On comparison of different approaches to the stability radius calculation

1. On comparison of different approaches to the stability radius calculation Olga Karelkina Department of Mathematics University of Turku MCDM 2011

2. Outline • Preliminaries • Problem statement • Exact method for calculation stability radius proposed by Chakravarti and Wagelmans • NSGA-II adaptation for calculation stability radius • Illustration and comparison of two approaches

3. Two major directions of investigation can be single out • quantitative • bounds for feasible changes in initial data, which preserve some pre-assigned properties of optimal solutions • deriving algorithms for the bounds calculation • qualitative • conditions under which the set of optimal solutions of the problem possesses a certain pre-assigned property of invariance to external influence on initial data of the problem

4. Shortest path problem (SP) Given a directed graph and – a nonnegative cost associated with each edge Problem: find a directed path from a source node to a distinguished terminal node , with the minimum total cost. The feasible set is the set of all sequences , these sequences are directed paths from to in .

5. SP as LP Vector of ordered edges costs

6. Perturbation of the problem We define norms and in for any finite dimension The perturbation of the problem parameters is modeled by adding to the cost vector perturbing vector The set of the perturbing vectors is denoted by

7. Stability radius Let be the set of feasible solutions to the shortest path problem Let be the set of optimal solutions to the shortest path problem with cost vector . An optimal solution is called stable if Stability radius of an optimal solution

8. Stability radius V. A. Emelichev, D.P. Podkopaev, Quantitative stability analysis for vector problems of 0 – 1 programming, Discrete Optimization. 7 (2010) 48 – 63 (1) The largest such that for

9. Calculating the stability radii of an optimal solution to the linear problem of 0-1 programming (2) Theorem Let be an optimal solution to (2). The stability radius of is the maximum number satisfying the following inequality : (3)

10. is the maximal satisfying the inequality : From here taking into account we get

11. Let us denote D. Gusfield, Parametric combinatorial computing and a problem of program module distribution, J. Assoc. Comput. Mach. 30 (1983) 551 – 563 is a continuous, piecewise linear and concave function of Lemma The number of linear pieces of is

12. Chakravarti and Wagelmans polinomial algorithm Construction of on • Compute and • The optimal solutions associated with these values each defines a linear function on • If these functions are identical, then is simply this linear function • Otherwise, we have two linear functions which intersect at a unique value • If coincides with the intersection point, then is the concave lower envelope of the two linear functions • Otherwise, the optimal solution associated with defines a third linear function which intersects each of the other linear functions on

13. Chakravarti and Wagelmans polinomial algorithm

14. A fast and elitist multi-objective genetic algorithm: NSGA-II • Modules • A fast non-dominated sorting approach • Diversity presentation • Density estimation • Crowded comparison operator • The main loop

15. Begin Initialize Population gen=0 Evaluation Assign Fitness Cond? No Reproduction Yes gen=gen+1 Stop Crossover Mutation

16. Implementation of NSGA-II into calculation stability radius Pareto set

17. Representation • Graph is represented by costs matrix (vector) • Every variable (feasible solution) is coded in a fixed length binary string • Initialization • Breadth First Search • Evaluation • A fast non-dominated sorting approach find-nondominated-front(P) include first member in for each take one soltion at a time include in temporarily for each compare with other members of if , then if dominates a member of , delete it else if , then if is dominated by other members of , do not include in

18. Assign fitness • Density estimation • Crowding distance is an estimate of the size of the largest cuboid enclosing the point without including any other point in the population • Crowded comparison operator

19. Reproduction • The tournament selection scheme • The strings with minimum front number and minimum value of ratios • are selected to the mating pool. A directed graph on 10 nodes

20. Crossovers • One-Node crossover • 5 is a common node for both parents • One-Edge crossover • Edges (2,3) and (1,7) are used as links

21. One-Node-Two-Edges crossover • Nodes 4 and 8 do not belong to any of the parents, subpaths ((3,4),(4,6)) and ((6,8),(8,7)) are used as links

22. Mutation • The search of genetic algorithm is mainly guided by crossover operators, even though mutation • operator is also used to maintain diversity in the population. • Scheme of two mutation types

23. Pareto fronts 5 generations 10 generations 15 generations 20 generations

24. Simulation results We consider a family of randomly generated directed graphs on 100 vertices and with approximately 5000 edges. Weight range is [1, 50]. The population size is set to 100 (number of vertices), while the probabilities of the one-node, one-edge and one-node-two-edges crossovers are 0.2, 0.3 and 0.5 correspondingly, mutation probability increases with the number of generations. Tests show that in average NSGA-II converges in 80% cases and gives the exact solution after 5 – 20 generations. Calculation results were compared with solutions obtained by exact method proposed by N. Chakravarti and A. P. M. Wagelmans in Calculation of stability radii for combinatorial optimization problems, OR Letters. 23 (1998) 1 – 7. Complexity of the exact method is NSGA-II complexity is Here n is the number of vertices and k is the number of generations.

25. References V.A. Emelichev, V.N. Krichko, D.P. Podkopaev, On the radius of stability of a vector problem of linear Boolean programming, Discrete Math. Appl. 10 (2000) 103 – 108 N. Chakravarti, A. P.M. Wagelmans, Calculation of stability radii for combinatorial optimization problems, OR Letters. 23 (1998) 1 – 7 D. Gusfield, Parametric combinatorial computing and a problem of program module distribution, J. Assoc. Comput. Mach. 30 (1983) 551 – 563 V. A. Emelichev, D.P. Podkopaev, Quantitative stability analysis for vector problems of 0 – 1 programming, Discrete Optimization. 7 (2010) 48 – 63 K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multi-objective genetic algorithm: NSGA-II, Evolutionary Computation. 6 (2) (2002), 182 – 197

26. Thank You for Your interest