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Optimization Basics: Maximizing Efficiency in Search Spaces with Evolutionary Algorithms

This course introduces mathematical modeling and optimization, focusing on maximizing or minimizing objective functions while considering constraints. Topics include Evolutionary Algorithms, Constraint Handling, and Multi-objective Optimization methods. Learn how to implement penalty functions, feasibility ranking, and scalarization approaches to efficiently optimize solutions in various contexts.

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Optimization Basics: Maximizing Efficiency in Search Spaces with Evolutionary Algorithms

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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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