Solver and modelling support for stochastic programming H.I. Gassmann, Dalhousie University Happy...

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Solver and modelling support for stochastic programming

H.I. Gassmann, Dalhousie University

Happy Birthday András, November 2009

Agenda• Stochastic programs• An Example• Algebraic modelling languages

– GAMS– MPL– AMPL

• Frontline Systems• SMI (Stochastic modelling interface)• Optimization Services• Other software

Stochastic programs• Two-stage recourse problems• Multistage recourse problems• Chance-constrained problems• Continuous distributions

An Example (Ferguson & Dantzig)

,,1,s.t.

jiijij

Jjzyxd

xij = # aircraft of type i assigned to route j

yj = # empty seats on route j

zj = # lost sales on route j

In the original formulation, d is defined by marginals

Words to ponder…• “Simple recourse is so previous millennium”• “Always build your scenario tree prior to calling

the solver”• “No need to bother with special algorithms ―

Cplex on the deterministic equivalent is best”• “Robust optimization is irrelevant”

GAMSSET equip := (E1, E2, E3, E4); route := (R1, R2, R3, R4, R5); scen := (S1*S720);

DATA avail[equip] := (E1,10, E2,19, E3,25, E4,15); price[route] := (R1,13, R2,13, R3, 7, R4, 7, R5, 1); capac[equip,route] := (...); cost [equip,route] := (...); demand[route,scen] := (...); prob[scen] := (...);

DECISION VARIABLES Assign[equip,route]; Empty[route,scen]; LostSales[route,scen];

MODEL MIN totalcost = SUM(equip,route: cost*Assign) + SUM(route,scen: prob*price*LostSales);

SUBJECT TO availability[equip]: SUM(route: Assign) <= avail;

passengers[route,scen]: SUM(equip: capac*Assign) – Empty + LostSales = demand;

GAMS• Solve deterministic equivalent • Two-stage problems can use DECIS• Discrete scenarios only

MPLINDEX aircraft : = (A,B,C,D);

STOCHASTIC

INDEP route := (NL1, NL2, ND0, ND1, NB0) -> (r1,r2,r3,r4,r45);

OUTCOME out5 := 1..5;

EVENT RouteOut[route,out5] := (NL1, 1..5, NL2, 1..2, ND0, 1..5, ND1, 1..5, NB0, 1..3);

PROBABILITIES p[route, out5 in RouteOut] := [NL1, 1, 0.2, NL1, 2, 0.05, ... ];

MPL (continued)RANDOM DATA Demand[route, out5 in RouteOut] := ...

DATA AircraftAvail[aircraft] := ...; TicketPrice[route] := ...; PotentialPass[aircraft,route] := ...; MonthlyCost [aircraft,route] := ...;

DECISION VARIABLES Assign[aircraft,route] -> x WHERE (PotentialPass > 0);

STAGE2 VARIABLES EmptySeats[route] -> y1; TurnedAway[route] -> y2;

MODEL MIN TotalCost = SUM(aircraft,route: MonthlyCost * Assign) + SUM(route: TicketPrice * TurnedAway);

SUBJECT TO AircraftCap[aircraft]: SUM(route: Assign) <= AircraftAvail; PassengerBal[route]: SUM(aircraft: PotentialPass * Assign) + TurnedAway[route] – EmptySeats[route] = Demand[route];

MPL capabilities• Solve deterministic equivalent (any MPL solver)• Built-in decomposition solver (two-stage - Cplex) • Communication with SMI • Planned extensions: nested decomposition

AMPLset aircraft := {A1, A2, A3, A4};set routes := {r1, r2, r3, r4, r5};

param avail{aircraft} >= 0;param ticketPrice{routes} > 0;param potentialPassenger{aircraft, routes} default 0;param MonthlyCost {a in aircraft, r in routes: potentialPassenger[a,r] > 0};

var demand{route} random;

var Assign{a in aircraft, r in routes: potentialPassenger[a,r] > 0} >= 0;var EmptySeats{route} suffix stage 2;var TurnedAway{route} suffix stage 2;

minimize totalCost: sum{a in aircraft, r in routes: potentialPassenger[a,r] > 0} MonthlyCost[a,r]*Assign[a,r] + sum{r in routes} TicketPrice[r]*TurnedAway[r];

subject to AircraftCap{a in aircraft}: sum{r in routes: potentialPassenger[a,r] > 0} Assign[a,r] <= avail[a];

subject to PassengerBalance{r in routes}: sum{a in aircraft: potentialPassenger[a,r] > 0} potentialPassenger[a,r] * Assign[a,r]

+ TurnedAway[r] - EmptySeats[r] = demand[r];

Frontline systems• Robust optimization• Chance constraints• Recourse models• Expected value, VaR, CVar objectives• One or two stages• Continuous distributions (automatic sampling)• Deterministic equivalent or simulation• Example

SMI (Stochastic Modeling Interface)

• Coin-OR project• API for interaction with stochastic models• Scenario-based• Discrete distributions

Optimization Services• The Optimization Services project aims to provide

A set of standards to facilitate communication between modeling languages, solvers, problem analyzers, simulation engines, and registry and discovery services in a distributed computing environment.

Solvers

Corporate databases

User interfaceData inter-

change

What Is Optimization Services (OS)?A set of XML-based standards for representing, among others,

– optimization instances (OSiL, also OSgL, OSnL and OSsL)– optimization results (OSrL and OSaL)– optimization solver options (OSoL)– communication between clients and solvers (OSpL)

• Open source libraries to work with these standards• A robust API for solver algorithms and modeling systems • Support for linear, integer, nonlinear and stochastic programs• A command line executable OSSolverService• Executables OSAmplClient and GAMSLinks for AMPL and GAMS • Utilities to convert MPS files and AMPL nl files into OSiL• Server software that works with Apache Tomcat and Apache Axis

Why a standard interface?

Modelling systems

Solvers

Modelling systems

Solvers

n*m hook-ups n+m hook-ups

Why XML?• Existing parsers to check syntax• Easy to generate automatically• Tree structure naturally mirrors expression trees for

nonlinear functions• Arbitrary precision and name space• Automatic attribute checking (e.g., nonnegativity)• Querying capabilities via XQuery• Encryption standards being developed• Easy integration into broader IT infrastructure

OSiL Schema – Deterministic data

Representation of uncertainty• Explicit event trees

– Scenario formulation– Only record data items that differ from parent scenario

• Implicit trees (distribution-based formulation)– Assume independence between stages– Distributions within stage discrete or continuous

• Probabilistic constraints– Simple chance constraints– Joint chance constraints

OSInstance: In-memory representation

• XML elements correspond to C++ classes• Child elements mapped as member classes

• set(), get() and calculate() methods

class OSInstance{public: OSInstance(); InstanceHeader *instanceHeader; InstanceData *instanceData;}; // class OSInstance

OSoL – OS option language• Solver options• Initializations of variables• System requirements• Job parameters• In-memory representation: OSOption• API: get(), set(), add() methods

OSrL and OSResult• Result of the optimization

– Solution status– Statistics– Value of primal and dual variables

• Can be displayed in a browser

Solver support• All versions of OS download with COIN-OR solvers

– Clp– Cbc– Ipopt– Bonmin– Couenne– Symphony

• Additiona solver support– Cplex– GLPK– Lindo

Future developments• Stochastic programming• Cone programming• Instance modification• Solution analysis

Other software• SLP-IOR• SPInE/SAMPL/SMPL• Fort-SP• DECIS• MSLiP• …

Conclusions• Slow but steady progress• Stochastic programming is more than scenario

trees and deterministic equivalents• SP-aware modeling systems

Solver and modelling support for stochastic programming H.I. Gassmann, Dalhousie University Happy...

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