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Optimization Methods Nelson Fonseca

Optimization Methods Nelson Fonseca. Significant part of the content of this file was furnished by Deep Medhi and it contains material in chapter 5 of the book Routing, Flow, and Capacity Design in Communication & Computer Networks. Optimization Problem. Optimization Problem.

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Optimization Methods Nelson Fonseca

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  1. Optimization Methods Nelson Fonseca RFaCD, dm, figs/tables

  2. Significant part of the content of this file was furnished by Deep Medhi and it contains material in chapter 5 of the book Routing, Flow, and Capacity Design in Communication & Computer Networks RFaCD, dm, figs/tables

  3. Optimization Problem RFaCD, dm, figs/tables

  4. Optimization Problem • S is called a convex set if for all points x, y the whole segment joining x and y belongs to S, i.e, when RFaCD, dm, figs/tables

  5. Optimization Problem • A valued function f defined on a convex set S is called convex function if • If F and S are convex the problem is a convex optimization problem. In this type of problem, local minimum are also global minimum RFaCD, dm, figs/tables

  6. Linear Programming (LP) RFaCD, dm, figs/tables

  7. Linear Programming • convex polytope, if bounded polyhedron • Extreme point or vertex a feasible point that cannot be expressed as a convex linear combination of other feasible points • If an LP problem has a bounded global minimum and the solution polytope contains at least one vertex, then the optimal solution set also contains at least one vertex RFaCD, dm, figs/tables

  8. An Example of an LP Problem RFaCD, dm, figs/tables

  9. Duality in LP RFaCD, dm, figs/tables

  10. Duality in LP • Equivalent problem, some time easier to solve • One-to-one correspondence between the dual variables and constraints RFaCD, dm, figs/tables

  11. Duality in LP • If the primal problem has a bounded optimal solution if and only if the dual also has a bounded optimal solution. Moreover, • Complementarity slackness conditions RFaCD, dm, figs/tables

  12. Simplex Method • A systematic, iterative algorithm of visiting consecutive basic feasible solutions, decreasing the objective function (minimization problem) during each iteraction, eventually identifying the minimum value • Although complexity is exponential, very efficient method RFaCD, dm, figs/tables

  13. Interior Point Method • Instead of considering only vertices for the solution polytope by moving along its edge (as in the simplex method) IPM algorithms follows a path through the interior of a polytope • IPM algorithms only approach the optimal solution asymptotically RFaCD, dm, figs/tables

  14. Mixed-integer Programming (MIP) • Only integer variables – Integer Programming (IP) • Mix of real and integer variables – MIP • Examples of MIP formulations: • Routing restriction (single-path routing, non-zero flows) • Modular links • MIP approximation of concave function RFaCD, dm, figs/tables

  15. Modular Link Cost RFaCD, dm, figs/tables

  16. Modular Link Cost RFaCD, dm, figs/tables

  17. Modular Link Cost RFaCD, dm, figs/tables

  18. Modular Link Cost RFaCD, dm, figs/tables

  19. Convex Function Cost • Delay curve M/M/1 RFaCD, dm, figs/tables

  20. Convex Function Cost RFaCD, dm, figs/tables

  21. Convex Function Cost RFaCD, dm, figs/tables

  22. F(z) 4 3 2 1 1 2 3 z Fig.4.5 Piece-wise linear approxima- tion of a convex function Piecewise Linear Approximation RFaCD, dm, figs/tables

  23. Piecewise Linear Approximation of Convex Function RFaCD, dm, figs/tables

  24. Concave Function RFaCD, dm, figs/tables

  25. F(z) 4 3 2 1 1 2 3 4 5 6 7 8 9 z Fig.4.7. Piece-wise linear approxima- tion of a concave function Piecewise Linear Approximation of Concave Function • Erlang B-loss Formula Based Concave RFaCD, dm, figs/tables

  26. Piecewise Linear Approximation of Concave Function RFaCD, dm, figs/tables

  27. Piecewise Linear Approximation of Concave Function RFaCD, dm, figs/tables

  28. The Branch-and-Bound (BB) • set of indices corresponding to unspecified values of binary variables, the binary requirement for these variable is relaxed so that for j , xj is a continuous variable from interval [0, 1] • set of indices corresponding to binary variables equal to 0 • set of indices corresponding to binary variables equal to 1 RFaCD, dm, figs/tables

  29. The Branch-and-Bound (BB) RFaCD, dm, figs/tables

  30. The Branch-and-Bound (BB) • Integral variables not necessarily binary, but can be non-negative integer. • Transform the integer-valued MIP into its binary counterpart. Replace variable xjby the expression 20uj0+ 21uj1+ 22uj2+ 2qujq where uj0 .. are binary variables and q is the smallest integer such that and K is the maximal (integer) value for xj RFaCD, dm, figs/tables

  31. The Branch-and-Bound (BB) RFaCD, dm, figs/tables

  32. The Branch-and-Cut (BC) Method • BB depends heavily on the quality of lower bounds obtained by solving the BB tree node problem • It may happen that it is advantageous to spend more time in a node and try to find a better lower bound that the one resulting from simple relaxation RFaCD, dm, figs/tables

  33. The Branch-and-Cut (BC) Method • The basic way to achieve better lower bounds is to construct valid inequalities in the BB tree nodes. Such inequalities are inserted into the relaxation problem • The idea is to exploit the integrality of variables in order to produce inequalities that are valid for all integral solutions and at the same time separate parts of the polyhedron containing non-integral optimal solutions RFaCD, dm, figs/tables

  34. Stochastic Heuristic Methods • Random element guiding the sequence of generated solution points • Suboptimal solutions to the problem • Local Search • Simmulated annealing • Evolutionary algorithm • Simulated Allocation • Tabu search RFaCD, dm, figs/tables

  35. Local Search • Any two points can be reached by tranversing its neighbours • can be trapped in local minimum RFaCD, dm, figs/tables

  36. Simulated Annealing • At each iteration, chooses a neighbour at random. • Moves to the neighbour in case the objective function does not increase and moves with an exponentially decreasing probability in case the objective function increases RFaCD, dm, figs/tables

  37. Simulated Annealing RFaCD, dm, figs/tables

  38. Simulated Annealing RFaCD, dm, figs/tables

  39. Evolutionary Algorithm • Maximize object function called fitness function • Iterative , stopping criteria chosen • Points seen as chromosomes (sequence of gens). • Parents chosen inversely proportional to their fitness function value • Crossover criteria dependent of problem RFaCD, dm, figs/tables

  40. Evolutionary Algorithm RFaCD, dm, figs/tables

  41. Evolutionary Algorithm RFaCD, dm, figs/tables

  42. Simmulated Allocation • Simpler, faster and some cases more efficient • Allocates a path with a certain probability and deacollate with its complement, paths are chosen randonly • Stopping criteria number of iterations of lower bound on the solution was found RFaCD, dm, figs/tables

  43. Simmulated Allocation RFaCD, dm, figs/tables

  44. Tabu Search • It is a neigghborhood search descendent method which avoids the “local minum trap” by accepting worse and constraining the current solution neighborhood by the solutions “search history” • The tabu contains changes of the solution attributes, called move, that cannot be applied in subsequent iterations • The tabu memory helps avoiding short-term cycles or subsequence of similar solutions RFaCD, dm, figs/tables

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