ZEIT4700 – S1, 2016 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.

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

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

Constraint handling - Penalty function method (Constrained) (Unconstrained) -Performance is sensitive to choice of parameters -No fixed way to generate penalty parameters -Scaling between different terms

Constraint handling – feasibility first techniques During the ranking, enforce the following relations: 1.Between two feasible solutions, the one with superior objective value is better. 2.Between a feasible and an infeasible solution, feasible is better 3.Between two infeasible solutions, the one with lower constraint violation is better. => All feasible solutions are ranked above infeasible solutions

Optimization – Multi-objective F1 (minimize) F2 (Minimize) The final set of non-dominated solutions should be: 1.Converged (close to the Pareto optimal front) 2.Diverse (should span entire range of solutions, preferably uniformly) For a problem to be multi-objective, the objectives must be conflicting, i.e, maximization of one of them must lead to minimization of other. The true optimum for this case is called Pareto Optimal Front (POF) A heuristic algorithm delivers a non-dominated set which preferably as close as possible to the POF.

Multiobjective Optimization – Scalarization approach f1 f2 -One solution per optimization search -Can only achieve convex fronts

f1 f2 -One solution per optimization search -Difficult to estimate c values

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

Evolutionary Algorithm (cntd) Minimize f(x) = (x-6)^2 0 ≤ x ≤ 31 Binary GAReal Parameter GA RepresentationBinaryReal Parent selectionBinary tournament/ Roulette wheel Binary tournament/ Roulette wheel CrossoverOne point/multi-pointSBX,PCX … MutationBinary flipPolynomial RankingSort / NDSort / ND / CD

Resources Course material and suggested reading can be accessed at

Rank 1 f1 f2 Rank 2 Rank 3