Design Search and Optimization_mynotes

7/29/2019 Design Search and Optimization_mynotes

1/59

Design Search and Optimization (DSO) (1 lecture)

First what is designsynthesis vs analysis

What is optimal design? Are all designs optimal (best)? Might we ever deliberately accept

sub-optimal design? To answer this we must be able to compare competing designs and say

which we prefer and hopefully why. (Examples of cars, fridgescost vs performance, trade

offs).


2/59

What is design and how does design search & optimisation fit into it?

Engineering Design is the use of analysis to support synthesis (creation) so as to adequately

define a product (or process).

Note difference between analysis and synthesis. Synthesis involves decisions whereas

analysis does not (ie decisions about the product) how big, what material, what

manufacturing method. Analysis provides the information for rational decisions to be made.

DSO is a formalism for carrying forward such decision making and is normally thought of as

an automated activity controlled by a computer code. It involves postulating (guessing) a

design, analysing it, deciding if the results are acceptable and if not deciding how to change

it. If it is to be changed the process is repeated until we have an acceptable design or we run

out of effort/time.

Design search and optimization is the term used to describe the use of formal optimization

methods in design.

Optimization concerns the finding of the inputs to functions that cause those functions to

reach extreme values.

Designers are often not interested in finding optimal designs in the strictest sense: rather, they

wish to improve their designs using the available resources within bounds set by the desire to

produce balanced, robust, safe and well-engineered products that can be made and sold

profitably. It is never possible to include all the factors of concern explicitly in computational

models and so any results produced by numerical optimization algorithms must be tempered

by judgments that allow for those aspects not encompassed by the models used. Moreover,

there will often be multiple ways of solving design issues that all have benefits and

drawbacks and no single clear winning design will emerge: instead, human judgment will be

needed. Thus, it is more accurate to say that designers use optimization methods in the search

for improved designshence the term Design Search and Optimization.

To set this up we adopt the ideas of:

Objectives

Design variables and their bounds

Constraints and their limits

Fixed parameters

External noise/uncertain parameters

Methods of analysis


3/59

Schemes for linking design variables to analysis

Schemes for linking analysis to objectives and constraints

Discuss the Nowacki beam problemdistribute his papergive reference to books.

Nowacki Beam (See Engineering Design via Surrogate Modelling) :


4/59

TAXONOMY OF OPTIMIZATION (METHODS)Optimization may be defined as the search for a set of inputs x (not necessarily numerical) that

minimize (or maximize) the outputs of a functionf (x) subject to inequality constraintsg(x) 0 andequality constraints h(x) = 0. The functions may be represented by simple expressions, complexcomputer simulations, analog devices or even large-scale experimental facilities.

Inputs can be either nonnumeric or numeric.Example of nonnumeric inputs, can be the selection of a compressor type as being either axial or

centrifugal or the selection of a material between aluminium alloy and fibre-reinforced plastic for usein wing construction.

If all the inputs are numeric, the next division that occurs in problems is between those with

continuous variable inputs and those that take discrete values (including the integers).Examples of discrete variables are things such as commercially available plate thicknesses,standardized bolt diameters, or the number of blades on a compressor disk. Continuous variableswould include wing-span and fan-blade length.

The outputs can also be categorized. There can be a single goal to be maximized or minimized (the

so-called objective function) or are there multiple objectives?If there are multiple goals, thesewill lead to what is termed a Pareto fr ont: a set of solutions that are all equally valid until some

weight or preference is expressedbetween the goals. For example, if it is desired to reduce weightand cost, these aims may pull the design in opposite directionsuntil it is known which is moreimportant or some weighting between them is chosen, it will not be possible to decide on the best

design.Next, one can categorize by the presence or absence of constraints. Such constraints may simplyapply to the upper and lower values the inputs may take (when they are commonly known as bounds)or they may involve extremely complex relationships with the inputs (as in the limits typically foundon stress levels in components).The next categorisation is the type offunctional relationship between the inputs and outputs.These can be linear, nonlinear or discontinuous. They can be stationary, time dependent or

stochastic in nature.o Discontinuous functions make optimization most difficult, and are the normo Linear relationships may lead to some very efficient solutions.

We can therefore categorize the Nowacki beam design problem as having continuous inputs,

one or two outputs, various constraints and stationary nonlinear (but not discontinuous)

relationships linking these quantities.Methods themselves may also be classified in a number of ways. The first division that can bemade is between methods that deal with:

1. Optimal selection2. Solve linear equations, and3. The rest.

Optimal selection routines commonly stem from the operational research (OR) community andtypically are set up to deal with idealized traveling salesman or knapsack problems.

Linear problems are nowadays almost universally solved using linear programming and the so-

called simplex or revised simplex methods, which can efficiently deal with thousands of

variables.

Optimal selection methods and linear programming, while valuable, essentially lie outside the scopeof this book, as they do not find much practical application in aerospace design, which is dominatedby large scale Computational Fluid Dynamics (CFD) and Computational Solid Mechanics (CSM)

models; they are not discussed further here.


5/59

The remaining methods that deal with nonlinear numeric problems form much the largest collectionof approaches. At the most basic level, such searches may be divided between those that need

information on the local gradient of the function being searched and those that do not. Searchesthat will workwithout any gradient information may be termed zeroth order while those needing

the first derivatives are first order, and so on. A further distinction may be made betweenapproaches that can deal with constraints, those that cannot and those that just need feasiblestarting points.Methods may also be categorized by whether they are deterministic in natureor have somestochastic element.

o Deterministic searches will always yield the same answers if started from the same initialconditions on a given problem; stochastic methods make no such guarantees.

o You may think that variation of results from run to run ought to be avoided but it turns outthat stochastic search methods are often very robust in nature: a straightforward random walkover the inputs is clearly not repeatable if truly random sequences are usednonetheless,such a very simple search is the only rational approach to take if absolutely no information isavailable on the functions being dealt with.

A random search is completely unbiased and therefore cannot be misled by features in the problem.

Although it is almost always the case in design that some prior information is available on thefunctions being dealt with, the pure random search can be surprisingly powerful and it also forms a

benchmark against which other methods can be measured: if a search cannot improve on a randomwalk, few would argue that it was a very appropriate method!

A distinction may further be made between searches that work with one design at a time and thosethat seek to manipulate populations or sets of designs.

Population-based search has gained much ground with the advent of cluster and parallel-basedcomputing architectures since the evaluation of groups of designs may be readily parallelized on suchsystems. Perhaps, the most well-known of such methods are those based on the so-called evolutionarymethods and those that use groups of calculations to construct approximations to the real objectivesand constraints, the so-called Design of Experiment and Response Surface methods.


6/59

SUMMARRY OF TAXONOMY OF OPTIMIZATION

inputs

some non-numeric all numeric

continuous some discrete

problems

outputs

Single goal Multiple goals (Pareto fronts)

constraints

unconstrained bounds constrained

methods

LINEAR PROBLEMS

LINEAR PROGRAMMING

OPTIMAL SELECTION

INTEGER PROGRAMMINGTHE REST

operational research

sorting/exchange methods

methods

simplex methods

search over vertex space

NOT DISCUSSED

FURTHER

NOT DISCUSSED

FURTHER

No gradients needed

(zero order)

gradients needed

(first or second order)

cope with constraints directly only unconstrained

deterministic stochastic

population based one at a time

the rest of the

course!

subject to

functions

linear non-linear discontinuous


7/59

Generally, optimization algorithms can be divided in two basic classes: deterministic and

probabilistic algorithms. Deterministic algorithms (see also Definition 30.11 on page 550) are

most often used if a clear relation between the characteristics of the possible solutions and

their utility for a given problem exists. Then, the search space can efficiently be explored

using for example a divide and conquer scheme. If the relation between a solution candidateand its fitness are not so obvious or too complicated, or the dimensionality of the searchspace is very high, it becomes harder to solve a problem deterministically.


8/59

Brief History of Optimization

Newton and classical gradients

144,42 2 xxdx

dyxxy

42

2

dx

yd 2min y

Pattern search and which way is down methods

Stochastic searchpopulation

DoE based search and RSMS - HYBRIDS

Nonlinear numerical optimization methods are a very large field, but may be grouped into

three main themes:1. The classical gradient-based methods and hill-climbers2. the evolutionary approaches3. The adaptation of Design of Experiment and Response Surface methods

First came classical calculus and Newtons method for dealing with functions where we

cannot solve explicitly. Calculus we are familiar with 0)(' xf and VExf )('' for a

minimum etc. Newton is basically root searching for 0)(' xf and is covered in a

subsequent lecture.

Next come Cauchy and steepest descentie find the downhill direction and move in that

direction until we start to go uphill againslow in valleys:-


9/59

Classical hill-climbers all seek to move uphill (or downhill) from some given starting point in

the direction of improved designs. Sometimes, they use simple heuristics to do this (as in the

Hooke and Jeeves search (Hooke and Jeeves) and sometimes they use the local gradient

(gradients obtained from finite differencing) to establish the best direction to move in (as in

quadratic programming).

Then came conjugate gradient methods and quasi-Newton methods that exploit local

curve fitting based on a curvature. Various ways of holding information on the local shape.

(The Hessian, or its inverse) ie approximation based on CxbAxxxfTT

21)( where x is

a vector and A is the Hessian.

Following these gradient based approaches were a series of Pattern searches which use

Heuristics. These include Hook & Jeeves and the simplex method (see plots of searches).

Then a series ofstochastic methods including SA, GA, ES and EP all using sequences of

random moves and schemes to exploit any gains made. Often working with population of

designs. The main benefit is that such methods can avoid becoming stuck in local basins of

attraction and can thus explore more globally. The downside is that they are often slower to

converge to any optima they do identify.

Finally come the explicit curve fitting methods based on designed experiments such as

polynomial curve fits, RBF schemes, Kriging etc. These can be either global with updatesor local with move limits and trust regions.

Design of Experiment and Response Surface methods are not really optimizersper se: rather,

they are techniques that allow complex and computationally expensive optimization problems

to be transformed into simpler tasks that can be tackled by gradient and evolutionary

methods. Essentially, these are curve fitting techniques that allow the designer to replace

calls to an expensive analysis code by calls to a curve fit that aims to mimic the real code.

Finally hybrids and meta searches built on these elements.

Key considerations are:

Staircase effect etc

Can be used for analytical forms

or numerically.


10/59

Ability to work with black box codes

Need for gradient information

Robustness to poor, noisy or incomplete data or badly shaped functions

Speed of convergence Ability to run calculations in parallel

Repeatability/average performance of stochastic methods

The Place of Optimization in Design Commercial Tools


11/59

GEOMETRY MODELLING & DESIGN PARAMETERIZATION-The Role of Parameterization in Design

-Discrete and Domain Element Parameterizations

-NACA Airfoils

-Spline Based Approaches

-Partial Differential Equation and Other Analytical Approaches

-Basis Function Representation

-Morphing

-Shape Grammars

-Mesh Based Evolutionary Encodings

-CAD Tools v's Dedicated Parameterization Methods

The purpose of design parameterization is to define geometric parameters, as well as to

collect a subset of these parameters as design variables that can vary during the design

process. During the design process, the design engineer will vary the design variables in order

to improve structural performances.

The need for parameterization design variables/design intent the things we are free to

change.

Flexibility versus no of variables & tacit knowledge of workable designs (SECT 2.1 of

book). Some exampleslook at the Nowacki beam & ask what are design choices? How do

we encapsulate (capture) them? What about choice of section? (Want a section that gives a

high second moment area). How do we parameterise a cross section? How many variables do

we need.

1) a circle - just need radius1 variable

2) a squarejust need side length1 variable

3) /4) an elipse or rectangle2 variables

5) an I section symmetric about N/A (neutral axis) or a box L or T sections in all cases

we need overall width and depth plus thickness of web and flange4 variables

Can we make a single parameterisation span square, rectangle, box, L, T and I? If so how

do we use an integer variable as an index plus 4 numbers or can we use the 4 numbers

themselves? Clearly square/rectangle/box can be linked:-


12/59

Not obvious how to deal with L, T or I together. If use an index then there is no obvious

ranking so search not simple. We can of course do

As the solids are when tf= depth/2

tw= width/2

or or

As continuous sets Simplest combined form is

for etc


13/59

LECTURE 2

To make this generate all shapes we consider 4 rectangles

We make the widths of :

T and B to equal width overall

L and R to equal webt

Height of T equal toflangetop

t

Height of B equal to flangebott

Height of L and R equal to height flangetopt flangebott and then add offsets of L and R from the

outer edges as offsetl or offsetr .

This needs seven variables but can now describe all our shapes as to get L or T we just set

fangebottomt to be zero.

Talk through the various figures in sect 2.1 and see PPT of External shape of UAV

Internal structure of UAV

Evolutionary optimisation of beams

T

L R

B


14/59

DEMO 1Excel & Nowacki Beam

How do we set up searches in Excel?

Try 142 xx (min at 2x )

4x 42 3x (min at211x )

Then look at Nowacki beam problem.


15/59

SINGLE VARIABLE OPTIMISERS, LINE SEARCH: - an optimal solution to a problemwhere there is just one real-valued variable are called line search problems

IntroductionOptimization methods may be classified according to the types of problems they are designed to deal

with. Problems can be classified according to the nature of the variables the designer is able to control(nonnumeric, discrete, real valued), the number and type of objectives (one, many, deterministic,probabilistic, static, dynamic), the presence of constraints (none, variable bounds, inequalities,

equalities) and the types of functional relationships involved (linear, nonlinear, discontinuous).Moreover, optimizers can be sub classified according to their general form of approach.They can be gradient based, rule based (heuristic) or stochasticoften, they will be a subtle blend ofall three. They can work with a single design point at a time or try to advance a group or populationof designs. They can be aiming to meet a single objective or to find a set of designs that togethersatisfy multiple objectives. In all cases, however, optimization may be defined as the search for a

set of inputs x that minimize (or maximize) the outputs of a function f (x)subject to constraints

gi (x) 0 and hj (x)= 0.

In essence, this involves trying to identify those regions of the design space (sets of designer chosen

variables) that give the best performance and then in accurately finding the minimum in any givenregion. If one thinks of trying to find the lowest point in a geographical landscape, then this amountsto identifying different river basins(sinks) and then tracking down to the river in the valley bottom andthen on to its final destinationin optimization texts, it is common to refer to such regions as basinsof attraction since, if one dropped a ball into such a region, the action of gravity would lead it to thesame point wherever in the basin it started. The search for the best basins is here termed global

optimization, while accurately locating the lowest point within a single basin is referred to aslocal optimization. Without wishing to stretch the analogy too far, constraints then act onthelandscape rather like national borders that must not be crossed. It will be obvious thatmany searcheswill thus end where the river crosses the border and not at the lowest pointin the basin. Of course,in design, we are often dealing with many more than two variables,but the analogy holds and we canstill define the idea of a basin of attraction in this way.

The aim of optimization is to try and accomplish such searches as quickly as possible in as robust afashion as possible, while essentially blindfoldedif we had a map of the design space showingcontours, we would not need to search at all, of course.Notice that even when dealing with multiple design goals it is usually possible to specify a single

objective in this way, either by weighting together the separate goals in some way, or by transformingthe problem into the search for a Pareto optimal set where the goal is then formally the production ofan appropriately spanning Pareto set (usually one that contains designs that represent a wide andevenly spread set of alternatives). It is also the case that suitable goals can often be specified bysetting a target performance for some quantity and then using an optimizer to minimize any deviations

from the target. This is often termed inverse design and has been applied with some success to airfoiloptimization where target pressure distributions are used to drive the optimization process.


16/59

LECTURE 3 CONT

Problemminimize a problem of a single real variable without constraints.

Approach 1given a functional form use calculus, ie

14)( 2 xxxf

42)( xxf

2)( xf , ie VE

Approach 2Bracket the minimum between two values and search inwards.

QHow do we find a bracket, ie a series of three values of )(xf such that )()( 12 xfxf

and ?)()( 32 xfxf

A Guess two values for x, calculate )(xf at these and then head downhill until the

function starts to rise and we have a bracket. If we have no knowledge then use

1,0 21 xx and 3x either -1 or 2 depending on the gradient (if )1()0( ff then 23 x

else -1). Given three points we use a quadratic curve fit and see if a minimum is

predicted (2nd diff is +VE) and if so jump to the minimum predicted and evaluate there.

If a maximum is predicted we simply increase the step size (by say a factor of 1.6180

golden section) and go on downhill, keeping the three lowest values of )(xf in either

case.

See code in Numerical Recipes for example. Another approach is just to keep doubling

the step size in the downhill direction until a bracket appears.

QGiven an initial bracket how do we trap the minimum efficiently.

There are a number of schemes for doing this and all aim to replace one or other (or both) of the outerbounds with ones closer to the optimum and thus produce a new, tighter bracketing triplet:

o Golden Section Search can yield sufficient accuracyo Fibonacci Search Faster, requires number of iterations

AApproach 3Golden section search (linear convergenceno use of gradients)

min at


17/59

Given 21, xx and 3x such that )()( 12 xfxf

)3()( 2 xfxf

We choose 4x so that it lies in the larger of the two intervals 1x to 2x and 2x to 3x and

such that either23

24

xx

xx

= 0.38197 or

12

14

xx

xx

= 1-0.38197 = 0.61803

Even if the initial bracket is not in the ratio 0.38197:0.61803 this process rapidly settles

on this ratio.

Note that (3 5)/2(0.38197).

Provided we start with a triplet where the internal point is 0.38197 times the triplet width fromone end, this will mean that whichever endpoint we give up when forming the new triplet, theremaining three points will still have this ratio of spacing. Moreover, the width of the triplet is

reduced by the same amount whichever endpoint is removed, and thus the search progresses at auniform (linear) rateConsider points spaced out atx1 = 0.0,x2 = 0.38197 andx3 = 1.0. With these spacings,x4 mustbe placed at 0.61804 (= 0.38197 (1.0 0.38197) + 0.38197) and the resulting triplet will thenbe eitherx1 = 0.0,x2 = 0.38197 andx4 = 0.61804 orx2 = 0.38197,x4 = 0.61804 andx3 = 1.0

This approach assumes nothing about the shape of the function and does not

require gradientsit is not quite so good as Fibonacci but does not require us to fix the

number of function calls apriori (which Fibonacci does).

X1 X2 X4

X3

Ratio 0.38197:0.61803

X1 X4 X2

X3

Ratio 0.61803: 0.38197


18/59

QGiven an initial bracket and assume the function is smooth so that at its minimum it will

behave quadratically.

AApproach 4inverse parabolic interpolation, quadratic search or quadratic interpolation.

Here we fit a parabola to the bracket and use this to estimate the location of the

minimum.

ie we assume ,)( 2 CBxAxxf so thatA

Bx2*

, Axf 2*)(

and we know 2211 )(,)( fxffxf and 33)( fxf such that 3212 , ffff and

321 xxx

ie CBxAxf 12

11

CBxAxf 22

22

CBxAxf 32

33

We solve these to get

)))(())(((2

))(())((

*23211223

2

2

2

321

2

1

2

223

xxffxxff

xxffxxff

x

and

))(())((

))(())((

23

2

1

2

212

2

2

2

3

23211223

xxxxxxxx

xxffxxffA

(which is always +VE so hence minimum)

For example, consider the function

42)( 34 xxxf with initial data at 2,1,2

1

x so that

42,31,8125.32

1321

fff , ie a bracket.

then

214286.1

)12)(38125.3()2

11)(34(2

)14)(38125.3()4

11)(34(

*

x

This may be compared to the analytical solution given from

064)( 23 xxxf when0

x

or2

11x . Note3125.2)5.1(

f


19/59

01212)( 2 xxxf at 0x inflexion

=9 at211x minimum

So the solution is improved

,5932164.2)214286.1( f ie less than all points in initial bracket.

So next triple is 1, 1.214286,2

With function values 3,2.5932164,4

this gives

317823.2,465064.1,385534.2,364454.1 *4*

4

*

2

*

2 fxfx

313928.2,482046.1,334451.2,427849.1 *5*

5

*

3

*

3 fxfx

OBSV 1.

Now you may ask why not start with x=0,1,2. Trouble with this is f=4,3,4 which is

symmetric so 1*1 x which does not help!!

OBSV 2.

This search approaches our goal from one side only and so is rather slow there are better

methods!

When dealing with discrete variables we use the integers as pointers and either use integer

programming or in mixed problems simply round variables to the nearest integer.

When our discretes have no natural order (ie materials selection) we in the end are forced

towards enumeration (listing).

Approach 5Newtons method

All will be familiar with the Newton-Raphson method (gradient based scheme) for finding

the root (or zero) of a function. If we apply this to the derivation of a function we can find

turning points instead, ie

)(

)(1

i

i

iixf

xfxx

NB this needs the second derivative.

Example: 42)( 34 xxxf starting at 2x


20/59

23 64)( xxxf

)1(12)( xxxf

)1(12

6423

1

ii

iiii

xx

xxxx

= ix )1(6

322

i

ii

x

xx

6

682)12(6

64222 21

xx

= 6667.13

21

52777.14

95

3

5

)3

2(6

5)925(2

3

213

x

*50000.1,50097.1 54 xxx


21/59

LECTURES 4&5 (LS110132)

Multi variable optimizers

Gradient Methods

We next consider multiple variables. Here in addition to finding the size of step to make we

must also fix the direction.

Perhaps the simplest approach to multi variable optimizing is to identify the direction of

steepest descent (Cauchy) and go in that direction until the function stops reducing (the

optimal step length*

i ) and then recomputed the direction of steepest descent, ie

)(*1 iiii xfxx where )( ixf are the gradients at ix

Exampleminimize 22212

12121 22),( xxxxxxxxf

starting at 1x =

0

0, 0)0,0( f

1

1)(

221

241

/

/1

21

21

2

1xf

xx

xx

xf

xff -this the direction vector

To get the optimal step length (how far you in a particular direction) we minimize

))(( 111 xfxf with respect to 1 , ie set 0/ 1 ddf

2

1

2

1

2

1111

1

1111 22)1

1

0

0

())((

ffxfxf

= 12

1 2

122*

11

1

d

df


22/59

so that

1

1

1

110

02x , 1)1()1)(1(2)1(211)1,1(

22 f (f is getting

smaller)

now

1

1

221

241)( 2xf

1

1)1

1

1

1())((

2

2

2222

ffxfxf

= 22222

222 )1()1)(1(2)1(2)1()1(

= 22222422 22

2

2

22

2

2

= 425 22

2

51210 *222

d

df

2.1

8.0

1

151

1

13x , 2.1)2.1,8.0( f , and so on to

5.1

1as the answer.


23/59

LECTURES 4&5 continued

CONJUGATE GRADIENT

The problem with steepest descent is that unless our function has circular contours the

direction of steepest descent never points at the final optimum. The conjugate gradient

approach seeks to improve over this with improved directions.

We start as per the steepest descent but at the second step use a direction conjugate to the

first, ie

)( 1*

112 xfxx but afterwards use iiii Sxx*

1

NB 0jT

i ASS if iS and jS are conjugate directions for a quadratic problem of the form

CxBAxxxf TT 21)(

Nota Bene -Abbreviation. NB: Used to direct attention to something particularly important.

where 121

2

i

i

i

ii Sf

ffS

and )( 11 xfS

Here iS takes the place of if used in steepest descent. Notice that iS accumulates

information from all previous steps this is good and bad good as the direction is

conjugate to all previous steps, bad as it can accumulate round off errors in practice

we restart from a steepest descent step after m steps where m is one more than the

number of design variables. If our function is quadratic this process converges in as

many steps as directions/dimensions in the problem.

Example: minimize 22212

12121 22),( xxxxxxxxf starting from

0

01x

Conjugate gradient search example.


24/59

Starting at

0

0and apply steepest descent gives as before

1

12x and 1

*

1 and 21

1

)(2

1

fxf

So

1

11S , ( )( 11 xfS ) 2

1

1)(

2

22

fxf

2

0

1

1

2

2

1

12S

2

0

1

1*

23 x

To find *2 we minimise )2

0

1

1()( 2222

fSxf

12

1

2f

2

222 )21()21(22)21(1

124 22

2

So4128

*

22

2

d

df

5.1

1

2

0

41

1

13x which is the solution

If we try another step we just find2

3f is zero and so the process stops, ie 3f is zero at the

minimum.


25/59

LECTURES 4& 5 continued

NEWTONs METHOD

Newtons method allows for direction and step size and is built on looking for the roots of

).(xf

First we approximate our function as a Taylor series.

)()(21)()()( 2 ii

T

ii

T

i xxHxxxxfxfxf

Where here iH is the matrix of second partial derivatives and is called the Hessian.

Now we set 0)(

jx

xfj=1,2,..n for n variables

So this gives 0)( iii xxHff

or iiii fHxx

1

1this requires a non singular Hessian of course. (So the Hessian both

modifies the search direction and sets the step length).

Exampleminimize 22212

12121 22),( xxxxxxxxf starting at

0

01x

1

2

2

2

12

221

2

2

1

2

1

xx

f

xx

f

xx

f

x

f

H

=

22

24for all ix

411

1

H

42

22=

12121

21

ix

ixf

xff

2

1

/

/=

1

1

221

241

21

21

ixxx

xx


26/59

So

1

2121

21

0

02x

23

1

1

1

1

12f

2

4

0

0

3

3

Hence MINIMUM in 1 STEP

This has converged in one step because )(xf is quadratic and so H is a constant. There are

however problems with this approach as we have to compute, invert and store H at each

step and this is fraught with difficulties on real problems. The most serious issue is obtaining

the second derivatives as these are very rarely available directly.

The Quasi Newton methods work with an approximation of either the Hessian or its inverse.

These are sometimes called variable metric methods

We already have )(11 iiii xfHxx

iH is the Hessian

Which we approximate by

)(* iiiii xfBxx

Here iB contains directional information and*

i the optimal step length. Note that this is the

steepest descent method if iB = I

There are then a number of schemes for updating iB without using second derivatives but

instead using approximations. None is perfect and they are known by the names of those

who proposed them such as BFGSBroyden Fletcher Goldfarb-Shanno, which is

iT

i

i

T

ii

i

T

i

T

iii

i

T

i

ii

T

i

i

T

i

T

iiiigd

Bgd

gd

dgB

gd

gBg

gd

ddBB )1(1

where iii xxd 1

iii ffg 1

We do not pursue such methods further here, they are very popular however.

See Figs 3.6,3.7 in book.


27/59

Noisy/Approximate Function Values- Gradient-based searches can rapidly identify optima in a fitness landscape with high precision.- Gradient-based methods only seek to follow the local gradient to the nearest optimum in thefunction, and so, if they are to be used where there are multiple optima, they must be restarted fromseveral different starting points to try and find these alternative solutions.- One great weakness in gradient-based searches, however, is their ratherpoor ability to deal withany noise in the objective function landscape.


28/59

Non gradient based search methods- establishing the gradient of the objective function in

all directions during a search can prove difficult, time consuming and sensitive to noise, a variety ofsearch methods have been developed that work without knowledge of gradients. The most commonmethods are pattern/direct searches and stochastic/evolutionary algorithms

-Pattern or direct search: Hooke and Jeeves, and Nelder and Mead

Advantages:

Simple

Robust (relatively)

No gradients necessary.

Disadvantages:

Can get stuck and special "tricks" are needed to get the search going again.

May take a lot of calculations

Nonlinear multi-objective multiplex implementations using pattern search

algor ithms have been made.

What do we do if we cannot calculate gradients (or do not wish to use finite differences

noise/speed).

This leads to Hook & Jeeves amongst others.

H&J method:

1 choose initial step length, set initial point to first base point

2 increase direction i by step and keep if better else decrease direction i by step and

keep if better


29/59

3 loop over all directions, if none improve then half step size and repeat unless

either step too small or run out of time in which case stop

4 explore must have helped so set current point to new base point

5 make pattern move equal to vector from previous base point to new base point

plus any previous successful pattern move still in use

6 if pattern move helps keep it if not go back to new base point and forget pattern

move.

7 repeat from step two

There are several themes here.

1 steps change in size for exploration

2 Directions and steps change for exploitation if the pattern moves help then they

accumulate so that moves get bigger and bolder until they fail. Siddall provides

full details and code, as does Schwefel.

Note that the pattern moves get larger at each exploratory search pattern and slowly alignsitself to the most profitable direction of move. This allows the search to make larger and


30/59

larger steps. Conversely, once a pattern move fails, the search resets and has to start

accumulating pattern moves from scratch


31/59

-Stochastic/Evolutionary search

In GA, every run yields a different answer, so need to take averages.

Run through flying circus slides on simple GAs.

Use web animation to demo a GA on the bump problem.

Typical search patterns from a GA, Simulated Annealing, an Evolution Strategy and

Evolutionary Programming.


32/59


33/59

Termination/Convergence

For local searches we stop at the optimum, ie when no further gains are being made-provided

we can afford to get that far.


34/59

For global search we use one of

1 a fixed or limited number of iterations

2 a fixed or limited elapsed time

3 when the search has stalled after a given number of iterations

4 when a given number of basins have been found and searched.

We rank searches by steepness of gradient/rate of improvement, final result or a balance

between the two.


35/59

LECTURES 6&7

CONSTRAINED OPTIMIZATION

In most real world engineering problems the designer has to satisfy various constraints as

well as meeting the desire for improved performance. Indeed performance is often set as a

constraint, ie reduce weight to below x, reduce drag to less than Y etc. Thus we need search

schemes to deal with constraints, ie

forxf )(min

nx

x

x

1

subject to bounds on ,x xxxL

and constraints 0)( xgi (inequality constraints)

0)( xhj (equality constraints)

Here we describe a number of approaches.

The simplest is to try and eliminate constraints by construction ie transform problem

variables using the constraints.

Example: minimise the surface area of a box of given volume.

ie min )(2),,( HWWBBHWBHf , whereL,B andH, are the length of the sides

and WBHv is fixed

So letBH

vW

we have

Minimize )(2),,( HBBH

VBHVBHf

B

V

H

VHB

222

So 02

22

H

VB

H

fwhen

B

VH

B

VH 2, or B= 2H

V


36/59

02

22

B

VH

B

fwhen

HVB or

HVB 2

Combining gives

3

4

2

VHH

V

H

V

3 VB (Substitute 3 VH into B= 2HV )

3 VW (Substitute 3 VH & 3 VB intoBH

vW

)

ie all sides of equal length is expected.

Another way we deal with inequalities is by deciding if they will be active at the

optimum or not. If so we replace by equality and if not we eliminate them. So if

inequalities are active then we replace with equality sign.

Often it is not possible to know which inequalities will be active or to eliminate using algebra

even if we do! Nonetheless we should not ignore this. It can be done numerically sometimes,

ie fixed LC calcs when angle of attack is a design variable for a wing or aerofoil.

GEOM & ALPHA

CFDOPTIMISE

with CL= fixed

CL,CD

GEOM

OPTIMISE Iterateon

alpha

CFD

CL

OR

CD


37/59

LAGRANGE MULTIPLIERS

In just the same way as there are formal analytic solutions to unconstrained optimization

problems the equivalent constrained solutions are based on Lagrange multipliers. This

approach essentially only works for equality constraints so for inequality constraints a

precursor step is to decide at any point if an inequality constant will be active, and if so

replace it with an equality.

So consider min ),( 21 xxf subject to 0),( 21 xxg

ie two variables and one equality constraint.

At a minimum it may be shown that

011

x

g

x

f

and 022

x

g

x

f

and 0),( 21 xxg

Here is the so called Lagrange Multiplier.

Now if we write gfL we get

111 x

g

x

f

x

L

all equal zero at the minimum

222 x

g

x

f

x

L

from the previous equations

gL


38/59

Thus if we seek the unconstrained minimun of L (more precisely, turning points of L) we can

locate the solution to the constrained problem. L is known as the Lagrange function.

For example minimize ),( yxf2xy

k

Subject to 0),( 222 ayxyxg (ie circle of radius a)

Here )(),,( 2222

ayxxy

kgfyxL

0222

xykx

x

L

423

22

xy

k

yx

k

2

yx

0222

ayxg

L

,

3

20

2

222

ayayy

Here yax ,3 3

2a

Note however that we cannot simply minimize L as the approach would admit of

saddlepoints or maxima for the gradients of L to be zero.

PENALTY FUNCTION METHODS

A more direct approach to dealing with constraints is via the use of penalty functions

we simply add penalties to the objective function when constraints are violated. There

are a number of ways of doing this, none of which is perfect:-

FIXED PENALTIES

Add a (very) large number to the objective if any constraint is broken

Add a (very) large number for each broken constraint

022 31

yykx

y

L


39/59

VARYING PENALTIES

FUNCTION OF DEGREE OF CONSTRAINT VIOLATION

Scales the penalties by the constraint violation

FUNCTION OF HOW LONG WE HAVE BEEN SEARCHING

Start with low penalties and gradually make more severe so that an essentially

unconstrained search becomes a fully constrained one

All these are taken to be exterior penalties, ie they only apply to broken constraints we can

also use interior penalties which come into effect as the search nears a constraint and then

gradually remove these as we progress so as to warn the search about nearby problems.

Sketch Penalty Types

COMBINED LAGRANGE + PENALTY FN METHOD

Since many penalty functions introduce discontinuities which make search with efficient

gradient descent methods difficult, a potentially attractive approach is to combine a penalty withthe method of Lagrange multipliers.

It is possible to combine the Lagrange scheme with a penalty approach to overcome some of

the difficulties of pure Lagrange methods. This is sometimes called the Augmented

Lagrange Multiplier method.

ie minimize )(xf subject to 0)( xhj pj .. .2,1

OF

STEP

INTERNAL EXTERNALX2

X1

OF OF

X1 X1 X1


40/59

)()(),(1

xhxfxL j

P

j

j

is the Lagrangian

We augment this with an exterior penalty

)()()(),,(1

2

1

xhrxhxfrxAP

d

jkj

P

j

jk

This model is therefore amenable to search by gradient-based methods provided the originalobjective and constraint functions are smooth.

It now turns out that minimizing A solves the original problem if we have the correct

for any kr . Note that kr is some, but not infinite penalty. However we can apply an iterative

scheme now that will allowj

andk

r to converge on a solution providedkk

rr 1

and we use

kjbkjkj xhr *)()1( 2 , note that k is the major iteration counter.

ie the new s' are added to by the (scaled) amount of violation of the constraints at the

previous minimum of A

This approach can also be extended to inequality constraints by setting up as follows; min

)(xf subject to

mixgpjxh ij ...1,0),(,...1,0)(

)()(),,(11

xhxfrxA j

p

j

jm

m

i

ii

p

d

jk

m

i

ik xhrr1

)(2

1

2

where i max

k

ii rxg2

),(


41/59

Sequential Quadratic Programming SQP

The use of sophisticated Lagrangian processes is now at its most complex and powerful in the

class of methods known as SQPthese use typically Quasi Newton methods to solve a series

of sub problems. They are the most powerful methods available for local minimization of

constrained smooth problems. Academic codes are available from the web. They are less

good for non-smooth functions and also they are local methods and so cannot find the best

basin of attraction to search.

(Chromosone) Repair

Repair is the process of dealing with a constrained optimization problem by substituting

feasible designs whenever infeasible ones occur during search. To do this a repair process is

invoked if any constraint is violated to find the nearest feasible design. Here nearness is

usually in the Euclidean sense of design variables. Having located such a design (perhaps by

a local search where the degree of infeasibility is set as a revised objective) the objective

function of the feasible design is used instead of that at the infeasible point and also

(optionally) the corrected design vector.

Replacing the design vector absorbs most information but can cause problems with the search

engine. This approach is most favoured in evolutionary or other zeroth order (non-gradient)

methods where gradients are not used at all.


42/59

LECTURE 8

META-MODELS + RSM

So far we have considered optimizers working with results coming from the evaluations of

design & constraint functions that have been presumed to be directly coupled to search codes.

These codes then build up a picture of how the function is changing with changes in the

design and seek improvements. Their internal models (we will call them meta-models to

distinguish from the actual user supplied design models) are implicit in their working.

We next consider schemes where the building & use of the meta model is explicit and

directly controlled by the user.

At its simplest this consists of running a few designs, collecting the results and curve fitting

to these. Then the curve fits can be used for design search. This would be a natural approach

for working with data from previous designs or from experiments or field trials it can also

be used with computer analysis codes.

We first plan where to run the code to generate data. This can aim to build either a local or a

global model depending on the range of the design variables. We use formal DoE (Design ofExperiment) methods for this (cf Taguchii). Having run the design points, often in parallel,

we curve fit. Here again we decide if we need a local (simple) fit or a global (complex) shape

& also if we need to regress (discuss noisy data). Curve fitting can be fast for simpler models

or very slow for large accurate ones.

We call the curve fit a Response Surface Model (RSM) or meta model. Examples include

Polynomial regression, radial basis functions and kriging and neural nets.

Having built a model we check its accuracy with test data (separate) or cross validation. We

then use it to search for a better design. Having found new candidate designs we run the full

computer code to check if they are good. If so we might stop. More usually we add these to

the curve fit & iterateupdating, until we run out of effect or we get convergence etc.

Give examples. Discuss exploit vs explore.

To summarise: the basic steps are


43/59

1) plan an experiment to sample the design space

2) run codes & build data-base of results (possibly in parallel)

3) choose & apply a curve fit, with or without regression

3a) refine curve fit by some tuning process

4) predict new interesting design points by searching the meta-model

5) run codes on new point(s) & update data-base (again possibly in parallel)

6) check the results from update points against predictions & then either stop or

move back to step 3)

Experience shows that for model building it can often take 10n initial designs to build a

reliable global model where n is the number of variables. There is also a trade between the

cost of building a meta-model and the usefulness of its predictions. (Show PPT on various

DoE designs & various RSM types)

Example 1a local trust region search

A very simple approach is to evaluate a small local experiments and then shift and shrink it

until a certain effort is used up.

Step one) choose initial area to search

Step two) sprinkle in 9 points useLPDoE

Step three) curve fit with quadratic regression polynomial

Step four) search within area over RSM to get new candidate design

Step five) shift search region centre to new candidate point

to solve the regression we used SVD to get a least squares solution to the over constrained non square matrix

equation bay where y are the function values, a are the design variable values and their powers

and b are the polynomial coefficients. So if the SVD of a is '.. VwU then it may be shown that

}{')]./1(diag.[}{ yUwVb j , see matlab for simple examples of SVD.


44/59

Step six) shrink search region by say 10%

Step seven) replace oldest design point with new result

Step eight) go to step three unless run out of time

Now this simple scheme has a number of faults:

1) it has no way of expanding the trust region if the data suggests it should be this

means it may be a), slow and b), fail to find a local minimum of the function.

2) The point being replaced, the oldest, may not be the most sensible one to discard

what about discarding the worst point for example.

Example 2a global RSM search

1) Here we first use 100 points in an LP array to sample the design space.

2) Then we construct a krig (stochastic or Gaussian process) RSM which has hyper

parameters which we tune.


45/59

3) We then search for peaks and return 10 likely locations

4) We add these to the original 100 pts to get 110 and we rebuild and retune the krig.

5) We then return to step 3 and repeat 3 times, ending with 130 points and the finalmodel.


46/59


47/59

Points to note

1) the use of a large initial DoE is warranted here because of the multi-modality of

the problem (the 20 points we might otherwise use).

2) the updates are added in groups of 10 because we wish to improve the model

globally and not just in one location, also krig training is costly.

3) the final surface is reasonable but still far from exact.

Meta Heuristics

It is clear from the two previous searches that what we have done is combined components

such as DoE sampling, RSM building and various searches to build a composite or meta-

search. It is of course possible to build more and bigger complexities into such approaches

and this leads to a whole family of meta heuristics

By way of example we can consider combing various gradient descent schemes & then

observing which works best and rewarding this with more of our finite budget of compute

resource. The development of such schemes is an art and also one must bear in mind that the

no free lunch theorem show that, averaged over all possible problems, all searches are as


48/59

good or bad as each otherso unless our efforts are based on tuning a method to the current

task they will be futilemoreover there will be a trade between performance on a specialised

task and general applicability.

Visualizationshow & describe HAT plots and parallel axis plots.

LECTURE 9

Multi-Objective Optimization

So far we have focussed on problems with a single goal or objective function. This is rarely

how real design problems occur, although it is quite common, even in industry, to treat them

this way. In reality most real design problems involve trading between multiple andconflicting goals. We therefore next turn to ways of tackling such problems.

Perhaps the simplest approach (and that most commonly used in industry) is to set all goals

except one in the form of constraints, ie instead of aiming for low weight or stress we set

upper limits on these goals and then ensure that our designs meet them. The difficulty with

this approach is deciding realistic but demanding targets: if they are two severe we may not

be able to satisfy them at all, if too loose they may not impact the design at all.

The next most simple way of proceeding is to use an aggregate or combined objective.

Typically we add all our goals together with some suitable weighting functions and minimize

this. This approach is a mimic of the function of money money is societys way of

allowing completely different things to be balanced against each other (the cost of a holiday v

a new car for example). It is the function of markets to establish the prices of items and

hence the weighting between them. Ideally the best approach to balancing competing goals

to a business engaged in design is to reduce all decisions to their impact on the companys

profits. Unfortunately this calculation is almost never possible so some surrogate is used.

This may be completely artificial or it may be some physical quantity such as SFC (aero

engine makers often use SFC).

It should be clear that if we consider two goals (say )(1 xf and ))(2 xf then depending on how

we weight them

)()()( 21 xBfxAfxf


49/59

then our final optimum will vary. For example consider

min 21 )( xxf

min xxxf 4)( 22

then BxxBABxBxAxxf 4)(4)( 222

now we only really need one weight hereA

BC so we get

cxxcAxf 4)1()( 2

04)1(2)( cxcAxf when)1(2

4

c

cx

So for each value of C we get a different solution and two different values of 1fand 2f :

Methods for combining goal functions

It will be clear from considering Pareto fronts and simple weighted sums of goals that

deciding how to combine goals will define the final designs selected. Before looking at more

advanced schemes for finding the front we briefly explore how a design team might decide

on a combined objective, either to reduce the problem to a single goal or to allow selections

to be made from those designs that are found to lie on the Pareto front. All such methods

attempt to formalise the process of assigning importance to the goals under review.

C 1x )(1 xf xf2

0 0 0 0

1 1 1 -3

23

119

71953

21 32

94

922

Good

designsPARETO FRONT


50/59

a) simple voting schemeseach design team member ranks the goals, the goals are then

given points from 1(lowest rank) to n (highest rank in n goals) and then these are

summed across the team to result in a weighting scheme. Before application all goals

are divided by the designers ideal value (or the ideal is subtracted from the goal) to

allow for the units in use. This simply ensures that all voices are heard.

b) The eigenvector method. Each pair of goals is ranked on a matrix by being given a

preference ratio, ie if goal i is three times more important than goal j we set 3ijp .

Then say goal j is twice as important as goal k we set 2jkp etc. To be consistent

we should of course say that 6ikp but in fact the method does not require this. In

any case we then form all the p values into the matrix P and seek W so that

wPw max , ie the eigenvectors of p are found.

We then take the eigenvector with the largest eigen value and use this as our weighting

scheme.

If we have

12161

2131

631

p

We get w=

111.0

222.0

667.0

The largest eigen value should equal the number of goals if the ps are consistent as

here (we get a value of 3).

Say however we were not consistent and used

12151

2131

631

p

Then we get a largest eigen value of 3.004 and the weight vector becomes


51/59

122.0

230.0

648.0

w

That is we decrease the importance of goal 1 and increase that of goals 2 and 3.

This scheme is simple and easy to use up to around 10 goals at most and is quite useful

for more than 4 goals where it is difficult to assign numerical values to the weights in a

consistent fashion. It is still the case however that the aggregate goal is a simple linear

sum of the individual functions.

One way of combining goals that is more elaborate is via the use of fuzzy logic. Thus

we define a series of linguistic terms that describe our goal and map these to a score:-

Thus a bad value scores nothing and a good value scores 1 while those in the indifferent

range have intermediate scores. If we do this to all functions the resulting scores can then be

combined by adding (essentially an average) or multiplication (a geometric average). We

then maximize the combined function. Various shapes for the so called membership

functions can be used but there seems little to be gained from going above the linear form

sketched here.

Score

1

0

BAD INDIFFERENT GOOD


52/59

When used these functions essentially allow non-linear combinations of goal functions which

clearly allow more complex combined goalshowever if taken too far they can obscure the

overall problem!

Example of Fuzzy Logic


53/59

Methods for finding Pareto sets

Sometimes we do not wish to combine goals without first examining the Pareto front itself.

Thus we need to construct the front. We then have 3 goals.

1) the designs we study truly lie on the front, ie they are well optimised

2) the front has many designs that span its full extent, ie it is well populated

3) the points are evenly spread on the front, ie we have smooth range of goals.

This is in fact an optimization task in its own right and may be tabled in a variety of ways.

A Perhaps the simplest is to construct a family of different combined goals with various

weighting schemes and then optimize these (including dealing with each goal on its

own). Although this does not tackle point 3 above it focuses on 1 and gives as many

points as desired for 2. It is however expensive and known to fail to evenly populate

the font, especially if the font is concave (using a linear sum of goals is equivalent to

finding the intersection of a target line and the front and targets only exist for convex

fronts).

1

a b

xa xb

f1

1 0Score

x

Score

1

0

f1

f2

x

xa xc xf xp

score= overall goal

2

1

x

xa xc xb xd

AND SIMILAR FOR


54/59

B The next best scheme is to use an optimizer to explore the design space placing any

new non-dominated points in an archive (and weeding out any dominated ones). Then

all new design points are given a goal value based on how much they improve the

archive ie how dominant they are. This means that the objective function is non-

stationary but provided our search is tolerant of this, this approach works fine (an

evolution strategy works quite well on this).

C Use a multi objective population based search (such as a GA). Here we aim to advance

the whole front in one go so the points in the population are scored against each other

and those that dominate most score mostthis is capable of meeting all three goals if

some pressure to spread out points is included. Its weakest aspect is probably in

finding the extreme ends of the front but these can be found from single objective

searches directly on each goal.

It is also possible to use response surface schemes to help reduce run times when finding

Pareto fronts but this is not covered here.


55/59

LECTURE 10

ROBUSTNESS IN OPTIMIZATION

When doing any kind of design work one must bear in mind that

1) the design as made will differ from nominal

2) the operating conditions may be unknown and differ from those used in design

3) the product may change/deteriorate in service

All these aspects are to some extent stochastic and thus not amenable to deterministic

approaches to design. One way of dealing with this uncertainty is to try and make designs

robust to such uncertainty, ie such that their performance changes little as the uncertain

quantities move over their likely ranges.

This kind of lack of robustness in design variables (type 1 above) in common when dealing

with constrained - designers typically stay well away from critical stress limits if they can for

example to avoid unexpected structural failures.

f(x)

vA

vB

x

xB

Design A is much more sensitive

than B even though f(xA) is

better than f(xB)

xA


56/59

More controlled ways of working all involve trying to simulate uncertainty in the design

calculations being used. A prerequisite is to know something of the real uncertainties

anticipated. This requires DATA.

If however we know something of the uncertainties inherent in the design, manufacture and

operation we can attempt to account for this.

The most obvious, simple and direct scheme is the Monte Carlo approach. We simply

generate a series of scenarios using suitably biased random numbers and run our design

calculations at each before working with mean or worst case designs unfortunately

generating such means or worst cases requires 100s of simulations this is usually way too

expensive.

The next approach is to replace full Monte Carlo sampling with limited size DoE variations

around each design point to gain some idea of local sensitivities. Typically one use between

10 and 30 variations at each design to try and characterize the issues.

A third approach is the so called noisy phenotype. In this case a normal design search is

carried out but at each iteration noise is added to the design variables. Then the resulting

perturbed design results are used to characterise the design. This makes all derived qualities

non-stationary during the search so a suitable method must be used that is tolerant of this.

Sometimes the nominal and perturbed designs are both evaluated and the worst used as the

characterising design. When used with a GA for example this tends to mean only robust

designs survive the evolutionary process.

A method increasingly popular in industry is to run a medium sized DoE and then build a

global response surface through the resulting data. This response surface is then used for

large scale Monte Carlo sampling to build models of robustness. The main weakness of this

scheme is that the more unusual design and events that lead to extremes of behaviour may not

be captured by this process. Therefore as design decisions focus in on promising areas

update points should be run and the RSM rebuilt and re-explored so that the surrogate model

is well set up where it needs to be.

The most complex approach to robustness is to build so called stochastic solvers. These arecodes like finite element codes which instead of reading in deterministic problem statements


57/59

for geometry and loadings can accept these specified in probabilistic form. They then

directly compute probability measures for the response quantities of interest. Such methods

are currently in their infancy but can be expected to become prevalent over the next 10-20

years.

In whatever form robustness is considered it invariably leads to a multi-objective design

problem because the designer will desire good performance for the nominal geometry AND

robustness to likely variations. This tends to lead to Pareto fronts with mean and standard

deviation as axes.

Dotted lines show variance

f(x)

xa xb

Range at xb

ROBUST

Non dominated

designs

PARETO FRONT

Locus of expected value and variance as x

increases

E[f(x)]

VAR[f(x)]

Range

at x1

Fragile


58/59

Construction of Pareto front from variance analysis.

Getting started

Assuming one has a reasonable toolkit of search methods, a parameterisation scheme and an

automated (or at least mechanistically repeatable) design analysis process it is then possible

to make some plans as to how to proceed. These will be dominated by the number of

designer chosen variables and the run time to evaluate a design. Other important aspects will

be the number of goals, the number and type of constraints and whether or not stochastic

measures of merit must be constructed using an essentially deterministic code.

The two flow charts in the PPT deck then give some advice on how to begin

Describe the various types of optimizers available to tackle non-linear

search problems and the range of typical problem types encountered in

design. Pay particular attention to speed accuracy, robustness and usability


59/59

Describe the ways in which optimization tools may be used to tackle

design problems with multiple goals, paying particular attention to how

goals may be combined or dealt with simultaneously

Date post:	14-Apr-2018
Category:	Documents
Upload:	adam-mahmood
View:	216 times
Download:	0 times

Design Search and Optimization_mynotes

Documents