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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. 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

  4. Constraint handling x2 x2 x1 x1 Optimum • Search space is reduced • Disconnected/constricted feasible regions possible • Feasibility of solutions to be considered in ranking Feasible Infeasible

  5. Constraint handling - Penalty function method Minimize (x) Subject to (Constrained) Minimize (Unconstrained) ,… are penalty parameters • Performance is sensitive to choice of parameters • No fixed way to generate penalty parameters • Scaling between different terms

  6. Constraint handling – feasibility first techniques During the ranking, enforce the following relations: Between two feasible solutions, the one with superior objective value is bettter. Between a feasible and an infeasible solution, feasible is better Between two infeasible solutions, the one with lower objective value is better. => All feasible solutions are ranked above infeasible solutions

  7. Optimization – Multi-objective The final set of non-dominated solutions should be: • Converged (to the Pareto optimal front) • Diverse (should span entire range of solutions • Uniformly) f2 f1

  8. Multiobjective Optimization – Scalarization approach Where f2 f1 • One solution per optimization search • Can only achieve convex fronts

  9. Multiobjective Optimization – – constraint method (for different values of c ) f2 f1 • One solution per optimization search • Difficult to estimate c values

  10. Multiobjective Optimization – Non-dominated sorting Convergence (nd-sort) f2 f2 f2 Diversity (crowding- distance sort) f1 f1 d2 f1 d1

  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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