Objective-Domain Dual Decomposition: An Effective Approach to Optimizing Partially Differentiable Objective Functions

This paper addresses a class of optimization problems in which either part of the objective function is differentiable while the rest is nondifferentiable or the objective function is differentiable in only part of the domain. Accordingly, we propose a dual-decomposition-based approach that includes both objective decomposition and domain decomposition. In the former, the original objective function is decomposed into several relatively simple subobjectives to isolate the nondifferentiable part of the objective function, and the problem is consequently formulated as a multiobjective optimization problem (MOP). In the latter decomposition, we decompose the domain into two subdomains, that is, the differentiable and nondifferentiable domains, to isolate the nondifferentiable domain of the nondifferentiable subobjective. Subsequently, the problem can be optimized with different schemes in the different subdomains. We propose a population-based optimization algorithm, called the simulated water-stream algorithm (SWA), for solving this MOP. The SWA is inspired by the natural phenomenon of water streams moving toward a basin, which is analogous to the process of searching for the minimal solutions of an optimization problem. The proposed SWA combines the deterministic search and heuristic search in a single framework. Experiments show that the SWA yields promising results compared with its existing counterparts.