Radial Epiderivative Based Line Search Methods in Nonconvex and Nonsmooth Box-Constrained Optimization

TL;DR

This work tackles box-constrained, nonsmooth, nonconvex optimization by introducing radial epiderivative-based line search methods that identify global descent directions without requiring differentiability. It develops two hybrid algorithms, RCC and RPSO, which merge cyclic coordinate descent and particle swarm optimization with descent-criterion testing via the radial epiderivative $f^{r}$, eliminating the need for extra univariate line searches. The authors prove convergence theorems for the general line-search framework and specifically for concave objective minimization, establishing global-minimization properties under appropriate conditions. Computational results on 29 benchmarks, including concave variants and SciPy comparisons, show RCC and RPSO achieving superior or near-optimal solutions with favorable robustness and efficiency compared to CC and PSO.

Abstract

In this paper, we develop a novel radial epiderivative-based line search methods for solving nonsmooth and nonconvex box-constrained optimization problems. The rationale for employing the concept of radial epiderivatives is that they provide necessary and sufficient conditions for both identifying global descent directions and achieving global minimum of nonconvex and nondifferentiable functions. These properties of radial epiderivatives are combined with line search methods to develop iterative solution algorithms. The proposed methods generate search directions at each iteration where global descent directions and stopping criteria are performed by using the abilities of the radial epiderivatives. We use two line search methods, that is cyclic coordinate and particle swarm optimization techniques to generate search directions, selecting only those that exhibit descent, as determined by using approximately computed radial epiderivatives at the current point. As a particular case, these methods are applied for minimizing concave functions. In the paper, two convergence theorems are proved. One of them deals with the general line search method and covers only the set of directions generated by the method. The second convergence theorem deals with minimizing concave functions which deals not only with the generated set of directions but covers the whole set of feasible solutions. The performance of the proposed method is evaluated by using well-known benchmark problems from the literature. The results demonstrate the advantages of the proposed approach in generating optimal or near-optimal solutions.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof
• Corollary 1
• Theorem 6