Eckart Zitzler ETH Zürich Dimo Brockhoff ETH Zurich Gene Expression Data Analysis 1 Computer Engineering and Networks Laboratory Dimensionality Reduction in Multiobjective Optimization: The Minimum Objective Subset Problem Dimo Brockhoff and Eckart Zitzler presented at Operations Research 2006 in Karlsruhe September 6, 2006
Motivation Dimensionality Reduction in Multiobjective Optimization 2 Multiobjective Problem Generating method values 3400 3200 3000 2800 (Approximation of) efficient set 2600 2400 2200 1 2 3 4 5 6 7 8 9 10 11 objectives 12 13 14 15 16 17 18 19 20 3200 reduce number of objectives values 3100 3000 2900 2800 2700 2600 2500 1 5 8 11 objectives 15 assist decision maker
Example Dimensionality Reduction in Multiobjective Optimization 3 values values omit still the same relations
Key questions Dimensionality Reduction in Multiobjective Optimization 4 Objective reduction possible without changing the problem? How to compute a minimum objective set? Applicable to real problems?
Related Work Dimensionality Reduction in Multiobjective Optimization 5 Omitting redundant objectives: Agrell (1997), Gal and Leberling (1977) Not suitable for black-box optimization PCA based objective reduction: Deb and Saxena (2005) Cannot guarantee preservation of dominance structure Various conflict definitions: Deb (2001); Tan et al. (2005) conflict as a property of the problem itself Purshouse and Fleming (2003): objective pairs conflict if the objective pair 2 solutions incomparable wrt
Open Questions Dimensionality Reduction in Multiobjective Optimization 6 Conflicts between arbitrary objective sets Objective reduction with preservation of problem structure in a black-box scenario Real problems
Key Contributions Dimensionality Reduction in Multiobjective Optimization 7 Generalization of Objective Conflicts The Minimum Objective Subset Problem Exact and heuristic algorithms Objective reduction for selected problems
Key Contributions Dimensionality Reduction in Multiobjective Optimization 8 Generalization of Objective Conflicts The Minimum Objective Subset Problem Exact and heuristic algorithms Objective reduction for selected problems
Relation Graphs and Dominance Dimensionality Reduction in Multiobjective Optimization 9 For a multiobjective problem, the question is to find the minimal elements of a given (pre)order Here, we restrict to the weak dominance relation indifferent incomparable minimal (reflexive and transitive edges are omitted)
Intersection of Linear (Pre)Orders Dimensionality Reduction in Multiobjective Optimization 10 Single objectives induce linear (pre)orders Their intersection yields Thus, the omission of objectives can only make incomparable solution pairs comparable and comparable solutions indifferent add edges in relation graph
Objective conflicts Dimensionality Reduction in Multiobjective Optimization 11 Objective sets conflict if they induce different relations Definition: nonconflicting with iff Omit objectives in if is nonconflicting with and preserve the dominance structure
Key Contributions Dimensionality Reduction in Multiobjective Optimization 12 Generalization of Objective Conflicts The Minimum Objective Subset Problem Exact and heuristic algorithms Objective reduction for selected problems
The Minimum Objective Subset Problem Dimensionality Reduction in Multiobjective Optimization 13 Minimum objective set and is called minimum if Minimum Objective Subset Problem (MOSS) Given: Set of solutions with weak dominance relations and Task: Compute a minimum objective set with MOSS is NP-hard reduction from set cover problem (SCP)
Algorithms for the MOSS Problem Dimensionality Reduction in Multiobjective Optimization 14 Exact algorithm Correctness proof Runtime: Worst case: Simple greedy heuristic Correctness proof Runtime: Best possible approximation ratio of
Key Contributions Dimensionality Reduction in Multiobjective Optimization 15 Generalization of Objective Conflicts The Minimum Objective Subset Problem Exact and heuristic algorithms Objective reduction for selected problems
Objective Reduction for Selected Problems Dimensionality Reduction in Multiobjective Optimization 16 Solutions with randomly chosen objective values (i.e., random orders as ): Objective reduction possible? Size of minimum set influenced by solution set size and number of objective? Greedy vs. exact algorithm Realistic scenarios for test problems
Varying A and k for Random Orders Dimensionality Reduction in Multiobjective Optimization 17 Various solution set sizes with random orders as number of objectives needed 14 12 10 8 6 4 A = 200 A =150 A =100 A = 50 A =25 2 2 4 6 8 10 12 14 16 number of objectives The more objectives, the smaller the minimum sets The more solutions in, the fewer objectives omissable
Greedy vs. Exact Algorithm for Random Orders Dimensionality Reduction in Multiobjective Optimization 18 Heuristic vs. exact algorithm on random orders with number of objectives needed 8 7 6 5 4 3 greedy exact 2 0 5 10 15 20 25 30 35 40 45 50 number of objectives The greedy algorithm's objective sets are not too large Greedy algorithm has clearly lower running time: can handle 50 objectives instead of 20 compared to exact algorithm within the same time
Realistic Scenarios for Test Problems Dimensionality Reduction in Multiobjective Optimization 19 Approximation of efficient set computed by evolutionary algorithm used as for and for Number of objectives in minimal set DTLZ2 DTLZ5 DTLZ7 KP100 KP250 KP500 problems k=15 k=25 Objective reduction of 50% possible for various test problems
Conclusions and Outlook Dimensionality Reduction in Multiobjective Optimization 20 Generalization of Objective Conflicts The MOSS Problem and algorithms Often: preservation of structure too strict Extension of approach to allow small changes in dominance structure: Brockhoff and Zitzler (2006) Method feasable for selected problems Also for real world problems? Method usable within generating methods?
Literature Dimensionality Reduction in Multiobjective Optimization 21 P. J. Agrell (1997): On redundancy in multi criteria decision making, European Journal of Operational Research Eur J Oper Res 98(3), p. 571-586 D. Brockhoff and E. Zitzler (2006): Are All Objectives Necessary? On Dimensionality Reduction in Evolutionary Multiobjective Optimization, Parallel Problem Solving from Nature IX - Proceedings K. Deb (2001) Multi-objective optimization using evolutionary algorithms, Wiley, UK K. Deb and D. K. Saxena (2005): On Finding Pareto-Optimal Solutions Through Dimensionality Reduction for Certain Large-Dimensional Multi-Objective Optimization Problems, KanGAL Report No. 2005011 T. Gal and H. Leberling (1977): Redundant objective functions in linear vector maximum problems and their determination, European Journal of Operational Research 1(3), p. 176-184 R. C. Purshouse and P. J. Fleming (2003): Conflict, Harmony, and Independence: Relationships in Evolutionary Multi-criterion Optimisation, EMO 2003 Proceedings, Springer Berlin, p. 16-30 K. C. Tan, E. F. Khor, and T. H. Lee (2005): Multiobjective Evolutionary Algorithms and Applications, Springer
Parallel Coordinates Plot for Example Dimensionality Reduction in Multiobjective Optimization 22