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ZEIT4700 – S1, 2014

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.

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ZEIT4700 – S1, 2014

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

  2. 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 . .

  3. Classical optimization techniques Section search (one variable) Gradient based Linear Programming Quadratic programming Simplex Drawbacks Assumptions on continuity/ derivability Limitation on variables In general find Local optimum only Constraint handling 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)

  4. 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.

  5. 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)

  6. 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) F(x) X X’ • “Greedy” • Local X X’

  7. 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) F(x) Attempts to escape local minima by accepting occasional ‘worse’ moves X X’ X X’

  8. 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 !

  9. Evolutionary Algorithms (EA) Initialization (population of solutions) Recombination / Crossover Parent selection “Evolve” childpop No Mutation Termination criterion met ? Output best solution obtained Yes Evaluate childpop Ranking (parent+child pop) Reduction

  10. Evolutionary Algorithms (contd.) Gen 25 Gen 1 Gen 100 Gen 50

  11. Evolutionary Algorithm (cntd) Minimize f(x) = (x-6)^2 0 ≤ x ≤ 31

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

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