Solving polyhedral d.c. optimization problems via concave minimization

• Published in: Journal of global optimization Vol. 78; no. 1; pp. 37 - 47
• Main Authors: vom Dahl, Simeon, Löhne, Andreas
• Format: Journal Article
• Language: English
• Published: New York, NY Springer US 01.09.2020; Springer; Springer Nature B.V
• Subjects: Algorithms; Computer Science; D.c. programming; Global optimization; Linear vector optimization; Mathematics; Mathematics and Statistics; Multi-objective linear programming; Operations Research/Decision Theory; Optimization; Real Functions; Global optimization; D.c. programming; 52B55; 90C29; Linear vector optimization; 90C26; Multi-objective linear programming
• ISSN: 1573-2916; 0925-5001; 1573-2916
• DOI: 10.1007/s10898-020-00913-z

The problem of minimizing the difference of two convex functions is called polyhedral d.c. optimization problem if at least one of the two component functions is polyhedral. We characterize the existence of global optimal solutions of polyhedral d.c. optimization problems. This result is used to show that, whenever the existence of an optimal solution can be certified, polyhedral d.c. optimization problems can be solved by certain concave minimization algorithms. No further assumptions are necessary in case of the first component being polyhedral and just some mild assumptions to the first component are required for the case where the second component is polyhedral. In case of both component functions being polyhedral, we obtain a primal and dual existence test and a primal and dual solution procedure. Numerical examples are discussed.