A partitioned optimization framework for structure-aware optimization

TL;DR

This paper introduces a novel partitioned optimization framework (POf) that simplifies complex problems by fixing variable combinations, and proposes a derivative-free method (DFPOm) to efficiently find optimal partitions, demonstrated on control and greybox problems.

Contribution

The paper formalizes the POf framework and develops the DFPOm algorithm, enabling efficient solutions for structured optimization problems with partitioned variables.

Findings

Successfully applied to an infinite-dimensional control problem.

Demonstrated improved performance on finite-dimensional greybox problems.

Proved the theoretical validity of the reformulation and the derivative-free approach.

Abstract

This work tackles a class of optimization problems in which fixing some well-chosen combinations of the variables makes the problem substantially easier to solve. We consider that the variables space may be partitioned into subsets that fix these combinations to given values, so that the restriction of the problem to any of the partition sets admits a tractable solution. Then, we exhibit a reformulation of the problem that consists in searching for the partition set index that minimizes the objective value of the solution to the restricted problem. We name partitioned optimization framework (POf) the formalization of this class of problems and this reformulation process. As we prove in this work, the POf allows solving the original problem by focusing on the reformulated problem: all solutions to the reformulated problem are partition indices for which the solution to the associated restricted problem is also a solution to the original problem. Second, we introduce a derivative-free partitioned optimization method (DFPOm) to efficiently solve problems that fit in the POf. We prove that the reformulated problem is nicely handled by derivative-free optimization (DFO) algorithms with a covering step. Then the DFPOm consists in solving the reformulated problem using such DFO algorithm with a covering step to obtain an optimal partition index, and to return the solution to the associated restricted problem as a solution to the initial problem. Finally, we illustrate how the DFPOm allows solving some classes of problems. We first focus on an infinite-dimensional case, by solving analytically an optimal control problem that challenges standard methods from the literature. Then, we apply the DFPOm on a class of finite-dimensional problems called composite greybox problems, and we highlight the gain in numerical performance provided by the DFPOm by comparing it to two DFO solvers.