ZEIT4700 – S1, 2015Mathematical Modeling and Optimization

School of Engineering and Information Technology

Optimization - basics

Maximization or minimization of given objective function(s), possibly subject to constraints, in a given search space

Minimize f1(x), . . . , fk(x) (objectives) Subject to gj(x) < 0, i = 1, . . . ,m (inequality constraints) hj(x) = 0, j = 1, . . . , p (equality constraints)

Xmin1 ≤ x1 ≤ Xmax1 (variable / search space)Xmin2 ≤ x2 ≤ Xmax2

. .

Classical optimization techniques Bracketing (Exhaustive search) / Region elimination methods Simplex based search Gradient based search

Nelder Mead simplex method(Image source : http://upload.wikimedia.org/wikipedia/commons/9/96/Nelder_Mead2.gif)

Interval halving method(Image source : K.Deb, Optimization for engineering design)

Classical optimization techniques (cntd.)Gradient based (Cauchy’s steepest descent method)

Image source : K. Deb, Multi-objective optimization using Evolutionary Algorithms, John Wiley and Sons, 2002.

Drawbacks of classical optimization methods

• Gradient based methods : Assumptions on continuity / derivability

• Region elimination methods : For single variable problem only

• Search for local optimum only

• Constraint handling is not inherently included

• No provisions for handling multiple objectives

Optimization – Heuristics/meta-heuristics

A heuristic is a technique which seeks good (i.e., near optimal) solutions at a reasonable computational cost without being able to guarantee either feasibility or optimality, or even in many cases to state how close to optimality a particular feasible solution is. - Reeves, C.R.: Modern Heuristic Techniques for Combinatorial Problems. Orient Longman (1993)

Simple “Hill climb”Start from random X

(while termination criterion not met){Perturb X to get a new point X’If F(X’) > F(X), move to X’, else not}

Maximize f(x)

X X’

F(x)

X X’• “Greedy”• Local

Simulated AnnealingStart from random X

(while termination criterion not met){Perturb X to get a new point X’ If F(X’) > F(X), move to X’, else Calculate P = exp(-(F(X) – F(X’))/T) move to X’ with probability P}

Maximize f(x)

X X’

F(x)

X X’

Attempts to escape local minimaby accepting occasional ‘worse’ moves

Genetic / Evolutionary algorithmsFrom point-to-point methods to population based methods..

• EAs are nature inspired optimization methods which search for the optimum solution(s) by evolving a population of solutions.

• Require no assumptions on differentiability / continuity of functions, hence can handle much more complex functions as compared to classical optimization techniques.

• Can deliver the whole Pareto Optimal Front in a single run as opposed to conventional methods.

• Its an Intelligent hit and trial !

Evolutionary Algorithms (EA)

Initialization (population of

solutions)Parent selection Recombination /

Crossover

Mutation

Ranking (parent+child pop)Reduction

Termination criterion met

? Yes

No

Output best solution obtained

“Evolve”childpop

Evaluate childpop

Gen 1 Gen 25

Gen 50 Gen 100

Evolutionary Algorithms (contd.)

Evolutionary Algorithm (cntd)

Minimize f(x) = (x-6)^20 ≤ x ≤ 31

Binary GA Real Parameter GA

Representation Binary RealParent selection Binary

tournatment/Roulett wheel

Binary tournatment/Roulett wheel

Crossover One point/multi-point

SBX,PCX …

Mutation Binary flip Polynomial

Resources

Course material and suggested reading can be accessed at http://seit.unsw.adfa.edu.au/research/sites/mdo/Hemant/design-2.html