A truncated epsilon-subdifferential method for global DC optimization

TL;DR

We address global optimization of a DC function with box constraints by formulating $f(\mathbf{x})=f_1(\mathbf{x})-f_2(\mathbf{x})$ as an unconstrained problem and introduce TESGO, a hybrid method combining local search with an escaping procedure based on truncated $\\varepsilon$-subdifferentials. TESGO leverages a global descent direction derived from $\\varepsilon$-subdifferentials to escape non-global minimizers and uses practical approximations $D_t f$ (and positive spanning sets) to enable implementable subproblems that generate better starting points for local search. The conceptual algorithm is proven to reach the global minimizer in finite steps under a finite critical-point assumption, and extensive experiments show TESGO often yields higher-quality solutions than four local DC solvers while outperforming two global solvers in terms of computational effort. The work demonstrates that integrating truncated $\\varepsilon$-subdifferentials with a structured escaping scheme provides a scalable, effective tool for global DC optimization across diverse problem classes.

Abstract

We consider the difference of convex (DC) optimization problem subject to box constraints. Utilizing epsilon-subdifferentials of DC components of the objective, we develop a new method for finding global solutions to this problem. The method combines a local search approach with a special procedure for escaping non-global solutions by identifying improved initial points for a local search. The method terminates when the solution cannot be improved further. The escaping procedure is designed using subsets of the epsilon-subdifferentials of DC components. We compute the deviation between these subsets and determine epsilon-subgradients, providing this deviation. Using these specific epsilon-subgradients, we formulate a subproblem with a convex objective function. The solution to this subproblem serves as a starting point for a local search. We study the convergence of the conceptual version of the proposed method and discuss its implementation. A large number of academic test problems demonstrate that the method requires reasonable computational effort to find higher-quality solutions than other local DC optimization methods. Additionally, we apply the new method to find global solutions to DC optimization problems and compare its performance with two benchmark global optimization solvers.

Figures (9)

• Figure 1: The graph of the function $f$ (blue) with its overestimation $\hat{f}$ (dashed red).
• Figure 2: Performance profiles for the accuracy of solutions with $\tau=0.25$.
• Figure 3: Performance profiles for the accuracy of solutions with $\tau=0.55$.
• Figure 4: Performance profiles for problems from Group 1 using CPU time.
• Figure 5: Performance profiles for problems from Group 1 using number of function evaluations.

Theorems & Definitions (33)

• Theorem 1
• Corollary 1
• Corollary 2
• Definition 1
• Theorem 2
• Proof 1
• Theorem 3
• Proof 2
• Example 1
• Theorem 4