ANALYSIS OF TWO ALGORITHMS FOR MULTIOBJECTIVE MINMAX 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 d, u Model of the System aleatory f(d, u) f pdf u u epistemic Bel/Pl bba m 3 m 1 m 2 u f 2

Evidence theory Belief and Plausibility that f(u) < ? q 1 m(q 1) = 0. 3 q 2 m(q 2) = 0. 1 q 3 m(q 3) = 0. 4 q 4 m(q 4) = 0. 2 Bel(f < ) = m(q 1) + m(q 2) = 0. 4 Pl(f < ) = m(q 1) + m(q 2) + m(q 3) = 0. 8 3

Methodology MACS 2 IDEA 4

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

MACS : Cross-check f 2 If agent in the population dominates or is dominated by the archive f 1 6

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

MACS : Validation f 2 Run global optimization over U until f 1 8

MACSminmax Initialization MO GLOBAL MINIMIZATION Performed by MACS 2 on d space, and uses u’s stored in U-archive (internal crosscheck). MO minimization MACS 2 Archive min solutions Archive max solutions ARCHIVE MAXIMA Store in U-archive solution of IDEA only if it is better than solution of MACS 2 (maximization might fail to find global optimum) Cross-Check FINAL CROSS-CHECK Local search to refine accuracy of U-archive SO maximizations IDEA SO GLOBAL MAXIMIZATION Performed by IDEA on u space, for each di solution of MO global minimization Cross-Check INITIALIZATION Initial population is randomly generated (LHS) and U-archive is initialized. Dominance 9

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

Comparison MACSν MACSminmax Local search vs. cross-check for every agent of the minimization Initialization Individualistic Actions MO minimization Cross-Check MACS 2 Cross-Check Min-Max Selection Social Actions Cross-Check Both implement similar mechanisms to increase probability of archiving global maxima Min-Max Selection Archive min solutions SO maximizations IDEA Archive max solutions Validation Cross-Check Subproblem selection Archive resize Global vs. local search, same purpose: make sure that each d is associate to a global maximum u Cross-Check Dominance 11

Performance metrics • Convergence • Spreading • Success rate 12

Test cases Settings MACS 2 • 200 n function evaluations • 10 agents • 5 (1/2) social agents • F=1 • CR = 0. 1 IDEA • 200 n function evaluations • 5 agents • F=1 • CR = 0. 1 13

Test case 1 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 100% 0. 2 1. 7 100 / 0. 5 79 / 2 MACSminmax 100% 0. 2 1. 3 100 / 0. 5 100 / 2 14

Test case 2 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 100% 65% 0. 5 16. 1 100 / 1 0/2 MACSminmax 100% 60% 0. 6 2. 0 100 / 1 64 / 2 15

Test case 3 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 100% 0. 6 7. 5 46 / 0. 5 3/2 MACSminmax 100% 0. 1 0. 3 100 / 0. 5 100 / 2 16

Test case 4 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 100% 91. 3% 0. 3 0. 9 83 / 0. 5 97 / 2 MACSminmax 100% 85. 7% 0. 4 1. 0 77 / 0. 5 91 / 2 17

Test case 5 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 98. 6% 54. 1% 1. 2 5. 8 48 / 1 60 / 6 MACSminmax 92. 8% 87. 6% 2. 7 8. 0 24 / 1 42 / 6 18

Test case 6 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 100% 0. 3 1. 2 95 / 0. 5 97 / 2 MACSminmax 100% 0. 3 2. 0 91 / 0. 5 63 / 2 19

Test case 7 Max f 1 Max f 2 Mconv Mspr pconv / tconv pspr / tspr MACS 100% 95. 3% 5. 0 9. 3 50 / 5 8/5 MACSminmax 100% 98. 3 4. 6 2. 1 66 / 5 100 / 5 20

Conclusions • Worst-case design – – • Two multi-objective algorithms – – – • Cross-checks to increase probability to find global maximum MACS : bi-level algorithm, modification of MACS 2 MACSminmax: restoration methodology, works with any MO/SO algorithm Test cases – – – • Evidence Theory to model epistemic uncertainty Maximization of Belief function: worst-case scenario design 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) 21

Evidence theory Belief and Plausibility that f(u) < ? 23

Computational approach 1. 2. 3. 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. 1 Pl 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 Bel 3 2 24

Results 25

Fronts 26