# ZEIT4700 – S1, 2014 Mathematical Modeling and Optimization School of Engineering and Information Technology.

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ZEIT4700 – S1, 2014 Mathematical 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  Section search (one variable)  Gradient based  Linear Programming  Quadratic programming  Simplex Drawbacks 1.Assumptions on continuity/ derivability 2.Limitation on variables 3.In general find Local optimum only 4.Constraint handling 5.Multiple objectives Newton’s Method (Image source : http://en.wikipedia.org/wiki/File:NewtonIteration_Ani.gif) Nelder Mead simplex method (Image source : http://upload.wikimedia.org/wikipedia/commons/9/96/Nelder_Mead2.gif)

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.

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) XX’ F(x) XX’ “Greedy” Local

Simulated Annealing 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 Calculate P = exp(-(F(X) – F(X’))/T) move to X’ with probability P } Maximize f(x) XX’ F(x) XX’ Attempts to escape local minima by accepting occasional ‘worse’ moves

Genetic / Evolutionary algorithms From 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)^2 0 ≤ x ≤ 31 Binary GAReal Parameter GA RepresentationBinaryReal Parent selectionBinary tournatment/ Roulett wheel Binary tournatment/ Roulett wheel CrossoverOne point/multi-pointSBX,PCX … MutationBinary flipPolynomial