Distributed Computation of Pareto Sets

Abstract

The needs of multidisciplinary engineering design have motivated the development of distributed solution approaches to computing efficient solutions to decomposable multiobjective optimization problems (MOPs). The decomposition is necessary due to the assumption that the overall MOP is not solvable since access to its solution space is subproblem-restricted. The state-of-the-art analyses for distributed solution approaches such as the alternating direction method of multipliers (ADMM) and the block coordinate descent (BCD) method were developed in the context of problem decomposition originating in a single objective setting and are not immediately applicable to MOPs. Applying certain scalarization techniques well-suited for nonconvex MOPs, the decomposable MOP is reformulated into a single objective problem (SOP) but the decomposability is not preserved and the SOP is not suitable for the application of ADMM. Furthermore, coupling between the subproblems makes BCD in its current form likewise inadequate. To address these challenges to distributed multiobjective optimization, existing theory is extended for (1) iterative augmented Lagrangian coordination techniques and (2) the block coordinate descent method. For the former, each augmented Lagrangian subproblem is subject to convex constraints and is solved approximately for solutions satisfying a necessary condition for optimality (but not necessarily optimal). For the latter, existing convergence proofs under generalized convexity assumptions are redeveloped so that new insights and convergence results become evident. An integration of the above two convergence analyses yields convergence results for a distributed solution approach that addresses coupling between subproblems and does not require convergence of the generated sequence of solution-multiplier pairs. Based on this study, a multi-objective decomposition algorithm is developed for the distributed generation of efficient solutions to nonconvex decomposable MOPs.