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Analysis of two algorithms for multi-objective min-max optimization

Analysis of two algorithms for multi-objective min-max optimization. Simone Alicino Prof. Massimiliano Vasile Department of Mechanical and Aerospace Engineering University of Strathclyde , Glasgow, UK BIOMA 2014 13 th September 2014, Ljubljana, Slovenia. Design under uncertainty. bba.

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Analysis of two algorithms for multi-objective min-max optimization

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  1. Analysis of two algorithms for multi-objective min-max optimization Simone Alicino Prof. MassimilianoVasile Department of Mechanical and Aerospace Engineering University of Strathclyde, Glasgow, UK BIOMA 2014 13th September 2014, Ljubljana, Slovenia

  2. Design under uncertainty bba Model of the System m3 m1 d, u f(d,u) aleatory u m2 f pdf epistemic u u Bel/Pl f

  3. Evidence theory Belief and Plausibility that f(u) <  ? q1 q2 q3 q4 Bel(f < ) = m(q1) + m(q2) = 0.4 Pl(f < ) = m(q1) + m(q2) + m(q3) = 0.8 m(q2) = 0.1 m(q4) = 0.2 m(q3) = 0.4 m(q1) = 0.3

  4. Methodology • MACS2 • Population-based search • Pareto ranking + Tchebycheffscalarization • Exploitation: sampling of neighborhood • Exploration: differential evolution • Archive • IDEA • Population-based search • Hybrid DE + MBH • Improves local convergence • Avoids stagnation • CROSS-CHECKS • To rank and check and • Increase prob. of finding global maxima MACS2 IDEA

  5. Initialization • MACS Individualistic Actions Cross-Check • INITIALIZATION • Initial population is randomly generated (LHS) in the search domain D. Min-Max Selection • INDIVIDUALISTIC ACTIONS • Child generated by random moves (pattern search) of each agent. Social individuals • Social Actions SUBPROBLEM SELECTION update of the composition of the social population and their associated scalar subproblems. All other individuals Cross-Check Min-Max Selection • SOCIAL ACTIONS • Child generated by interaction (DE) of agents with neighbours or global archive. Validation Cross-Check GLOBAL ARCHIVE an external repository in which non-dominated solutions are stored. The archive is kept below a maximum size. Subproblem selection Archive resize

  6. MACS: Cross-check f2 If agent in the populationdominates or is dominated by the archive f1

  7. MACS: Min-max selection f f u unew D U d dnew otherwise

  8. MACS: Validation f2 Run global optimization over U until f1

  9. MACSminmax Initialization • INITIALIZATION • Initial population is randomly generated (LHS) and • U-archive is initialized. MO minimization MACS2 Cross-Check • MO GLOBAL MINIMIZATION • Performed by MACS2 on d space, and uses u’s stored in U-archive (internal cross-check). ARCHIVE MAXIMA Store in U-archive solution of IDEA only if it is better than solution of MACS2 (maximization might fail to find global optimum) Archive min solutions SO maximizations IDEA • SO GLOBAL MAXIMIZATION • Performed by IDEA on u space, for each di solution of MO global minimization Archive max solutions Cross-Check FINAL CROSS-CHECK Local search to refine accuracy of U-archive Dominance

  10. MACSminmax: restoration Archived maximum Candidate minimum in d Selected minimum in d Solution

  11. Comparison Initialization Initialization Individualistic Actions MO minimization MACS2 Cross-Check Local search vs. cross-check for every agent of the minimization Cross-Check Min-Max Selection MACSminmax MACSν • Social Actions Bothimplement similar mechanisms to increase probability of archiving global maxima Archive min solutions Cross-Check SO maximizations IDEA Min-Max Selection Archive max solutions Validation Cross-Check Cross-Check Dominance Subproblem selection Global vs. local search, same purpose: make sure that each d is associate to a global maximum u Archive resize

  12. Performance metrics • Convergence • Spreading • Success rate

  13. Settings • MACS2 • 200n function evaluations • 10 agents • 5 (1/2) social agents • F = 1 • CR = 0.1 • IDEA • 200n function evaluations • 5 agents • F = 1 • CR = 0.1 Test cases

  14. Test case 1

  15. Test case 2

  16. Test case 3

  17. Test case 4

  18. Test case 5

  19. Test case 6

  20. Test case 7

  21. Conclusions • Worst-case design • Evidence Theory to model epistemic uncertainty • Maximization of Belief function: worst-case scenario design • Two multi-objective algorithms • Cross-checks to increase probability to find global maximum • MACS: bi-level algorithm, modification of MACS2 • MACSminmax: restoration methodology, works with any MO/SO algorithm • Test cases • 6 bi- and 1 three- objective cases, with different dimensions and complexity • Global fronts identified, with good to excellent accuracy • Comparable performance between MACS and MACSminmax • Limitations • Limited number of cases, objectives, and dimensions • Test suite: neither fronts, nor global maxima analytically known (difficult to assess performance)

  22. Evidence theory Belief and Plausibility that f(u) <  ?

  23. Computational approach 1 Pl • Worst-case solution(Bel= 1) (best d that gives the minimum of the maxima of f over u) • Above this point the design is certainly feasible given the current information. • Best possible solution (Pl = 0) • Below this point the design is certainly not possible • Belief and Plausibility of every intermediate solution between best and worst • Trade-off curve 3 Bel 2

  24. Results

  25. Fronts

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