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.

Collaborative multi-objective optimization for distributed design of complex products

“…The chosen parameters correspond to those suggested by the author of the library. These are as follows, (i) the number of hidden layers is set to 1, (ii) the number of hidden nodes in the hidden layer is set to 13, (iii) the Sigmoid symmetric 1 The fast arti cial neural network (FANN) library is used and a description is found in http://fann.sourceforge.net/fann_en.pdf . to the lowest consistency error obtained by at least 5% of the system solutions in the rst iteration of the distributed approach, and the regression power p is set to 3.…”

“…Dandurand et al [1], work with the individual subproblems to compute the Pareto-optimal solutions. For this, consistency constraints are used which rely on the existence of copies of the common variables and linking variables from the subproblems.…”

“…such that x 0 ∈ X 0 and x 1 ∈ X 1 (1) The subproblem contains two decision variables, x 0 and x 1 , where X 0 , X 1 ⊆ R 2 . A solution to the subproblem is given by x 1 = …”

Multiobjective optimization for interwoven systems

“…In Dandurand and Wiecek () and Guarneri and Wiecek (), two multiobjective decomposition algorithms are proposed to compute the Pareto optimal set of a multiobjective problem that is decomposable to multiobjective subproblems, each with different objective functions defined on their own feasible sets and subject to a common constraint allowing for passing information among the subproblems. Because the algorithms compute the Pareto optimal solutions to the original problem by working only with the subproblems, the approach is referred to as decentralized or distributed.…”

“…The new solutions are in turn passed back to the first subproblem. Lagrangian relaxation is applied to the consistency constraints of each SOP so that each of them is solved applying the block coordinate descent (Dandurand and Wiecek, ) or subgradient optimization (Guarneri and Wiecek, ). The process continues iteratively until the solutions to every subproblem are within a tolerance level of or as close as possible to the desired targets.…”