solving minimax problems using an heuristic pattern search algorithm

IntroductionPattern search method - Hooke and Jeeves

Heuristic Pattern SearchNumerical Experiments and Conclusions

Solving Minimax Problems using an HeuristicPattern Search Algorithm

Isabel A.C.P. Espírito Santo and Edite M.G.P. Fernandes

University of Minho, Braga, PORTUGALiapinho;[email protected]

ICCAM 2008Ghent, Belgium

July 7 - 11, 2008

IACP Espírito Santo and EMGP Fernandes Minimax Problems using an Heuristic Pattern Search 1 / 33



Outline

1 Introduction

2 Pattern search method - Hooke and Jeeves

3 Heuristic Pattern Search

4 Numerical Experiments and Conclusions




Outline

1 Introduction







The Minimax Optimization Problem

The bound constrained problems to be addressed are:

minx∈Ω

f(x)

where

f(x) = maxj=1,...,m

Fj(x),

Fj : IRn → IR, j = 1, . . . ,m are continuously differentiablefunctionsΩ = x ∈ IRn : l ≤ x ≤ u is a closed set.




Motivation

Minimax problems appear in many engineering areas:

optimal control,engineering design,discrete optimization,Chebyshev approximation,game theory,computer-aided design,circuit design.

See references in Laskari, Parsopoulos & Vrahatis (2002) and Xu(2001).




Motivation

Inequality constrained optimization problem

minx o(x)subject to gj(x) ≥ 0, j = 1, . . . ,m

can be transformed into a minimax problem

minx

maxj=1,...,m

Fj(x)

where

F1(x) = o(x)Fj(x) = o(x)− αjgj(x), αj > 0, j = 2, . . . ,m.

For sufficiently large αj , the minimizer of the minimax problemcoincides with solution of inequality problem.




Optimality conditions

Stationary point (Xu, 2001)

x∗ is a stationary point to the minimax problem, if there existelements λ∗j ≥ 0, j = 1, . . . ,m such that∑m

j=1 λ∗j∇Fj(x∗) = 0 and∑m

j=1 λ∗j = 1,

andλ∗j = 0 if Fj(x∗) < maxF1(x∗), . . . , Fm(x∗).

Theorem (Xu, 2001)

If x∗ is a local minimum of minimax problem, then it is a stationarypoint. Conversely, if f(x) is convex and x∗ is a stationary point,then it is a global minimum to the minimax problem.




Difficulties

Minimax problems are difficult to solve through traditionalgradient based algorithms.

First derivatives of f(x) are discontinuousat points where f(x) = Fj(x)

for two or more values of j in the set 1, . . . ,m,even if all the functions Fj(x) have continuous first derivatives.




Tools of Heuristic Pattern Search Method

Our proposal:1 uses a derivative-free method, known as pattern search

method, as outline in Lewis & Torczon (1999);2 is based on the Hooke and Jeeves moves - exploratory move +

pattern move - Hooke & Jeeves (1961);3 and uses an heuristic move - a random descent walk - Hedar

& Fukushima (2004)

to obtain high accuracy solutions

⇒ the pattern move is followed by a random descent walk, when asuccessful iterate is encountered.




Outline

1 Introduction







Hooke and Jeeves pattern search method

Let xk ∈ IRn be the iterate at iteration k;Let ∆k be the step length;

The Hooke and Jeeves (HJ) method performs two types of moves:the exploratory move carries out a coordinate search - asearch along the coordinate axes - about a selected iterate,with a step size ∆k;when xk is a successful iterate, the pattern move - apromising direction - is defined by xk − xk−1.




Hooke and Jeeves moves

When iterate xk is successful ⇒ pattern move, followed by anexploratory move about xk + (xk − xk−1):

in IR2




Hooke and Jeeves moves

When pattern move is unsuccessful ⇒ exploratory move about xk:

in IR2




Outline

1 Introduction







Heuristic pattern move

Heuristic pattern search method performs two types of moves:

the exploratory moveis a coordinate search about a selected iterate, with a step size∆k;

a two-stage moveis a pattern move followed by an approximate descentrandom search - defined by xk − xk−1 + dk - when xk is asuccessful iterate.




Approximate descent random search

Let pk = xk + (xk − xk−1)based on two points y1 and y2, randomly generated from theneighborhood of pk;an approximate descent search for f at pk is

dk = − 1∑2j=1 |∆fj |

2∑i=1

(∆fi)pk − yi

‖pk − yi‖,

where ∆fj = f(pk)− f(yj).




Heuristic descent move

When iterate xk is successful ⇒ a pattern move followed by adescent random search dk at xk + (xk − xk−1):

in IR2




Heuristic descent move

When pattern move is unsuccessful ⇒ exploratory move about xk:

in IR2




Constraining for feasibility

To maintain iterate in Ω, a reflexion into the feasible region iscarried out - componentwise (for i = 1, . . . , n)

xk,i =

li + (li − xk,i) if xk,i < li

xk,i if li ≤ xk,i ≤ ui

ui − (xk,i − ui) if xk,i > ui

If a component of an iterate still is out of the bounds thenxk,i = (li + ui)/2.




Outline

1 Introduction







Details

Algorithms coded in C programming language with AMPL interfaceto read problems coded in AMPL:

22 benchmark bound constrained minimax problems - Lukšan &Vlček (2000); Petalas, Parsopoulos & Vrahatis (2007)

16 minimax problems;6 problems - mm10, mm12, mm2, mm20, mm3 and mm30 areinequality constrained problems that were transformed intominimax problems;




Details

Termination conditionsThe found solution has objective function value within 1% of theoptimal objective value (best known):if |f∗| ≤ 10−12 then

|f(xk)− f∗| ≤ 0.012 |1 + f(xk)|

else|f(xk)− f∗| ≤ 0.01 |f(xk)|

or

number of function evaluations ≤ 20000.




Details

Initialization of the step lengthThe step length ∆k is a vector - to handle variables of differentorder.

The step length initialization (k = 0) is defined componentwise by

∆0,i = γ∆x0,i, i = 1, . . . , n

(x0 published in literature)

Tested values:γ∆ = 0.01 γ∆ = 0.1 γ∆ = 1γ∆ = 10 γ∆ = 100 γ∆ = 1000




Table of results - HJ pattern search / Heuristic patternsearch

P - Problem; f∗ - best known solution in the literature;

HJ pattern search (deterministic method):Nit− number of iterations to achieve the desired accuracyNfe− number of objective function evaluationssolution - obtained solution according to termination conditions.

Heuristic pattern search (stochastic method) - each problemwas run 100 times:

AvNit− average number of iterations, over the 100 runsAvNfe− average number of function evaluations, over the 100 runssolution - best of the solutions found in the 100 runsaverage - average of the solutions found in the 100 runs.




Tests with the heuristic pattern search algorithm

With the heuristic pattern search we solve1 each problem 100 times, using the 6 values of γ∆ - total of

600 runs for each problem;2 for each γ∆, choose the best run (over the 100 runs) - with

solution closest to f∗;3 choose the γ∆ that gives solution closest to f∗;




Results #1 - Heuristic pattern search

P f∗ γ∆ solution AvNit AvNfe average

mm1 1.952225 0.01 1.957691 27 237 1.964877mm10 7.20 1000 7.201657 257 1454 7.260439mm11 2 1 2 3 22 2mm12 -1.414214 1 -1.41395 10 65 -1.40479mm13 0 1 0 1 102 0mm14 0 1 0 1 6 0mm15 3.59972 1000 3.600187 1079 7972 3.675277mm16 115.706 0.01 116.0436 1303 19206 141.0696mm17 0.002636 0.01 0.002749 1709 20005 0.053114mm2 -44 0.1 -43.8542 171 1744 -43.6039mm20 -40.10 0.1 -43.9738 129 1296 -40.3967




Results #2 - Heuristic pattern search

P f∗ γ∆ solution AvNit AvNfe average

mm21 0.147e-7 1000 0.467807 472 20021 0.495673mm22 3.88719 0.01 3.708345 63 1240 3.904433mm3 680.630 1000 684.6938 75 1983 686.9234mm30 247 1000 247.7281 874 14236 253.4493mm4 0 0.1 8.26e-05 2824 19472 0.01632mm5 0 0.01 0.003266 864 20012 0.011521mm51 0 0.1 3.71e-05 800 17128 46.17007mm6 0 1 3.66e-07 825 9607 0.000474mm7 0.002016 0.01 0.005054 2004 20005 0.06983mm8 0 1 0 1 6 0mm9 -3 100 -2.99623 20 130 -2.98209




Tests: HJ pattern search vs Heuristic pattern search

When comparing the HJ pattern search with the heuristic patternsearch we solve

1 each problem by pattern search algorithm, using the 6 valuesof γ∆;

2 choose the best run - solution closest to f∗;3 use this γ∆ to solve each problem by heuristic pattern search

algorithm, 100 times;4 choose the best run (over the 100 runs) - solution closest to

f∗.




Comparison - minimax problemsPattern search Heuristic pattern search

P γ∆ solution Nit Nfe solution AvNit AvNfe

mm1 10 1.95719 17 79 1.957188 144 786mm11 1 2 3 14 2 3 22mm13 1 0 1 102 0 1 102mm14 1 0 1 6 0 1 6mm15 1 3.62122 14 93 3.602588 551 4204mm16 0.1 116.73 250 1847 116.3001 1292 19064mm17 1000 0.084704 2236 20004 0.017755 1736 20004mm21 0.1 0.801571 585 20026 0.475883 467 20023mm22 1000 3.92475 84 1608 3.710802 37 738mm4 100 0.333333 4009 20001 0.000357 2762 20003mm5 100 1.60864 960 20012 0.004962 843 20013mm51 100 2.58774 960 20012 4.9e-05 546 12543mm6 1 0.12282 4995 20001 3.66e-07 825 9607mm7 1 0.086806 2462 20006 0.007452 1883 20005mm8 1 0 1 6 0 1 6mm9 1 -3 3 14 -2.99191 16 133




Comparison - Inequality constrained problems

Comparison with α = αj = 1 and α = αj = 10 (j = 2, . . . ,m)

Pattern search Heuristic pattern search

P α γ∆ solution Nit Nfe solution AvNit AvNfe

mm10 1 0.01 7.2576 129 520 7.24468 3263 1680610 100 7.34863 4013 20004 7.213753 222 1151

mm12 1 10 -1.41101 9 43 -1.40234 13 8510 100 -1.40479 17 84 -1.40479 17 108

mm2 1 1000 -43.5742 11 57 -53.3333 2205 2000510 1 -43.6279 14 119 -43.5998 2024 18230

mm20 1 1 -52.3333 2228 20001 -40.1463 1255 1140710 0.01 -39.7264 115 822 -43.9834 117 1171

mm3 1 0.1 687.308 3 39 682.8735 4 6710 1000 687.086 25 283 684.6938 75 1083

mm30 1 0.01 154.295 1490 20005 247.9039 842 1444410 10 267.999 1345 20003 248.8527 1044 16861




Conclusions and Future Work

We presented a derivative-free pattern search search methodthat incorporates an heuristic descent random walk, when asuccessful iterate is found:

1 it improves solution accuracy;

2 it solves difficult non-differentiable problems - bound minimaxproblems;

3 it is easy to implement.

Future development: extend heuristic pattern search to equality andinequality constrained problems, using an augmented Lagrangianfunction - penalty multiplier method.




References

A.-R. Hedar and M. Fukushima, Heuristic pattern search and its hybridizationwith simulated annealing for nonlinear global optimization, OptimizationMethods and Software, 19 (2004) 291–308.

R. Hooke and T. A. Jeeves, Direct search solution of numerical and statisticalproblems, Journal on Associated Computation, 8 (1961) 212–229.

E. C. Laskari, K. E. Parsopoulos and M. N. Vrahatis, Particle swarmoptimization for minimax problems, Proceedings of IEEE 2002 Congress onEvolutionary Computation, ISBN: 0-7803-7278-6, 1576–1581, 2002.

R. M. Lewis and V. Torczon, Pattern search algorithms for bound constrainedminimization, SIAM Journal on Optimization, 9 (1999) 1082–1099.

L. Lukšan and J. Vlček, Test problems for nonsmooth unconstrained and linearlyconstrained optimization, TR 798, ICS, Academy of Science of the CzechRepublic, January 2000.

V. Torczon, On the convergence of pattern search algorithms, SIAM Journal onOptimization, 7 (1997) 1–25.

S. Xu, Smoothing method for minimax problems, Computational Optimizationand Applications, 20 (2001) 267–279.




Thanks for your attention

Isabel A.C.P. Espírito [email protected]

Edite M.G.P. [email protected]

www.norg.uminho.pt/NSOS/


solving minimax problems using an heuristic pattern search algorithm

Documents