0.138624

pinf: 0.0155736

dinf: 0.0234493

dgap: 0.278065

Iteration: 4

pobj: 0.324781

dobj: 0.240768

pinf: 0.00213858

dinf: 0.00322009

dgap: 0.0869943

Iteration: 5

pobj: 0.304854

dobj: 0.281612

pinf: 0.000325256

dinf: 0.000489741

dgap: 0.0234594

Iteration: 6

pobj: 0.295994

dobj: 0.28797

pinf: 6.22552e-005

dinf: 9.37384e-005

dgap: 0.00805627

Iteration: 7

pobj: 0.291318

dobj: 0.289908

pinf: 3.5414e-006

dinf: 5.33232e-006

dgap: 0.00141155

Iteration: 8

pobj: 0.290252

dobj: 0.29005

pinf: 3.48668e-007

dinf: 5.24993e-007

dgap: 0.000202496

Iteration: 9

pobj: 0.290074

dobj: 0.29006

pinf: 2.12652e-008

dinf: 3.20192e-008

dgap: 1.45576e-005

Iteration: 10

pobj: 0.290061

dobj: 0.29006

pinf: 1.08233e-009

dinf: 1.62968e-009

dgap: 7.5469e-007

Iteration: 11

pobj: 0.290061

dobj: 0.29006

pinf: 5.41673e-011

dinf: 8.15601e-011

dgap: 3.78077e-008

Optimal solution found.

Iteration: 0

pobj: 0.123232

dobj: -0.961476

pinf: 2.22045e-016

dinf: 3.81415

dgap: 1.08471

Iteration: 1

pobj: 0.121498

dobj: 0.00278445

pinf: 2.22045e-016

dinf: 0.415683

dgap: 0.118714

Iteration: 2

pobj: 0.11584

dobj: 0.0819563

pinf: 6.66134e-016

dinf: 0.107489

dgap: 0.0338839

Iteration: 3

pobj: 0.110571

dobj: 0.0978265

pinf: 9.99201e-016

dinf: 0.0331768

dgap: 0.0127444

Iteration: 4

pobj: 0.105146

dobj: 0.102043

pinf: 1.9984e-015

dinf: 0.000770381

dgap: 0.00310324

Iteration: 5

pobj: 0.103416

dobj: 0.102824

pinf: 3.88578e-015

dinf: 4.96563e-005

dgap: 0.000591971

Iteration: 6

pobj: 0.103052

dobj: 0.102966

pinf: 7.54952e-015

dinf: 2.50231e-006

dgap: 8.59119e-005

Iteration: 7

pobj: 0.103009

dobj: 0.102999

pinf: 1.46549e-014

dinf: 1.30899e-007

dgap: 1.08648e-005

Iteration: 8

pobj: 0.103006

dobj: 0.103005

pinf: 2.85327e-014

dinf: 6.69656e-009

dgap: 8.78254e-007

Iteration: 9

pobj: 0.103006

dobj: 0.103006

pinf: 5.56222e-014

dinf: 3.35782e-010

dgap: 4.72658e-008

Optimal solution found.

Iteration: 0

pobj: 0.351675

dobj: -0.837846

pinf: 3.37288

dinf: 5.07138

dgap: 17.2281

Iteration: 1

pobj: 0.407214

dobj: -0.622043

pinf: 0.360178

dinf: 0.541556

dgap: 2.64917

Iteration: 2

pobj: 0.412212

dobj: -0.163113

pinf: 0.0866316

dinf: 0.130257

dgap: 0.906931

Iteration: 3

pobj: 0.371608

dobj: 0.137345

pinf: 0.015385

dinf: 0.0231325

dgap: 0.275101

Iteration: 4

pobj: 0.322178

dobj: 0.238962

pinf: 0.00212085

dinf: 0.00318886

dgap: 0.0861418

Iteration: 5

pobj: 0.302614

dobj: 0.278608

pinf: 0.000351124

dinf: 0.000527943

dgap: 0.0242462

Iteration: 6

pobj: 0.293291

dobj: 0.285236

pinf: 6.33175e-005

dinf: 9.52028e-005

dgap: 0.00808745

Iteration: 7

pobj: 0.288546

dobj: 0.287156

pinf: 3.73612e-006

dinf: 5.61755e-006

dgap: 0.00139091

Iteration: 8

pobj: 0.287471

dobj: 0.287289

pinf: 3.34871e-007

dinf: 5.03505e-007

dgap: 0.000182441

Iteration: 9

pobj: 0.287309

dobj: 0.287297

pinf: 1.91704e-008

dinf: 2.88242e-008

dgap: 1.19711e-005

Iteration: 10

pobj: 0.287298

dobj: 0.287297

pinf: 9.67931e-010

dinf: 1.45536e-009

dgap: 6.11765e-007

Iteration: 11

pobj: 0.287297

dobj: 0.287297

pinf: 4.84209e-011

dinf: 7.28043e-011

dgap: 3.06229e-008

Optimal solution found.

Iteration: 0

pobj: 0.123672

dobj: -0.958803

pinf: 0

dinf: 3.80571

dgap: 1.08247

Iteration: 1

pobj: 0.122002

dobj: 0.00508167

pinf: 7.69561e-017

dinf: 0.409394

dgap: 0.11692

Iteration: 2

pobj: 0.116563

dobj: 0.0834193

pinf: 1.11022e-016

dinf: 0.105204

dgap: 0.0331438

Iteration: 3

pobj: 0.111506

dobj: 0.0990009

pinf: 2.22045e-016

dinf: 0.0324776

dgap: 0.0125053

Iteration: 4

pobj: 0.106235

dobj: 0.103226

pinf: 3.33067e-016

dinf: 0.00030351

dgap: 0.00300823

Iteration: 5

pobj: 0.104614

dobj: 0.104014

pinf: 6.66134e-016

dinf: 1.97027e-005

dgap: 0.000599909

Iteration: 6

pobj: 0.104264

dobj: 0.104166

pinf: 1.33227e-015

dinf: 9.01238e-007

dgap: 9.73663e-005

Iteration: 7

pobj: 0.104215

dobj: 0.104201

pinf: 2.66454e-015

dinf: 4.55005e-008

dgap: 1.42392e-005

Iteration: 8

pobj: 0.104208

dobj: 0.104207

pinf: 5.10703e-015

dinf: 2.3187e-009

dgap: 1.21067e-006

Iteration: 9

pobj: 0.104208

dobj: 0.104208

pinf: 9.99201e-015

dinf: 1.1635e-010

dgap: 6.57644e-008

Optimal solution found.

Iteration: 0

pobj: 0.356938

dobj: -0.834683

pinf: 3.36383

dinf: 5.0603

dgap: 17.1499

Iteration: 1

pobj: 0.410054

dobj: -0.619

pinf: 0.353629

dinf: 0.531974

dgap: 2.61503

Iteration: 2

pobj: 0.413344

dobj: -0.154362

pinf: 0.0829905

dinf: 0.124845

dgap: 0.882619

Iteration: 3

pobj: 0.371293

dobj: 0.140296

pinf: 0.0151086

dinf: 0.0227283

dgap: 0.270588

Iteration: 4

pobj: 0.322159

dobj: 0.240021

pinf: 0.00207286

dinf: 0.00311827

dgap: 0.0849452

Iteration: 5

pobj: 0.302798

dobj: 0.279666

pinf: 0.000323197

dinf: 0.000486195

dgap: 0.0233401

Iteration: 6

pobj: 0.293872

dobj: 0.286032

pinf: 5.80907e-005

dinf: 8.73876e-005

dgap: 0.00786931

Iteration: 7

pobj: 0.289283

dobj: 0.287889

pinf: 3.54359e-006

dinf: 5.33073e-006

dgap: 0.00139609

Iteration: 8

pobj: 0.288218

dobj: 0.288026

pinf: 3.34809e-007

dinf: 5.03664e-007

dgap: 0.000192126

Iteration: 9

pobj: 0.288048

dobj: 0.288035

pinf: 1.97232e-008

dinf: 2.96702e-008

dgap: 1.30708e-005

Iteration: 10

pobj: 0.288036

dobj: 0.288035

pinf: 9.98953e-010

dinf: 1.50275e-009

dgap: 6.71042e-007

Iteration: 11

pobj: 0.288035

dobj: 0.288035

pinf: 4.9981e-011

dinf: 7.5188e-011

dgap: 3.35985e-008

Optimal solution found.

Iteration: 0

pobj: 0.123828

dobj: -0.958768

pinf: 0

dinf: 3.80567

dgap: 1.0826

Iteration: 1

pobj: 0.122163

dobj: 0.00512226

pinf: 8.29771e-017

dinf: 0.409766

dgap: 0.117041

Iteration: 2

pobj: 0.116729

dobj: 0.0835203

pinf: 4.55014e-017

dinf: 0.105423

dgap: 0.0332091

Iteration: 3

pobj: 0.111667

dobj: 0.099081

pinf: 1.11022e-016

dinf: 0.0327148

dgap: 0.0125857

Iteration: 4

pobj: 0.106334

dobj: 0.103347

pinf: 1.11022e-016

dinf: 0.000148925

dgap: 0.00298678

Iteration: 5

pobj: 0.104712

dobj: 0.104116

pinf: 2.22045e-016

dinf: 9.79927e-006

dgap: 0.000595376

Iteration: 6

pobj: 0.104362

dobj: 0.104265

pinf: 5.55112e-016

dinf: 4.55414e-007

dgap: 9.74638e-005

Iteration: 7

pobj: 0.104313

dobj: 0.104298

pinf: 9.99201e-016

dinf: 2.26524e-008

dgap: 1.47448e-005

Iteration: 8

pobj: 0.104307

dobj: 0.104305

pinf: 1.9984e-015

dinf: 1.14862e-009

dgap: 1.35756e-006

Iteration: 9

pobj: 0.104306

dobj: 0.104306

pinf: 3.88578e-015

dinf: 5.76651e-011

dgap: 7.64563e-008

Optimal solution found.

Iteration: 0

pobj: 0.354456

dobj: -0.837507

pinf: 3.35771

dinf: 5.05822

dgap: 17.1142

Iteration: 1

pobj: 0.407408

dobj: -0.621565

pinf: 0.348215

dinf: 0.524568

dgap: 2.58953

Iteration: 2

pobj: 0.410091

dobj: -0.152485

pinf: 0.0799313

dinf: 0.120412

dgap: 0.864013

Iteration: 3

pobj: 0.366946

dobj: 0.138263

pinf: 0.0147853

dinf: 0.0222733

dgap: 0.266957

Iteration: 4

pobj: 0.317962

dobj: 0.236951

pinf: 0.00202432

dinf: 0.00304953

dgap: 0.0837088

Iteration: 5

pobj: 0.298775

dobj: 0.275807

pinf: 0.000320591

dinf: 0.000482953

dgap: 0.0231729

Iteration: 6

pobj: 0.289855

dobj: 0.282122

pinf: 5.67655e-005

dinf: 8.55142e-005

dgap: 0.00776052

Iteration: 7

pobj: 0.285313

dobj: 0.283941

pinf: 3.54152e-006

dinf: 5.33511e-006

dgap: 0.00137392

Iteration: 8

pobj: 0.28426

dobj: 0.284074

pinf: 3.32216e-007

dinf: 5.00466e-007

dgap: 0.000186246

Iteration: 9

pobj: 0.284095

dobj: 0.284082

pinf: 1.94344e-008

dinf: 2.92769e-008

dgap: 1.24804e-005

Iteration: 10

pobj: 0.284083

dobj: 0.284083

pinf: 9.83431e-010

dinf: 1.48149e-009

dgap: 6.39364e-007

Iteration: 11

pobj: 0.284083

dobj: 0.284083

pinf: 4.9202e-011

dinf: 7.41203e-011

dgap: 3.20087e-008

Optimal solution found.

Iteration: 0

pobj: 0.120572

dobj: -0.963489

pinf: 2.22045e-016

dinf: 3.80865

dgap: 1.08406

Iteration: 1

pobj: 0.11892

dobj: 0.000848444

pinf: 4.44089e-016

dinf: 0.413097

dgap: 0.118072

Iteration: 2

pobj: 0.113618

dobj: 0.0799412

pinf: 6.66134e-016

dinf: 0.106839

dgap: 0.0336768

Iteration: 3

pobj: 0.108609

dobj: 0.0974965

pinf: 1.11022e-015

dinf: 0.0282408

dgap: 0.0111121

Iteration: 4

pobj: 0.104363

dobj: 0.101038

pinf: 2.22045e-015

dinf: 0.0024385

dgap: 0.00332474

Iteration: 5

pobj: 0.102658

dobj: 0.101973

pinf: 4.44089e-015

dinf: 6.13585e-005

dgap: 0.000685151

Iteration: 6

pobj: 0.102245

dobj: 0.102123

pinf: 8.65974e-015

dinf: 2.655e-006

dgap: 0.000121783

Iteration: 7

pobj: 0.102174

dobj: 0.102155

pinf: 1.68754e-014

dinf: 1.15466e-007

dgap: 1.91016e-005

Iteration: 8

pobj: 0.102166

dobj: 0.102163

pinf: 3.28626e-014

dinf: 5.84258e-009

dgap: 2.60925e-006

Iteration: 9

pobj: 0.102165

dobj: 0.102165

pinf: 6.40599e-014

dinf: 2.95638e-010

dgap: 2.20304e-007

Iteration: 10

pobj: 0.102165

dobj: 0.102165

pinf: 1.249e-013

dinf: 1.48168e-011

dgap: 1.19855e-008

Optimal solution found.

Iteration: 0

pobj: 0.355536

dobj: -0.837486

pinf: 3.35801

dinf: 5.03847

dgap: 17.0503

Iteration: 1

pobj: 0.406298

dobj: -0.623431

pinf: 0.347674

dinf: 0.521661

dgap: 2.58085

Iteration: 2

pobj: 0.40821

dobj: -0.153157

pinf: 0.0789282

dinf: 0.118427

dgap: 0.856748

Iteration: 3

pobj: 0.365108

dobj: 0.136459

pinf: 0.0147352

dinf: 0.0221091

dgap: 0.266264

Iteration: 4

pobj: 0.316844

dobj: 0.236044

pinf: 0.00202041

dinf: 0.00303149

dgap: 0.0834158

Iteration: 5

pobj: 0.297495

dobj: 0.277706

pinf: 0.0002203

dinf: 0.000330545

dgap: 0.0199108

Iteration: 6

pobj: 0.290699

dobj: 0.283204

pinf: 5.38778e-005

dinf: 8.08401e-005

dgap: 0.00751782

Iteration: 7

pobj: 0.286699

dobj: 0.285187

pinf: 2.36363e-006

dinf: 3.54646e-006

dgap: 0.00151324

Iteration: 8

pobj: 0.285716

dobj: 0.285416

pinf: 2.40962e-007

dinf: 3.61548e-007

dgap: 0.000299523

Iteration: 9

pobj: 0.285478

dobj: 0.285441

pinf: 1.30473e-008

dinf: 1.95765e-008

dgap: 3.67768e-005

Iteration: 10

pobj: 0.285445

dobj: 0.285443

pinf: 7.12605e-010

dinf: 1.06922e-009

dgap: 2.85218e-006

Iteration: 11

pobj: 0.285443

dobj: 0.285443

pinf: 3.60473e-011

dinf: 5.40867e-011

dgap: 1.5117e-007

Optimal solution found.

Iteration: 0

pobj: 0.119834

dobj: -0.963069

pinf: 0

dinf: 3.80328

dgap: 1.0829

Iteration: 1

pobj: 0.118243

dobj: 0.00110299

pinf: 7.00794e-017

dinf: 0.409733

dgap: 0.11714

Iteration: 2

pobj: 0.113155

dobj: 0.0799458

pinf: 3.53816e-017

dinf: 0.105445

dgap: 0.0332088

Iteration: 3

pobj: 0.108351

dobj: 0.097328

pinf: 2.22045e-016

dinf: 0.0281211

dgap: 0.0110234

Iteration: 4

pobj: 0.104218

dobj: 0.10094

pinf: 2.22045e-016

dinf: 0.00229415

dgap: 0.00327804

Iteration: 5

pobj: 0.102588

dobj: 0.101903

pinf: 4.44089e-016

dinf: 5.84166e-005

dgap: 0.000684432

Iteration: 6

pobj: 0.102192

dobj: 0.10207

pinf: 8.88178e-016

dinf: 2.57387e-006

dgap: 0.000122657

Iteration: 7

pobj: 0.102124

dobj: 0.102106

pinf: 1.77636e-015

dinf: 1.18507e-007

dgap: 1.7633e-005

Iteration: 8

pobj: 0.102115

dobj: 0.102113

pinf: 3.55271e-015

dinf: 6.1043e-009

dgap: 1.59537e-006

Iteration: 9

pobj: 0.102114

dobj: 0.102114

pinf: 6.66134e-015

dinf: 3.06637e-010

dgap: 8.73248e-008

Optimal solution found.

Iteration: 0

pobj: 0.352235

dobj: -0.840882

pinf: 3.35497

dinf: 5.02725

dgap: 16.9985

Iteration: 1

pobj: 0.401996

dobj: -0.627414

pinf: 0.34653

dinf: 0.519257

dgap: 2.57143

Iteration: 2

pobj: 0.403933

dobj: -0.15606

pinf: 0.0781291

dinf: 0.117072

dgap: 0.851064

Iteration: 3

pobj: 0.36086

dobj: 0.13277

pinf: 0.0146947

dinf: 0.0220192

dgap: 0.265347

Iteration: 4

pobj: 0.312815

dobj: 0.232108

pinf: 0.00201562

dinf: 0.00302031

dgap: 0.0832983

Iteration: 5

pobj: 0.293472

dobj: 0.273739

pinf: 0.000215875

dinf: 0.000323477

dgap: 0.019851

Iteration: 6

pobj: 0.286152

dobj: 0.279249

pinf: 4.21906e-005

dinf: 6.32204e-005

dgap: 0.00691984

Iteration: 7

pobj: 0.282389

dobj: 0.281014

pinf: 2.10034e-006

dinf: 3.14725e-006

dgap: 0.00137536

Iteration: 8

pobj: 0.281459

dobj: 0.281204

pinf: 1.99725e-007

dinf: 2.99278e-007

dgap: 0.000255898

Iteration: 9

pobj: 0.281251

dobj: 0.281222

pinf: 1.17218e-008

dinf: 1.75644e-008

dgap: 2.96122e-005

Iteration: 10

pobj: 0.281225

dobj: 0.281222

pinf: 6.33569e-010

dinf: 9.49369e-010

dgap: 2.0657e-006

Iteration: 11

pobj: 0.281223

dobj: 0.281222

pinf: 3.19088e-011

dinf: 4.78138e-011

dgap: 1.06887e-007

Optimal solution found.

Iteration: 0

pobj: 0.117959

dobj: -0.965968

pinf: 2.22045e-016

dinf: 3.80402

dgap: 1.08393

Iteration: 1

pobj: 0.116408

dobj: -0.00120716

pinf: 3.33067e-016

dinf: 0.411025

dgap: 0.117615

Iteration: 2

pobj: 0.111886

dobj: 0.0791923

pinf: 6.66134e-016

dinf: 0.103513

dgap: 0.032694

Iteration: 3

pobj: 0.108142

dobj: 0.0986186

pinf: 1.22125e-015

dinf: 0.0236912

dgap: 0.00952298

Iteration: 4

pobj: 0.105629

dobj: 0.102493

pinf: 2.22045e-015

dinf: 0.00298173

dgap: 0.0031366

Iteration: 5

pobj: 0.104677

dobj: 0.103975

pinf: 4.44089e-015

dinf: 1.46869e-016

dgap: 0.000702338

Iteration: 6

pobj: 0.104493

dobj: 0.104337

pinf: 8.65974e-015

dinf: 1.17757e-016

dgap: 0.00015564

Iteration: 7

pobj: 0.104458

dobj: 0.104425

pinf: 1.67644e-014

dinf: 1.46869e-016

dgap: 3.28108e-005

Iteration: 8

pobj: 0.104451

dobj: 0.104443

pinf: 3.21965e-014

dinf: 1.54537e-016

dgap: 7.4908e-006

Iteration: 9

pobj: 0.104449

dobj: 0.104447

pinf: 5.80647e-014

dinf: 1.44222e-016

dgap: 2.03276e-006

Iteration: 10

pobj: 0.104448

dobj: 0.104447

pinf: 1.05915e-013

dinf: 8.77708e-017

dgap: 5.2115e-007

Iteration: 11

pobj: 0.104448

dobj: 0.104448

pinf: 2.06501e-013

dinf: 1.17757e-016

dgap: 8.7804e-008

Optimal solution found.

Iteration: 0

pobj: 0.363208

dobj: -0.831093

pinf: 3.33886

dinf: 4.97613

dgap: 16.7531

Iteration: 1

pobj: 0.405647

dobj: -0.619192

pinf: 0.336719

dinf: 0.501835

dgap: 2.50551

Iteration: 2

pobj: 0.405875

dobj: -0.142641

pinf: 0.0734616

dinf: 0.109485

dgap: 0.816231

Iteration: 3

pobj: 0.363526

dobj: 0.140595

pinf: 0.0140557

dinf: 0.0209482

dgap: 0.257183

Iteration: 4

pobj: 0.319154

dobj: 0.240886

pinf: 0.00188664

dinf: 0.00281179

dgap: 0.0805058

Iteration: 5

pobj: 0.302186

dobj: 0.28447

pinf: 0.000143818

dinf: 0.000214342

dgap: 0.0177788

Iteration: 6

pobj: 0.297486

dobj: 0.291089

pinf: 3.44784e-005

dinf: 5.13855e-005

dgap: 0.00640617

Iteration: 7

pobj: 0.295016

dobj: 0.293756

pinf: 6.77545e-007

dinf: 1.00979e-006

dgap: 0.00126035

Iteration: 8

pobj: 0.294482

dobj: 0.294252

pinf: 5.49922e-008

dinf: 8.19586e-008

dgap: 0.000230268

Iteration: 9

pobj: 0.294385

dobj: 0.294311

pinf: 1.54543e-008

dinf: 2.30325e-008

dgap: 7.38351e-005

Iteration: 10

pobj: 0.294341

dobj: 0.294325

pinf: 2.95609e-009

dinf: 4.40567e-009

dgap: 1.57884e-005

Iteration: 11

pobj: 0.29433

dobj: 0.294329

pinf: 2.68477e-010

dinf: 4.0013e-010

dgap: 1.51719e-006

Iteration: 12

pobj: 0.294329

dobj: 0.294329

pinf: 1.46059e-011

dinf: 2.1768e-011

dgap: 8.34121e-008

Optimal solution found.

Iteration: 0

pobj: 0.117461

dobj: -0.964915

pinf: 0

dinf: 3.79704

dgap: 1.08238

Iteration: 1

pobj: 0.115984

dobj: -0.000549955

pinf: 2.22045e-016

dinf: 0.407162

dgap: 0.116534

Iteration: 2

pobj: 0.111511

dobj: 0.0788346

pinf: 2.22045e-016

dinf: 0.103723

dgap: 0.0326761

Iteration: 3

pobj: 0.10771

dobj: 0.0974338

pinf: 4.44089e-016

dinf: 0.0258255

dgap: 0.0102759

Iteration: 4

pobj: 0.105485

dobj: 0.101423

pinf: 8.88178e-016

dinf: 0.00687743

dgap: 0.00406256

Iteration: 5

pobj: 0.104204

dobj: 0.103265

pinf: 1.77636e-015

dinf: 9.89593e-005

dgap: 0.000938682

Iteration: 6

pobj: 0.103908

dobj: 0.103706

pinf: 3.33067e-015

dinf: 6.15003e-006

dgap: 0.000202417

Iteration: 7

pobj: 0.103838

dobj: 0.10379

pinf: 6.21725e-015

dinf: 7.30475e-007

dgap: 4.77859e-005

Iteration: 8

pobj: 0.103818

dobj: 0.103806

pinf: 1.15463e-014

dinf: 1.1318e-007

dgap: 1.20983e-005

Iteration: 9

pobj: 0.103811

dobj: 0.103807

pinf: 2.08722e-014

dinf: 2.28524e-008

dgap: 3.37276e-006

Iteration: 10

pobj: 0.103808

dobj: 0.103808

pinf: 3.9968e-014

dinf: 1.90124e-009

dgap: 4.65286e-007

Iteration: 11

pobj: 0.103808

dobj: 0.103808

pinf: 7.77156e-014

dinf: 1.0103e-010

dgap: 3.34638e-008

Optimal solution found.

Iteration: 0

pobj: 0.385274

dobj: -0.811764

pinf: 3.34789

dinf: 4.96184

dgap: 16.7501

Iteration: 1

pobj: 0.425933

dobj: -0.60281

pinf: 0.344618

dinf: 0.510751

dgap: 2.53942

Iteration: 2

pobj: 0.427563

dobj: -0.131613

pinf: 0.0765528

dinf: 0.113457

dgap: 0.838096

Iteration: 3

pobj: 0.386747

dobj: 0.158417

pinf: 0.0145386

dinf: 0.0215473

dgap: 0.263701

Iteration: 4

pobj: 0.341808

dobj: 0.261327

pinf: 0.00199159

dinf: 0.0029517

dgap: 0.0828522

Iteration: 5

pobj: 0.324467

dobj: 0.303467

pinf: 0.000243478

dinf: 0.000360853

dgap: 0.021118

Iteration: 6

pobj: 0.318583

dobj: 0.312041

pinf: 4.74853e-005

dinf: 7.03771e-005

dgap: 0.00655609

Iteration: 7

pobj: 0.315852

dobj: 0.314476

pinf: 3.20168e-006

dinf: 4.74514e-006

dgap: 0.00137698

Iteration: 8

pobj: 0.315215

dobj: 0.314771

pinf: 8.31961e-007

dinf: 1.23303e-006

dgap: 0.000443206

Iteration: 9

pobj: 0.314924

dobj: 0.314825

pinf: 1.39334e-007

dinf: 2.06505e-007

dgap: 9.95035e-005

Iteration: 10

pobj: 0.314848

dobj: 0.314836

pinf: 1.43918e-008

dinf: 2.13298e-008

dgap: 1.19034e-005

Iteration: 11

pobj: 0.314838

dobj: 0.314837

pinf: 8.57183e-010

dinf: 1.27041e-009

dgap: 7.43637e-007

Iteration: 12

pobj: 0.314837

dobj: 0.314837

pinf: 4.33867e-011

dinf: 6.43032e-011

dgap: 3.77926e-008

Optimal solution found.

Iteration: 0

pobj: 0.116791

dobj: -0.967377

pinf: 1.11022e-016

dinf: 3.80334

dgap: 1.08417

Iteration: 1

pobj: 0.115279

dobj: -0.00262861

pinf: 2.22045e-016

dinf: 0.411907

dgap: 0.117908

Iteration: 2

pobj: 0.110732

dobj: 0.0774768

pinf: 4.44089e-016

dinf: 0.105455

dgap: 0.0332556

Iteration: 3

pobj: 0.106893

dobj: 0.096494

pinf: 6.66134e-016

dinf: 0.0261378

dgap: 0.0103991

Iteration: 4

pobj: 0.104649

dobj: 0.100586

pinf: 9.99201e-016

dinf: 0.00685832

dgap: 0.0040624

Iteration: 5

pobj: 0.103386

dobj: 0.102436

pinf: 1.9984e-015

dinf: 0.000136677

dgap: 0.000949866

Iteration: 6

pobj: 0.103096

dobj: 0.102896

pinf: 3.88578e-015

dinf: 7.68168e-006

dgap: 0.000199878

Iteration: 7

pobj: 0.103032

dobj: 0.102989

pinf: 7.43849e-015

dinf: 8.37687e-007

dgap: 4.2728e-005

Iteration: 8

pobj: 0.103016

dobj: 0.103002

pinf: 1.31006e-014

dinf: 1.92874e-007

dgap: 1.38698e-005

Iteration: 9

pobj: 0.103009

dobj: 0.103005

pinf: 2.40918e-014

dinf: 2.90994e-008

dgap: 3.67126e-006

Iteration: 10

pobj: 0.103006

dobj: 0.103006

pinf: 4.69624e-014

dinf: 1.43561e-009

dgap: 6.08212e-007

Iteration: 11

pobj: 0.103006

dobj: 0.103006

pinf: 9.18154e-014

dinf: 6.71023e-011

dgap: 8.06956e-008

Optimal solution found.

Iteration: 0

pobj: 0.38066

dobj: -0.816103

pinf: 3.34714

dinf: 4.95893

dgap: 16.7365

Iteration: 1

pobj: 0.420876

dobj: -0.606924

pinf: 0.343878

dinf: 0.509469

dgap: 2.53425

Iteration: 2

pobj: 0.422235

dobj: -0.134453

pinf: 0.0759615

dinf: 0.11254

dgap: 0.833128

Iteration: 3

pobj: 0.381545

dobj: 0.154297

pinf: 0.0144642

dinf: 0.0214293

dgap: 0.262381

Iteration: 4

pobj: 0.336996

dobj: 0.256943

pinf: 0.0019833

dinf: 0.00293834

dgap: 0.0824029

Iteration: 5

pobj: 0.319809

dobj: 0.298993

pinf: 0.000241351

dinf: 0.000357572

dgap: 0.0209314

Iteration: 6

pobj: 0.313918

dobj: 0.307667

pinf: 4.41234e-005

dinf: 6.53705e-005

dgap: 0.00626419

Iteration: 7

pobj: 0.311327

dobj: 0.309963

pinf: 3.80142e-006

dinf: 5.63195e-006

dgap: 0.00136453

Iteration: 8

pobj: 0.310693

dobj: 0.310233

pinf: 1.03309e-006

dinf: 1.53056e-006

dgap: 0.000459589

Iteration: 9

pobj: 0.31038

dobj: 0.310291

pinf: 1.36903e-007

dinf: 2.02828e-007

dgap: 8.92301e-005

Iteration: 10

pobj: 0.310312

dobj: 0.310301

pinf: 1.34995e-008

dinf: 2.00001e-008

dgap: 1.02076e-005

Iteration: 11

pobj: 0.310303

dobj: 0.310302

pinf: 7.81876e-010

dinf: 1.15838e-009

dgap: 6.19152e-007

Iteration: 12

pobj: 0.310302

dobj: 0.310302

pinf: 3.94785e-011

dinf: 5.84893e-011

dgap: 3.13779e-008

Optimal solution found.

Iteration: 0

pobj: 0.115995

dobj: -0.968196

pinf: 2.22045e-016

dinf: 3.80337

dgap: 1.08419

Iteration: 1

pobj: 0.114487

dobj: -0.00342417

pinf: 2.22045e-016

dinf: 0.411915

dgap: 0.117911

Iteration: 2

pobj: 0.109964

dobj: 0.076731

pinf: 4.44089e-016

dinf: 0.105409

dgap: 0.0332335

Iteration: 3

pobj: 0.106061

dobj: 0.0963413

pinf: 8.88178e-016

dinf: 0.0242878

dgap: 0.00971954

Iteration: 4

pobj: 0.103422

dobj: 0.100283

pinf: 1.55431e-015

dinf: 0.00311373

dgap: 0.00313903

Iteration: 5

pobj: 0.102405

dobj: 0.101732

pinf: 3.10862e-015

dinf: 2.40896e-005

dgap: 0.000673131

Iteration: 6

pobj: 0.10219

dobj: 0.102056

pinf: 5.9952e-015

dinf: 1.35684e-006

dgap: 0.000134059

Iteration: 7

pobj: 0.102141

dobj: 0.102112

pinf: 1.13243e-014

dinf: 1.60625e-007

dgap: 2.96809e-005

Iteration: 8

pobj: 0.102128

dobj: 0.102118

pinf: 1.9762e-014

dinf: 4.1199e-008

dgap: 1.00488e-005

Iteration: 9

pobj: 0.102122

dobj: 0.102119

pinf: 3.66374e-014

dinf: 6.07175e-009

dgap: 2.20192e-006

Iteration: 10

pobj: 0.10212

dobj: 0.10212

pinf: 7.06102e-014

dinf: 4.33201e-010

dgap: 2.35553e-007

Iteration: 11

pobj: 0.10212

dobj: 0.10212

pinf: 1.37446e-013

dinf: 2.25349e-011

dgap: 1.38522e-008

Optimal solution found.

Iteration: 0

pobj: 0.380213

dobj: -0.816458

pinf: 3.35571

dinf: 4.94216

dgap: 16.72

Iteration: 1

pobj: 0.419197

dobj: -0.609183

pinf: 0.349697

dinf: 0.51502

dgap: 2.55486

Iteration: 2

pobj: 0.421981

dobj: -0.140946

pinf: 0.0786301

dinf: 0.115803

dgap: 0.849094

Iteration: 3

pobj: 0.383437

dobj: 0.153208

pinf: 0.0148188

dinf: 0.0218246

dgap: 0.266296

Iteration: 4

pobj: 0.33938

dobj: 0.2578

pinf: 0.00205631

dinf: 0.00302845

dgap: 0.0840279

Iteration: 5

pobj: 0.321604

dobj: 0.303741

pinf: 0.000135538

dinf: 0.000199615

dgap: 0.0179219

Iteration: 6

pobj: 0.316139

dobj: 0.310232

pinf: 2.64637e-005

dinf: 3.89746e-005

dgap: 0.0059151

Iteration: 7

pobj: 0.313487

dobj: 0.312476

pinf: 9.80003e-007

dinf: 1.44331e-006

dgap: 0.00101092

Iteration: 8

pobj: 0.312944

dobj: 0.312593

pinf: 2.66492e-007

dinf: 3.92479e-007

dgap: 0.000351696

Iteration: 9

pobj: 0.312672

dobj: 0.312631

pinf: 1.88438e-008

dinf: 2.77524e-008

dgap: 4.11368e-005

Iteration: 10

pobj: 0.312638

dobj: 0.312634

pinf: 1.4419e-009

dinf: 2.12357e-009

dgap: 3.4519e-006

Iteration: 11

pobj: 0.312635

dobj: 0.312635

pinf: 7.56953e-011

dinf: 1.11481e-010

dgap: 1.83903e-007

Optimal solution found.

Iteration: 0

pobj: 0.115338

dobj: -0.968327

pinf: 0

dinf: 3.80189

dgap: 1.08366

Iteration: 1

pobj: 0.113838

dobj: -0.00366892

pinf: 9.2316e-017

dinf: 0.410564

dgap: 0.117507

Iteration: 2

pobj: 0.109368

dobj: 0.0764341

pinf: 7.24442e-017

dinf: 0.104415

dgap: 0.0329334

Iteration: 3

pobj: 0.105623

dobj: 0.0961807

pinf: 1.11022e-016

dinf: 0.0233323

dgap: 0.00944239

Iteration: 4

pobj: 0.103241

dobj: 0.100131

pinf: 8.45578e-017

dinf: 0.00317813

dgap: 0.00311025

Iteration: 5

pobj: 0.102326

dobj: 0.101662

pinf: 1.27305e-016

dinf: 2.47515e-005

dgap: 0.000664339

Iteration: 6

pobj: 0.102133

dobj: 0.102

pinf: 2.22045e-016

dinf: 1.5577e-006

dgap: 0.000133547

Iteration: 7

pobj: 0.102087

dobj: 0.102055

pinf: 2.22045e-016

dinf: 2.05964e-007

dgap: 3.22562e-005

Iteration: 8

pobj: 0.102072

dobj: 0.102062

pinf: 4.44089e-016

dinf: 4.44937e-008

dgap: 1.04288e-005

Iteration: 9

pobj: 0.102067

dobj: 0.102065

pinf: 8.88178e-016

dinf: 1.71902e-009

dgap: 1.8093e-006

Iteration: 10

pobj: 0.102066

dobj: 0.102066

pinf: 1.77636e-015

dinf: 6.13393e-011

dgap: 4.21972e-007

Iteration: 11

pobj: 0.102066

dobj: 0.102066

pinf: 3.44169e-015

dinf: 1.17757e-016

dgap: 9.26229e-008

Optimal solution found.

Iteration: 0

pobj: 0.376361

dobj: -0.820006

pinf: 3.35392

dinf: 4.93227

dgap: 16.6782

Iteration: 1

pobj: 0.414421

dobj: -0.613273

pinf: 0.350025

dinf: 0.514746

dgap: 2.55217

Iteration: 2

pobj: 0.417204

dobj: -0.146965

pinf: 0.0795125

dinf: 0.116931

dgap: 0.853255

Iteration: 3

pobj: 0.379064

dobj: 0.148053

pinf: 0.0149737

dinf: 0.0220204

dgap: 0.267468

Iteration: 4

pobj: 0.335041

dobj: 0.252963

pinf: 0.00207423

dinf: 0.00305036

dgap: 0.0845503

Iteration: 5

pobj: 0.317407

dobj: 0.299617

pinf: 0.000124761

dinf: 0.000183474

dgap: 0.0178442

Iteration: 6

pobj: 0.312192

dobj: 0.306338

pinf: 2.35626e-005

dinf: 3.46512e-005

dgap: 0.0058604

Iteration: 7

pobj: 0.309767

dobj: 0.308611

pinf: 1.55944e-006

dinf: 2.29331e-006

dgap: 0.00115604

Iteration: 8

pobj: 0.309164

dobj: 0.308771

pinf: 3.81572e-007

dinf: 5.61141e-007

dgap: 0.000393093

Iteration: 9

pobj: 0.308865

dobj: 0.308827

pinf: 1.28258e-008

dinf: 1.88616e-008

dgap: 3.86656e-005

Iteration: 10

pobj: 0.308833

dobj: 0.30883

pinf: 8.54524e-010

dinf: 1.25666e-009

dgap: 3.3425e-006

Iteration: 11

pobj: 0.30883

dobj: 0.30883

pinf: 4.41087e-011

dinf: 6.48665e-011

dgap: 1.83208e-007

Optimal solution found.

Iteration: 0

pobj: 0.113578

dobj: -0.970611

pinf: 1.11022e-016

dinf: 3.80243

dgap: 1.08419

Iteration: 1

pobj: 0.112099

dobj: -0.00572224

pinf: 1.11022e-016

dinf: 0.411499

dgap: 0.117821

Iteration: 2

pobj: 0.107756

dobj: 0.074632

pinf: 1.11022e-016

dinf: 0.105031

dgap: 0.0331242

Iteration: 3

pobj: 0.104226

dobj: 0.0951043

pinf: 2.22045e-016

dinf: 0.0224643

dgap: 0.00912126

Iteration: 4

pobj: 0.102363

dobj: 0.0989338

pinf: 4.44089e-016

dinf: 0.00532085

dgap: 0.00342873

Iteration: 5

pobj: 0.101491

dobj: 0.100737

pinf: 7.77156e-016

dinf: 0.000138659

dgap: 0.000753594

Iteration: 6

pobj: 0.101337

dobj: 0.101145

pinf: 1.55431e-015

dinf: 2.07781e-005

dgap: 0.000192457

Iteration: 7

pobj: 0.10129

dobj: 0.101226

pinf: 2.66454e-015

dinf: 4.62642e-006

dgap: 6.40981e-005

Iteration: 8

pobj: 0.101268

dobj: 0.10126

pinf: 5.21805e-015

dinf: 1.8804e-007

dgap: 7.69986e-006

Iteration: 9

pobj: 0.101264

dobj: 0.101263

pinf: 1.0103e-014

dinf: 1.04771e-008

dgap: 8.33675e-007

Iteration: 10

pobj: 0.101264

dobj: 0.101263

pinf: 1.96509e-014

dinf: 5.41637e-010

dgap: 5.89179e-008

Optimal solution found.

Iteration: 0

pobj: 0.374999

dobj: -0.819766

pinf: 3.36109

dinf: 4.91558

dgap: 16.6536

Iteration: 1

pobj: 0.412775

dobj: -0.614757

pinf: 0.3547

dinf: 0.518747

dgap: 2.56685

Iteration: 2

pobj: 0.416925

dobj: -0.151292

pinf: 0.081188

dinf: 0.118737

dgap: 0.862867

Iteration: 3

pobj: 0.38059

dobj: 0.147736

pinf: 0.0151612

dinf: 0.0221732

dgap: 0.269653

Iteration: 4

pobj: 0.337333

dobj: 0.254622

pinf: 0.00212441

dinf: 0.00310694

dgap: 0.0852274

Iteration: 5

pobj: 0.319892

dobj: 0.304305

pinf: 5.30192e-005

dinf: 7.75404e-005

dgap: 0.0156074

Iteration: 6

pobj: 0.315548

dobj: 0.31102

pinf: 9.60636e-006

dinf: 1.40493e-005

dgap: 0.00452997

Iteration: 7

pobj: 0.313995

dobj: 0.312431

pinf: 2.33809e-006

dinf: 3.41944e-006

dgap: 0.00156415

Iteration: 8

pobj: 0.313136

dobj: 0.312869

pinf: 4.9422e-009

dinf: 7.22795e-009

dgap: 0.000266974

Iteration: 9

pobj: 0.312988

dobj: 0.312943

pinf: 4.17764e-010

dinf: 6.10978e-010

dgap: 4.50174e-005

Iteration: 10

pobj: 0.312963

dobj: 0.312958

pinf: 2.27116e-011

dinf: 3.32153e-011

dgap: 5.13114e-006

Iteration: 11

pobj: 0.312961

dobj: 0.312961

pinf: 1.16222e-012

dinf: 1.69991e-012

dgap: 4.02392e-007

Iteration: 12

pobj: 0.312961

dobj: 0.312961

pinf: 5.09592e-013

dinf: 8.49645e-014

dgap: 2.17216e-008

Optimal solution found.

Iteration: 0

pobj: 0.115856

dobj: -0.964936

pinf: 1.11022e-016

dinf: 3.7895

dgap: 1.08079

Iteration: 1

pobj: 0.114505

dobj: -0.00064014

pinf: 2.22045e-016

dinf: 0.402157

dgap: 0.115145

Iteration: 2

pobj: 0.110545

dobj: 0.0786897

pinf: 3.33067e-016

dinf: 0.101424

dgap: 0.031855

Iteration: 3

pobj: 0.10727

dobj: 0.0987058

pinf: 4.44089e-016

dinf: 0.0214124

dgap: 0.00856395

Iteration: 4

pobj: 0.105381

dobj: 0.102591

pinf: 8.88178e-016

dinf: 0.00369991

dgap: 0.0027898

Iteration: 5

pobj: 0.104625

dobj: 0.104105

pinf: 1.88738e-015

dinf: 0.000140424

dgap: 0.000520473

Iteration: 6

pobj: 0.10446

dobj: 0.104275

pinf: 3.10862e-015

dinf: 3.73256e-005

dgap: 0.000185431

Iteration: 7

pobj: 0.104362

dobj: 0.104322

pinf: 6.10623e-015

dinf: 3.28397e-006

dgap: 4.00393e-005

Iteration: 8

pobj: 0.104338

dobj: 0.104331

pinf: 1.16573e-014

dinf: 2.14965e-007

dgap: 6.97598e-006

Iteration: 9

pobj: 0.104333

dobj: 0.104332

pinf: 2.28706e-014

dinf: 9.66859e-009

dgap: 1.08288e-006

Iteration: 10

pobj: 0.104332

dobj: 0.104332

pinf: 4.4631e-014

dinf: 4.73064e-010

dgap: 1.04238e-007

Optimal solution found.

Iteration: 0

pobj: 0.386136

dobj: -0.80738

pinf: 3.38111

dinf: 4.88128

dgap: 16.6285

Iteration: 1

pobj: 0.420447

dobj: -0.605035

pinf: 0.371231

dinf: 0.535943

dgap: 2.62531

Iteration: 2

pobj: 0.428152

dobj: -0.15625

pinf: 0.0918502

dinf: 0.132603

dgap: 0.921431

Iteration: 3

pobj: 0.398811

dobj: 0.158827

pinf: 0.0163072

dinf: 0.0235426

dgap: 0.280851

Iteration: 4

pobj: 0.356177

dobj: 0.268852

pinf: 0.00236023

dinf: 0.00340744

dgap: 0.0903166

Iteration: 5

pobj: 0.338016

dobj: 0.322296

pinf: 2.22343e-016

dinf: 1.18018e-015

dgap: 0.0157196

Iteration: 6

pobj: 0.33352

dobj: 0.328293

pinf: 2.22045e-016

dinf: 6.47366e-016

dgap: 0.0052267

Iteration: 7

pobj: 0.331078

dobj: 0.329859

pinf: 4.44089e-016

dinf: 9.35491e-016

dgap: 0.0012186

Iteration: 8

pobj: 0.330452

dobj: 0.330197

pinf: 8.88178e-016

dinf: 5.43896e-016

dgap: 0.000254928

Iteration: 9

pobj: 0.330271

dobj: 0.330237

pinf: 1.55431e-015

dinf: 8.7419e-016

dgap: 3.37471e-005

Iteration: 10

pobj: 0.330242

dobj: 0.330239

pinf: 3.10862e-015

dinf: 1.35974e-015

dgap: 2.55041e-006

Iteration: 11

pobj: 0.33024

dobj: 0.330239

pinf: 6.21725e-015

dinf: 1.45604e-015

dgap: 1.33747e-007

Optimal solution found.

Iteration: 0

pobj: 0.116146

dobj: -0.96426

pinf: 0

dinf: 3.78728

dgap: 1.08041

Iteration: 1

pobj: 0.114817

dobj: 9.0969e-006

pinf: 5.63718e-017

dinf: 0.400898

dgap: 0.114808

Iteration: 2

pobj: 0.110957

dobj: 0.0792982

pinf: 7.63278e-017

dinf: 0.100825

dgap: 0.0316592

Iteration: 3

pobj: 0.107808

dobj: 0.0991303

pinf: 1.22958e-016

dinf: 0.0218602

dgap: 0.00867721

Iteration: 4

pobj: 0.105946

dobj: 0.103137

pinf: 1.11022e-016

dinf: 0.0036967

dgap: 0.00280938

Iteration: 5

pobj: 0.105199

dobj: 0.104673

pinf: 3.33067e-016

dinf: 0.000138618

dgap: 0.000525296

Iteration: 6

pobj: 0.105032

dobj: 0.104853

pinf: 4.44089e-016

dinf: 3.42385e-005

dgap: 0.000178496

Iteration: 7

pobj: 0.104941

dobj: 0.104897

pinf: 7.77156e-016

dinf: 4.12481e-006

dgap: 4.42235e-005

Iteration: 8

pobj: 0.104915

dobj: 0.104908

pinf: 1.66533e-015

dinf: 2.58393e-007

dgap: 7.91354e-006

Iteration: 9

pobj: 0.104911

dobj: 0.104909

pinf: 3.10862e-015

dinf: 1.03816e-008

dgap: 1.53242e-006

Iteration: 10

pobj: 0.104909

dobj: 0.104909

pinf: 6.21725e-015

dinf: 4.09348e-010

dgap: 2.49718e-007

Iteration: 11

pobj: 0.104909

dobj: 0.104909

pinf: 1.19904e-014

dinf: 1.98054e-011

dgap: 2.22062e-008

Optimal solution found.

Iteration: 0

pobj: 0.385849

dobj: -0.807609

pinf: 3.36414

dinf: 4.8757

dgap: 16.5322

Iteration: 1

pobj: 0.41963

dobj: -0.604388

pinf: 0.357598

dinf: 0.518272

dgap: 2.56214

Iteration: 2

pobj: 0.425749

dobj: -0.147987

pinf: 0.0856397

dinf: 0.124119

dgap: 0.884085

Iteration: 3

pobj: 0.394405

dobj: 0.159673

pinf: 0.015689

dinf: 0.0227382

dgap: 0.27314

Iteration: 4

pobj: 0.352432

dobj: 0.267564

pinf: 0.00226345

dinf: 0.00328044

dgap: 0.087645

Iteration: 5

pobj: 0.334806

dobj: 0.319322

pinf: 8.19094e-006

dinf: 1.18712e-005

dgap: 0.0154874

Iteration: 6

pobj: 0.33039

dobj: 0.325292

pinf: 1.64836e-006

dinf: 2.38899e-006

dgap: 0.00509793

Iteration: 7

pobj: 0.32805

dobj: 0.326833

pinf: 6.21725e-015

dinf: 1.02958e-015

dgap: 0.00121633

Iteration: 8

pobj: 0.327445

dobj: 0.327186

pinf: 1.17684e-014

dinf: 1.01754e-015

dgap: 0.000258891

Iteration: 9

pobj: 0.327267

dobj: 0.32723

pinf: 2.28706e-014

dinf: 1.27555e-015

dgap: 3.7215e-005

Iteration: 10

pobj: 0.327235

dobj: 0.327232

pinf: 4.41869e-014

dinf: 7.19507e-016

dgap: 3.18587e-006

Iteration: 11

pobj: 0.327232

dobj: 0.327232

pinf: 8.61533e-014

dinf: 8.15844e-016

dgap: 1.72663e-007

Optimal solution found.

Iteration: 0

pobj: 0.117079

dobj: -0.962103

pinf: 0

dinf: 3.78066

dgap: 1.07918

Iteration: 1

pobj: 0.115808

dobj: 0.00200619

pinf: 1.11022e-016

dinf: 0.397167

dgap: 0.113802

Iteration: 2

pobj: 0.112228

dobj: 0.081182

pinf: 6.65555e-017

dinf: 0.0988887

dgap: 0.0310461

Iteration: 3

pobj: 0.109478

dobj: 0.101359

pinf: 1.24755e-016

dinf: 0.0204133

dgap: 0.00811975

Iteration: 4

pobj: 0.107975

dobj: 0.10534

pinf: 2.22045e-016

dinf: 0.00375517

dgap: 0.00263493

Iteration: 5

pobj: 0.107366

dobj: 0.106896

pinf: 3.33067e-016

dinf: 0.000179177

dgap: 0.000470257

Iteration: 6

pobj: 0.107234

dobj: 0.10708

pinf: 6.66134e-016

dinf: 4.35616e-005

dgap: 0.000153871

Iteration: 7

pobj: 0.10717

dobj: 0.107145

pinf: 1.22125e-015

dinf: 1.30558e-006

dgap: 2.4619e-005

Iteration: 8

pobj: 0.107161

dobj: 0.107157

pinf: 2.33147e-015

dinf: 1.30306e-007

dgap: 3.51796e-006

Iteration: 9

pobj: 0.107159

dobj: 0.107159

pinf: 4.55191e-015

dinf: 9.18165e-009

dgap: 2.89914e-007

Iteration: 10

pobj: 0.107159

dobj: 0.107159

pinf: 8.88178e-015

dinf: 4.8166e-010

dgap: 1.56171e-008

Optimal solution found.

Iteration: 0

pobj: 0.393465

dobj: -0.79853

pinf: 3.33552

dinf: 4.86131

dgap: 16.3522

Iteration: 1

pobj: 0.426762

dobj: -0.59399

pinf: 0.333281

dinf: 0.485736

dgap: 2.448

Iteration: 2

pobj: 0.430131

dobj: -0.120357

pinf: 0.0735086

dinf: 0.107134

dgap: 0.809646

Iteration: 3

pobj: 0.395132

dobj: 0.171582

pinf: 0.0142669

dinf: 0.0207932

dgap: 0.256665

Iteration: 4

pobj: 0.355217

dobj: 0.27596

pinf: 0.00202361

dinf: 0.00294928

dgap: 0.0815358

Iteration: 5

pobj: 0.33927

dobj: 0.3246

pinf: 2.43872e-005

dinf: 3.55429e-005

dgap: 0.0146787

Iteration: 6

pobj: 0.335384

dobj: 0.330328

pinf: 5.00583e-006

dinf: 7.29568e-006

dgap: 0.00505709

Iteration: 7

pobj: 0.333437

dobj: 0.332259

pinf: 7.77156e-016

dinf: 9.28879e-016

dgap: 0.00117847

Iteration: 8

pobj: 0.332974

dobj: 0.332683

pinf: 1.33227e-015

dinf: 8.00593e-016

dgap: 0.000291088

Iteration: 9

pobj: 0.332831

dobj: 0.332777

pinf: 2.55351e-015

dinf: 6.28037e-016

dgap: 5.43746e-005

Iteration: 10

pobj: 0.332806

dobj: 0.332795

pinf: 4.996e-015

dinf: 1.09906e-015

dgap: 1.04756e-005

Iteration: 11

pobj: 0.332803

dobj: 0.332801

pinf: 9.99201e-015

dinf: 1.01754e-015

dgap: 2.12542e-006

Iteration: 12

pobj: 0.332802

dobj: 0.332802

pinf: 1.95399e-014

dinf: 1.19575e-015

dgap: 3.44637e-007

Iteration: 13

pobj: 0.332802

dobj: 0.332802

pinf: 3.81917e-014

dinf: 1.21111e-015

dgap: 2.99936e-008

Optimal solution found.

Iteration: 0

pobj: 0.119434

dobj: -0.961719

pinf: 1.11022e-016

dinf: 3.78683

dgap: 1.08115

Iteration: 1

pobj: 0.118122

dobj: 0.00270501

pinf: 3.33067e-016

dinf: 0.402671

dgap: 0.115417

Iteration: 2

pobj: 0.114453

dobj: 0.0826542

pinf: 4.44089e-016

dinf: 0.101095

dgap: 0.0317989

Iteration: 3

pobj: 0.111704

dobj: 0.103444

pinf: 8.88178e-016

dinf: 0.0206162

dgap: 0.00826024

Iteration: 4

pobj: 0.110379

dobj: 0.107455

pinf: 1.33227e-015

dinf: 0.00514287

dgap: 0.00292366

Iteration: 5

pobj: 0.109702

dobj: 0.10915

pinf: 2.66454e-015

dinf: 0.000250018

dgap: 0.000552299

Iteration: 6

pobj: 0.109532

dobj: 0.109366

pinf: 4.77396e-015

dinf: 5.00946e-005

dgap: 0.00016602

Iteration: 7

pobj: 0.109462

dobj: 0.109429

pinf: 9.32587e-015

dinf: 1.88675e-006

dgap: 3.32356e-005

Iteration: 8

pobj: 0.10945

dobj: 0.109444

pinf: 1.79856e-014

dinf: 1.21345e-007

dgap: 6.14172e-006

Iteration: 9

pobj: 0.109448

dobj: 0.109447

pinf: 3.54161e-014

dinf: 4.76901e-009

dgap: 8.19382e-007

Iteration: 10

pobj: 0.109448

dobj: 0.109448

pinf: 6.91669e-014

dinf: 2.27442e-010

dgap: 9.04688e-008

Optimal solution found.

Iteration: 0

pobj: 0.397362

dobj: -0.796393

pinf: 3.31572

dinf: 4.89165

dgap: 16.3646

Iteration: 1

pobj: 0.431916

dobj: -0.587479

pinf: 0.315443

dinf: 0.46537

dgap: 2.37876

Iteration: 2

pobj: 0.433101

dobj: -0.0968538

pinf: 0.0637383

dinf: 0.0940325

dgap: 0.751997

Iteration: 3

pobj: 0.393641

dobj: 0.179585

pinf: 0.0128501

dinf: 0.0189577

dgap: 0.242945

Iteration: 4

pobj: 0.35405

dobj: 0.279949

pinf: 0.00176741

dinf: 0.00260745

dgap: 0.0759839

Iteration: 5

pobj: 0.33866

dobj: 0.324075

pinf: 5.69979e-005

dinf: 8.40883e-005

dgap: 0.0146048

Iteration: 6

pobj: 0.33464

dobj: 0.329599

pinf: 1.13094e-005

dinf: 1.66846e-005

dgap: 0.0050443

Iteration: 7

pobj: 0.332671

dobj: 0.331461

pinf: 1.12025e-016

dinf: 1.03555e-015

dgap: 0.00121005

Iteration: 8

pobj: 0.332188

dobj: 0.33188

pinf: 2.22084e-016

dinf: 1.52226e-015

dgap: 0.000308424

Iteration: 9

pobj: 0.33202

dobj: 0.331962

pinf: 1.55171e-017

dinf: 1.05325e-015

dgap: 5.81311e-005

Iteration: 10

pobj: 0.331981

dobj: 0.331971

pinf: 3.33356e-016

dinf: 1.03555e-015

dgap: 1.02146e-005

Iteration: 11

pobj: 0.331973

dobj: 0.331972

pinf: 1.11954e-016

dinf: 1.02358e-015

dgap: 1.19184e-006

Iteration: 12

pobj: 0.331972

dobj: 0.331972

pinf: 1.119e-016

dinf: 8.59975e-016

dgap: 7.76165e-008

Optimal solution found.

Iteration: 0

pobj: 0.118627

dobj: -0.962724

pinf: 2.22045e-016

dinf: 3.78784

dgap: 1.08135

Iteration: 1

pobj: 0.117306

dobj: 0.00177466

pinf: 4.44089e-016

dinf: 0.403095

dgap: 0.115532

Iteration: 2

pobj: 0.113611

dobj: 0.0817504

pinf: 6.66134e-016

dinf: 0.101286

dgap: 0.0318609

Iteration: 3

pobj: 0.11081

dobj: 0.102659

pinf: 1.33227e-015

dinf: 0.0201934

dgap: 0.00815118

Iteration: 4

pobj: 0.109439

dobj: 0.106523

pinf: 2.22045e-015

dinf: 0.00490821

dgap: 0.00291623

Iteration: 5

pobj: 0.108761

dobj: 0.108207

pinf: 4.21885e-015

dinf: 0.000229887

dgap: 0.000554443

Iteration: 6

pobj: 0.108593

dobj: 0.108425

pinf: 7.54952e-015

dinf: 4.71651e-005

dgap: 0.000167987

Iteration: 7

pobj: 0.10852

dobj: 0.108487

pinf: 1.4877e-014

dinf: 1.63072e-006

dgap: 3.36787e-005

Iteration: 8

pobj: 0.108508

dobj: 0.108502

pinf: 2.88658e-014

dinf: 9.17565e-008

dgap: 6.07344e-006

Iteration: 9

pobj: 0.108506

dobj: 0.108505

pinf: 5.66214e-014

dinf: 3.66359e-009

dgap: 7.99799e-007

Iteration: 10

pobj: 0.108506

dobj: 0.108506

pinf: 1.09912e-013

dinf: 2.17213e-010

dgap: 7.12718e-008

Optimal solution found.

Iteration: 0

pobj: 0.396213

dobj: -0.796955

pinf: 3.31709

dinf: 4.88067

dgap: 16.3339

Iteration: 1

pobj: 0.430546

dobj: -0.589339

pinf: 0.316202

dinf: 0.46525

dgap: 2.37906

Iteration: 2

pobj: 0.431926

dobj: -0.098464

pinf: 0.0634511

dinf: 0.0933601

dgap: 0.750465

Iteration: 3

pobj: 0.392846

dobj: 0.178609

pinf: 0.0128326

dinf: 0.0188815

dgap: 0.242841

Iteration: 4

pobj: 0.353657

dobj: 0.279594

pinf: 0.00177424

dinf: 0.00261057

dgap: 0.0759265

Iteration: 5

pobj: 0.338475

dobj: 0.32437

pinf: 4.61853e-005

dinf: 6.79556e-005

dgap: 0.0141207

Iteration: 6

pobj: 0.33464

dobj: 0.3298

pinf: 9.38175e-006

dinf: 1.3804e-005

dgap: 0.00484233

Iteration: 7

pobj: 0.332742

dobj: 0.331622

pinf: 1.79856e-014

dinf: 1.15378e-015

dgap: 0.00111915

Iteration: 8

pobj: 0.3323

dobj: 0.33202

pinf: 3.35287e-014

dinf: 1.9767e-015

dgap: 0.00028012

Iteration: 9

pobj: 0.332161

dobj: 0.332105

pinf: 6.55032e-014

dinf: 1.06489e-015

dgap: 5.55974e-005

Iteration: 10

pobj: 0.332131

dobj: 0.33212

pinf: 1.29674e-013

dinf: 9.15513e-016

dgap: 1.13708e-005

Iteration: 11

pobj: 0.332126

dobj: 0.332123

pinf: 2.58127e-013

dinf: 1.00535e-015

dgap: 2.44323e-006

Iteration: 12

pobj: 0.332124

dobj: 0.332124

pinf: 5.13145e-013

dinf: 1.19575e-015

dgap: 5.87982e-007

Iteration: 13

pobj: 0.332124

dobj: 0.332124

pinf: 1.02618e-012

dinf: 1.43901e-015

dgap: 1.24635e-007

Optimal solution found.

Iteration: 0

pobj: 0.118736

dobj: -0.962961

pinf: 0

dinf: 3.78935

dgap: 1.0817

Iteration: 1

pobj: 0.117401

dobj: 0.00158224

pinf: 9.15303e-017

dinf: 0.404122

dgap: 0.115819

Iteration: 2

pobj: 0.113658

dobj: 0.081688

pinf: 2.22045e-016

dinf: 0.101598

dgap: 0.0319702

Iteration: 3

pobj: 0.110815

dobj: 0.102575

pinf: 2.22045e-016

dinf: 0.0204246

dgap: 0.00824055

Iteration: 4

pobj: 0.109417

dobj: 0.106464

pinf: 4.44089e-016

dinf: 0.00494811

dgap: 0.00295323

Iteration: 5

pobj: 0.108728

dobj: 0.108162

pinf: 8.88178e-016

dinf: 0.000234885

dgap: 0.000566548

Iteration: 6

pobj: 0.108557

dobj: 0.108382

pinf: 1.55431e-015

dinf: 4.97748e-005

dgap: 0.000174904

Iteration: 7

pobj: 0.10848

dobj: 0.108444

pinf: 3.10862e-015

dinf: 1.6261e-006

dgap: 3.54034e-005

Iteration: 8

pobj: 0.108466

dobj: 0.108459

pinf: 5.9952e-015

dinf: 8.34651e-008

dgap: 6.431e-006

Iteration: 9

pobj: 0.108464

dobj: 0.108463

pinf: 1.16573e-014

dinf: 3.25765e-009

dgap: 9.14054e-007

Iteration: 10

pobj: 0.108463

dobj: 0.108463

pinf: 2.28706e-014

dinf: 1.72456e-010

dgap: 7.94426e-008

Optimal solution found.

Iteration: 0

pobj: 0.392365

dobj: -0.800785

pinf: 3.3168

dinf: 4.87906

dgap: 16.3271

Iteration: 1

pobj: 0.426576

dobj: -0.59326

pinf: 0.316057

dinf: 0.464925

dgap: 2.37787

Iteration: 2

pobj: 0.427997

dobj: -0.101583

pinf: 0.0630009

dinf: 0.0926752

dgap: 0.747776

Iteration: 3

pobj: 0.388761

dobj: 0.174835

pinf: 0.0127871

dinf: 0.01881

dgap: 0.242352

Iteration: 4

pobj: 0.349519

dobj: 0.275554

pinf: 0.00176838

dinf: 0.00260131

dgap: 0.0758171

Iteration: 5

pobj: 0.334306

dobj: 0.320292

pinf: 4.26914e-005

dinf: 6.27997e-005

dgap: 0.0140288

Iteration: 6

pobj: 0.330479

dobj: 0.325645

pinf: 8.71698e-006

dinf: 1.28228e-005

dgap: 0.00483663

Iteration: 7

pobj: 0.328572

dobj: 0.327448

pinf: 1.11022e-015

dinf: 1.48122e-015

dgap: 0.00112388

Iteration: 8

pobj: 0.328126

dobj: 0.327846

pinf: 2.10942e-015

dinf: 5.43896e-016

dgap: 0.000280319

Iteration: 9

pobj: 0.327984

dobj: 0.327929

pinf: 4.10783e-015

dinf: 7.36439e-016

dgap: 5.55484e-005

Iteration: 10

pobj: 0.327954

dobj: 0.327942

pinf: 8.21565e-015

dinf: 6.93334e-016

dgap: 1.12352e-005

Iteration: 11

pobj: 0.327947

dobj: 0.327945

pinf: 1.63203e-014

dinf: 1.07066e-015

dgap: 2.33535e-006

Iteration: 12

pobj: 0.327946

dobj: 0.327945

pinf: 3.23075e-014

dinf: 1.06489e-015

dgap: 4.95477e-007

Iteration: 13

pobj: 0.327946

dobj: 0.327945

pinf: 6.36158e-014

dinf: 1.29948e-015

dgap: 7.05679e-008

Optimal solution found.

Iteration: 0

pobj: 0.119679

dobj: -0.961681

pinf: 2.22045e-016

dinf: 3.78626

dgap: 1.08136

Iteration: 1

pobj: 0.118384

dobj: 0.00272727

pinf: 2.22045e-016

dinf: 0.403377

dgap: 0.115657

Iteration: 2

pobj: 0.114706

dobj: 0.0825212

pinf: 4.44089e-016

dinf: 0.102386

dgap: 0.032185

Iteration: 3

pobj: 0.111867

dobj: 0.103416

pinf: 8.88178e-016

dinf: 0.0213116

dgap: 0.00845108

Iteration: 4

pobj: 0.110293

dobj: 0.107551

pinf: 1.55431e-015

dinf: 0.00386282

dgap: 0.00274137

Iteration: 5

pobj: 0.109629

dobj: 0.109107

pinf: 3.10862e-015

dinf: 0.000197921

dgap: 0.00052235

Iteration: 6

pobj: 0.109439

dobj: 0.109312

pinf: 5.77316e-015

dinf: 2.60714e-005

dgap: 0.000127236

Iteration: 7

pobj: 0.109374

dobj: 0.109344

pinf: 1.08802e-014

dinf: 1.79711e-006

dgap: 2.95826e-005

Iteration: 8

pobj: 0.109356

dobj: 0.109351

pinf: 2.17604e-014

dinf: 4.06545e-008

dgap: 5.02064e-006

Iteration: 9

pobj: 0.109352

dobj: 0.109351

pinf: 4.24105e-014

dinf: 1.89586e-009

dgap: 7.21367e-007

Iteration: 10

pobj: 0.109352

dobj: 0.109351

pinf: 8.26006e-014

dinf: 1.05037e-010

dgap: 6.15178e-008

Optimal solution found.

Iteration: 0

pobj: 0.400811

dobj: -0.794445

pinf: 3.31634

dinf: 4.89251

dgap: 16.3718

Iteration: 1

pobj: 0.435729

dobj: -0.58661

pinf: 0.31574

dinf: 0.465803

dgap: 2.38302

Iteration: 2

pobj: 0.436902

dobj: -0.0924833

pinf: 0.0618415

dinf: 0.0912331

dgap: 0.74363

Iteration: 3

pobj: 0.396171

dobj: 0.182021

pinf: 0.0126606

dinf: 0.0186778

dgap: 0.242216

Iteration: 4

pobj: 0.355871

dobj: 0.282165

pinf: 0.00174574

dinf: 0.00257544

dgap: 0.0755348

Iteration: 5

pobj: 0.339889

dobj: 0.326098

pinf: 3.55689e-005

dinf: 5.24738e-005

dgap: 0.0138033

Iteration: 6

pobj: 0.335674

dobj: 0.331253

pinf: 6.18329e-006

dinf: 9.12204e-006

dgap: 0.00442275

Iteration: 7

pobj: 0.333785

dobj: 0.332737

pinf: 1.9762e-014

dinf: 1.55034e-015

dgap: 0.00104714

Iteration: 8

pobj: 0.333341

dobj: 0.333071

pinf: 3.61933e-014

dinf: 1.15378e-015

dgap: 0.000269813

Iteration: 9

pobj: 0.333183

dobj: 0.333138

pinf: 7.01661e-014

dinf: 1.42178e-015

dgap: 4.4317e-005

Iteration: 10

pobj: 0.333152

dobj: 0.333146

pinf: 1.37002e-013

dinf: 1.04148e-015

dgap: 6.50229e-006

Iteration: 11

pobj: 0.333148

dobj: 0.333147

pinf: 2.68452e-013

dinf: 1.7519e-015

dgap: 1.11416e-006

Iteration: 12

pobj: 0.333147

dobj: 0.333147

pinf: 5.24025e-013

dinf: 1.95475e-015

dgap: 1.64139e-007

Optimal solution found.

Iteration: 0

pobj: 0.114165

dobj: -0.966275

pinf: 2.22045e-016

dinf: 3.78424

dgap: 1.08044

Iteration: 1

pobj: 0.112872

dobj: -0.00204408

pinf: 2.22045e-016

dinf: 0.400969

dgap: 0.114916

Iteration: 2

pobj: 0.108962

dobj: 0.0769199

pinf: 2.22045e-016

dinf: 0.102178

dgap: 0.0320425

Iteration: 3

pobj: 0.105721

dobj: 0.0934422

pinf: 4.44089e-016

dinf: 0.032301

dgap: 0.012279

Iteration: 4

pobj: 0.103022

dobj: 0.0998585

pinf: 8.88178e-016

dinf: 0.00211667

dgap: 0.00316377

Iteration: 5

pobj: 0.102134

dobj: 0.101455

pinf: 1.77636e-015

dinf: 0.000122782

dgap: 0.000678932

Iteration: 6

pobj: 0.10187

dobj: 0.101733

pinf: 3.33067e-015

dinf: 1.15584e-005

dgap: 0.00013755

Iteration: 7

pobj: 0.101798

dobj: 0.101771

pinf: 6.21725e-015

dinf: 1.67093e-006

dgap: 2.72901e-005

Iteration: 8

pobj: 0.101783

dobj: 0.101774

pinf: 1.06581e-014

dinf: 4.65597e-007

dgap: 8.87898e-006

Iteration: 9

pobj: 0.101776

dobj: 0.101775

pinf: 2.02061e-014

dinf: 4.9579e-008

dgap: 1.34903e-006

Iteration: 10

pobj: 0.101775

dobj: 0.101775

pinf: 3.88578e-014

dinf: 3.98419e-009

dgap: 1.51041e-007

Iteration: 11

pobj: 0.101775

dobj: 0.101775

pinf: 7.54952e-014

dinf: 2.43325e-010

dgap: 1.08439e-008

Optimal solution found.

Iteration: 0

pobj: 0.399783

dobj: -0.794994

pinf: 3.31488

dinf: 4.88105

dgap: 16.3266

Iteration: 1

pobj: 0.434178

dobj: -0.588119

pinf: 0.314923

dinf: 0.463713

dgap: 2.37577

Iteration: 2

pobj: 0.435608

dobj: -0.0929461

pinf: 0.0610592

dinf: 0.0899077

dgap: 0.738909

Iteration: 3

pobj: 0.395512

dobj: 0.182253

pinf: 0.0125265

dinf: 0.0184449

dgap: 0.240708

Iteration: 4

pobj: 0.355861

dobj: 0.28295

pinf: 0.00175211

dinf: 0.00257992

dgap: 0.0747253

Iteration: 5

pobj: 0.341244

dobj: 0.320681

pinf: 0.000266868

dinf: 0.000392954

dgap: 0.0206834

Iteration: 6

pobj: 0.335556

dobj: 0.329728

pinf: 3.02395e-005

dinf: 4.45267e-005

dgap: 0.00583581

Iteration: 7

pobj: 0.333705

dobj: 0.331908

pinf: 5.47769e-006

dinf: 8.06572e-006

dgap: 0.00179878

Iteration: 8

pobj: 0.332879

dobj: 0.332499

pinf: 2.90878e-014

dinf: 1.21619e-015

dgap: 0.000379786

Iteration: 9

pobj: 0.332692

dobj: 0.332604

pinf: 5.4845e-014

dinf: 1.09344e-015

dgap: 8.79728e-005

Iteration: 10

pobj: 0.332633

dobj: 0.33262

pinf: 1.06803e-013

dinf: 1.16969e-015

dgap: 1.23297e-005

Iteration: 11

pobj: 0.332622

dobj: 0.332621

pinf: 2.08056e-013

dinf: 5.97873e-016

dgap: 1.14505e-006

Iteration: 12

pobj: 0.332621

dobj: 0.332621

pinf: 4.05453e-013

dinf: 9.86788e-016

dgap: 6.83197e-008

Optimal solution found.

Iteration: 0

pobj: 0.11327

dobj: -0.96457

pinf: 0

dinf: 3.77327

dgap: 1.07784

Iteration: 1

pobj: 0.112092

dobj: -0.000673744

pinf: 6.58281e-017

dinf: 0.393348

dgap: 0.112766

Iteration: 2

pobj: 0.108576

dobj: 0.0775103

pinf: 1.11022e-016

dinf: 0.0993329

dgap: 0.031066

Iteration: 3

pobj: 0.105643

dobj: 0.0938764

pinf: 1.11022e-016

dinf: 0.0311067

dgap: 0.0117666

Iteration: 4

pobj: 0.103234

dobj: 0.100227

pinf: 2.22045e-016

dinf: 0.00206712

dgap: 0.0030073

Iteration: 5

pobj: 0.102495

dobj: 0.101849

pinf: 4.44089e-016

dinf: 0.000120302

dgap: 0.000645755

Iteration: 6

pobj: 0.10231

dobj: 0.102173

pinf: 8.88178e-016

dinf: 1.17049e-005

dgap: 0.000137607

Iteration: 7

pobj: 0.102266

dobj: 0.102244

pinf: 1.66533e-015

dinf: 9.16852e-007

dgap: 2.17722e-005

Iteration: 8

pobj: 0.10226

dobj: 0.102255

pinf: 2.9976e-015

dinf: 2.07813e-007

dgap: 5.65601e-006

Iteration: 9

pobj: 0.102258

dobj: 0.102257

pinf: 5.44009e-015

dinf: 4.18585e-008

dgap: 1.41695e-006

Iteration: 10

pobj: 0.102258

dobj: 0.102258

pinf: 1.05471e-014

dinf: 1.9544e-009

dgap: 1.75332e-007

Iteration: 11

pobj: 0.102258

dobj: 0.102258

pinf: 2.04281e-014

dinf: 8.59725e-011

dgap: 3.60605e-008

Optimal solution found.

Iteration: 0

pobj: 0.395682

dobj: -0.799794

pinf: 3.31125

dinf: 4.8802

dgap: 16.308

Iteration: 1

pobj: 0.430324

dobj: -0.593347

pinf: 0.311919

dinf: 0.459714

dgap: 2.36364

Iteration: 2

pobj: 0.430759

dobj: -0.0996301

pinf: 0.0623997

dinf: 0.091966

dgap: 0.745774

Iteration: 3

pobj: 0.391726

dobj: 0.178522

pinf: 0.0126767

dinf: 0.0186832

dgap: 0.241254

Iteration: 4

pobj: 0.352444

dobj: 0.279045

pinf: 0.00179454

dinf: 0.00264483

dgap: 0.0753015

Iteration: 5

pobj: 0.337886

dobj: 0.317213

pinf: 0.000270728

dinf: 0.000399006

dgap: 0.0207992

Iteration: 6

pobj: 0.33229

dobj: 0.326359

pinf: 3.08463e-005

dinf: 4.5462e-005

dgap: 0.00593959

Iteration: 7

pobj: 0.330519

dobj: 0.328551

pinf: 6.26728e-006

dinf: 9.23685e-006

dgap: 0.00196944

Iteration: 8

pobj: 0.329668

dobj: 0.329214

pinf: 1.77636e-015

dinf: 7.61131e-016

dgap: 0.000454159

Iteration: 9

pobj: 0.329481

dobj: 0.329373

pinf: 3.33067e-015

dinf: 1.18018e-015

dgap: 0.000107128

Iteration: 10

pobj: 0.329427

dobj: 0.329407

pinf: 6.43929e-015

dinf: 1.02958e-015

dgap: 1.98478e-005

Iteration: 11

pobj: 0.329416

dobj: 0.329412

pinf: 1.26565e-014

dinf: 8.9509e-016

dgap: 3.90656e-006

Iteration: 12

pobj: 0.329413

dobj: 0.329412

pinf: 2.4869e-014

dinf: 1.20601e-015

dgap: 6.37612e-007

Iteration: 13

pobj: 0.329412

dobj: 0.329412

pinf: 4.86278e-014

dinf: 1.21619e-015

dgap: 6.99926e-008

Optimal solution found.

Iteration: 0

pobj: 0.114706

dobj: -0.96241

pinf: 1.11022e-016

dinf: 3.77192

dgap: 1.07712

Iteration: 1

pobj: 0.113529

dobj: 0.00144866

pinf: 1.11022e-016

dinf: 0.391088

dgap: 0.11208

Iteration: 2

pobj: 0.110034

dobj: 0.0794452

pinf: 1.11022e-016

dinf: 0.0978975

dgap: 0.0305892

Iteration: 3

pobj: 0.107094

dobj: 0.0958683

pinf: 3.33067e-016

dinf: 0.0297254

dgap: 0.0112254

Iteration: 4

pobj: 0.104634

dobj: 0.101833

pinf: 6.66134e-016

dinf: 0.0017786

dgap: 0.00280165

Iteration: 5

pobj: 0.103814

dobj: 0.103204

pinf: 1.33227e-015

dinf: 8.40376e-005

dgap: 0.000610204

Iteration: 6

pobj: 0.103589

dobj: 0.103439

pinf: 2.55351e-015

dinf: 7.05916e-006

dgap: 0.000149631

Iteration: 7

pobj: 0.103526

dobj: 0.103502

pinf: 5.10703e-015

dinf: 3.9425e-008

dgap: 2.4386e-005

Iteration: 8

pobj: 0.103517

dobj: 0.103511

pinf: 9.4369e-015

dinf: 5.93066e-009

dgap: 6.41972e-006

Iteration: 9

pobj: 0.103514

dobj: 0.103512

pinf: 1.70974e-014

dinf: 1.16637e-009

dgap: 2.24055e-006

Iteration: 10

pobj: 0.103513

dobj: 0.103513

pinf: 3.37508e-014

dinf: 2.35238e-011

dgap: 4.19046e-007

Iteration: 11

pobj: 0.103513

dobj: 0.103513

pinf: 6.55032e-014

dinf: 1.41201e-012

dgap: 6.01232e-008

Optimal solution found.

Iteration: 0

pobj: 0.398638

dobj: -0.796719

pinf: 3.31318

dinf: 4.88079

dgap: 16.3186

Iteration: 1

pobj: 0.432431

dobj: -0.589456

pinf: 0.314857

dinf: 0.46383

dgap: 2.37497

Iteration: 2

pobj: 0.432625

dobj: -0.0999254

pinf: 0.0651942

dinf: 0.0960405

dgap: 0.759542

Iteration: 3

pobj: 0.394498

dobj: 0.179915

pinf: 0.0131196

dinf: 0.0193271

dgap: 0.244279

Iteration: 4

pobj: 0.354809

dobj: 0.279984

pinf: 0.00186453

dinf: 0.00274673

dgap: 0.0768724

Iteration: 5

pobj: 0.339842

dobj: 0.31936

pinf: 0.000257823

dinf: 0.000379811

dgap: 0.0206025

Iteration: 6

pobj: 0.334275

dobj: 0.328237

pinf: 3.03067e-005

dinf: 4.46462e-005

dgap: 0.00604621

Iteration: 7

pobj: 0.332443

dobj: 0.330413

pinf: 6.21768e-006

dinf: 9.15954e-006

dgap: 0.00203112

Iteration: 8

pobj: 0.331495

dobj: 0.331002

pinf: 1.13243e-014

dinf: 1.21619e-015

dgap: 0.000492231

Iteration: 9

pobj: 0.331266

dobj: 0.331155

pinf: 2.16493e-014

dinf: 1.54636e-015

dgap: 0.000110334

Iteration: 10

pobj: 0.331204

dobj: 0.331186

pinf: 4.22995e-014

dinf: 5.76889e-016

dgap: 1.80047e-005

Iteration: 11

pobj: 0.331191

dobj: 0.331189

pinf: 8.24896e-014

dinf: 9.48575e-016

dgap: 2.06416e-006

Iteration: 12

pobj: 0.331189

dobj: 0.331189

pinf: 1.60871e-013

dinf: 1.14304e-015

dgap: 1.38534e-007

Optimal solution found.

Iteration: 0

pobj: 0.11477

dobj: -0.961072

pinf: 0

dinf: 3.76706

dgap: 1.07584

Iteration: 1

pobj: 0.113632

dobj: 0.00263219

pinf: 2.22045e-016

dinf: 0.387308

dgap: 0.111

Iteration: 2

pobj: 0.110322

dobj: 0.0805013

pinf: 2.22045e-016

dinf: 0.095544

dgap: 0.0298208

Iteration: 3

pobj: 0.107587

dobj: 0.0965035

pinf: 4.44089e-016

dinf: 0.0295032

dgap: 0.0110831

Iteration: 4

pobj: 0.105282

dobj: 0.102542

pinf: 8.88178e-016

dinf: 0.00180192

dgap: 0.00273982

Iteration: 5

pobj: 0.10454

dobj: 0.103948

pinf: 1.77636e-015

dinf: 9.80713e-005

dgap: 0.000592444

Iteration: 6

pobj: 0.104348

dobj: 0.10421

pinf: 3.33067e-015

dinf: 8.12006e-006

dgap: 0.000138232

Iteration: 7

pobj: 0.104303

dobj: 0.104281

pinf: 6.43929e-015

dinf: 3.19953e-007

dgap: 2.13694e-005

Iteration: 8

pobj: 0.104297

dobj: 0.104295

pinf: 1.24345e-014

dinf: 2.46719e-008

dgap: 2.67608e-006

Iteration: 9

pobj: 0.104297

dobj: 0.104296

pinf: 2.22045e-014

dinf: 5.00253e-009

dgap: 6.11958e-007

Iteration: 10

pobj: 0.104297

dobj: 0.104297

pinf: 4.13003e-014

dinf: 6.97906e-010

dgap: 1.06644e-007

Iteration: 11

pobj: 0.104297

dobj: 0.104297

pinf: 7.92699e-014

dinf: 5.72651e-011

dgap: 1.18903e-008

Optimal solution found.

Iteration: 0

pobj: 0.396694

dobj: -0.797435

pinf: 3.31531

dinf: 4.86496

dgap: 16.2746

Iteration: 1

pobj: 0.429327

dobj: -0.591061

pinf: 0.316147

dinf: 0.463922

dgap: 2.37423

Iteration: 2

pobj: 0.430326

dobj: -0.102681

pinf: 0.0657035

dinf: 0.0964149

dgap: 0.761109

Iteration: 3

pobj: 0.393718

dobj: 0.179723

pinf: 0.0131117

dinf: 0.0192405

dgap: 0.243563

Iteration: 4

pobj: 0.355241

dobj: 0.2805

pinf: 0.00187089

dinf: 0.00274539

dgap: 0.0767899

Iteration: 5

pobj: 0.340985

dobj: 0.320463

pinf: 0.000260152

dinf: 0.000381753

dgap: 0.0206442

Iteration: 6

pobj: 0.335827

dobj: 0.329761

pinf: 3.0625e-005

dinf: 4.49398e-005

dgap: 0.00607401

Iteration: 7

pobj: 0.334207

dobj: 0.332124

pinf: 6.53239e-006

dinf: 9.58579e-006

dgap: 0.00208414

Iteration: 8

pobj: 0.333389

dobj: 0.332899

pinf: 2.84217e-014

dinf: 5.08768e-016

dgap: 0.000490514

Iteration: 9

pobj: 0.333219

dobj: 0.333109

pinf: 5.37348e-014

dinf: 9.08757e-016

dgap: 0.000110014

Iteration: 10

pobj: 0.333181

dobj: 0.333165

pinf: 1.05027e-013

dinf: 1.12127e-015

dgap: 1.65065e-005

Iteration: 11

pobj: 0.333176

dobj: 0.333173

pinf: 2.05169e-013

dinf: 1.41308e-015

dgap: 2.15942e-006

Iteration: 12

pobj: 0.333174

dobj: 0.333174

pinf: 4.00346e-013

dinf: 1.39553e-015

dgap: 2.62284e-007

Optimal solution found.

Pfaff (Invesco) MCO RFinance 2016 18 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: backtest, part III

> W <- list("MCO" = Wmco, "MSR" = Wmsr, "MDP" = Wmdp,

+ "GMV" = Wgmv, "ERC" = Werc)

> E <- lapply(W, function(x)

+ wTs <- timeSeries(x, charvec = ep)

+ wTsL1 <- lag(wTs, 1)

+ RetFac <- 1 + rowSums(R[ep, ] * wTsL1) / 100.0

+ RetFac[1] <- 100

+ timeSeries(cumprod(RetFac), charvec = ep)

+ )

> cols <- topo.colors(6)

> plot(E[[1]], lwd = 2,

+ ylab = "Index", xlab = "", col = cols[1],

+ main = "Comparison of Allocation Strategies")

> lines(E[[2]], col = cols[2])

> lines(E[[3]], col = cols[3])

> lines(E[[4]], col = cols[4])

> lines(E[[5]], col = cols[5])

> legend("topleft",

+ legend = c("MCO", "MSR", "MDP", "GMW", "ERC"),

+ col = cols, lty = 1, lwd = 2)

> abline(h = 100, col = "gray")

Comparison of Allocation Strategies

Inde

x

2010−01−01 2010−07−01 2011−01−01 2011−07−01

100

102

104

106

108

110

112

MCOMSRMDPGMWERC

Pfaff (Invesco) MCO RFinance 2016 19 / 24

Example: Multi-asset class portfolio

Example: Multi-asset class portfolioR code: backtest, part IV

> Rstrat <- matrix(unlist(lapply(E, Return.calculate)), ncol = 5)

> RstratTs <- na.omit(xts(Rstrat, order.by = as.Date(ep)))

> Bench <- xts(rep(0, nrow(RstratTs)), order.by = as.Date(ep)[-1])

> S1 <- as.matrix(table.AnnualizedReturns(RstratTs, Rf = Bench,

+ scale = 12))

> S2 <- VaR(RstratTs)

> ans <- rbind(S1, -100 * S2)

> colnames(ans) <- c("MCO", "MSR", "MDP", "GMV", "ERC")

> rownames(ans) <- c("Return (p.a.)", "StdDev. Risk (p.a.)",

+ "Sharpe Ratio", "VaR (p.a.)")

> round(ans, 3)

MCO MSR MDP GMV ERC

Return (p.a.) 0.061 0.060 0.058 0.051 0.060

StdDev. Risk (p.a.) 0.038 0.039 0.037 0.034 0.034

Sharpe Ratio 1.605 1.532 1.585 1.513 1.746

VaR (p.a.) 1.301 1.380 1.308 1.211 1.017

Pfaff (Invesco) MCO RFinance 2016 20 / 24

Summary

Summary

Aiding decision makers by making portfolio choices for conflictingobjectives (a posteriori analysis).

Allows amendment of classical portfolio optimization formulations(e.g. GMV, ERC, MDP and/or MSR) by additional goals.

MCDM: For tri-criterion formulations, depiction of solutions by afourth portfolio characteristic/measure is feasible by means of ternaryplots.

Caveat/strength of EMO: It is at the user’s discretion to chose his’optimal’ allocation out of the Pareto efficient set, which might be achallenge on its own.

Pfaff (Invesco) MCO RFinance 2016 21 / 24

Bibliography

Bibliography I

Akima, H. and A. Gebhardt (2015). akima: Interpolation of Irregularly andRegularly Spaced Data. R package version 0.5-12.

Deb, K. (2001). Multi-Objective Optimization using Evolutionary Algorithms.Chichester, UK: John Wiley & Sons, Ltd.

Deb, K. (2011). Multi-objective Evolutionary Optimisation for Product Designand Manufacturing, Chapter Multi-Objective Optimization Using EvolutionaryAlgorithms: An Introduction, pp. 3–34. New York, NY: Springer.

Deb, K., R. Steuer, R. Tewari, and R. Tewari (2011). On the effectiveness of ansga-ii local search approach customized for portfolio optimization. KanGALReport 2011007, Indian Institute of Technology Kanpur, Kanpur, India.

Ehrgott, M. (2005). Multicriteria Optimization (second ed.). Berlin, Heidelberg,New York: Springer.

Goldberg, D. (1989). Genetic Algorithms in Search, Optimization, and MachineLearning. New York, NY: Addison-Wesley.

Pfaff (Invesco) MCO RFinance 2016 22 / 24

Bibliography

Bibliography II

Hamilton, N. (2015). ggtern: An Extension to ’ggplot2’, for the Creation ofTernary Diagrams. R package version 1.0.6.1.

Hirschberger, M., R. Steuer, S. Utz, and M. Wimmer (2013, January–February).Computing the nondominated surface in tri-criterion portfolio selection.Operations Research 61(1), 169–183.

Jones, D. and M. Tamiz (2010). Pratical Goal Programming. International Seriesin Opertations Research and Management Science. New York: Springer.

K., D., A. Pratap, S. Agarwal, and T. Meyarivan (2002). A fast and elitistmultiobjective genetic algorithm: Nsga-ii. IEEE Transaction on EvolutionaryComputation 6(2), 181–197.

Ligges, U. and M. Machler (2003). Scatterplot3d - an r package for visualizingmultivariate data. Journal of Statistical Software 8(11), 1–20.

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