Hybridizing two linear relaxation techniques in interval-based solvers

• Published in: Journal of global optimization Vol. 91; no. 3; pp. 437 - 456
• Main Authors: Araya, Ignacio, Messine, Frédéric, Ninin, Jordan, Trombettoni, Gilles
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2025; Springer Nature B.V; Springer Verlag
• Subjects: Approximation; Codes; Computer Science; Global optimization; Lower bounds; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Optimization algorithms; Optimization and Control; Polytopes; Propagation; Real Functions; Transcendental functions; Interval analysis; Branch-and-bound; Global optimization; Linear relaxation; Reliable computing
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-024-01449-2

In deterministic global optimization, techniques for linear relaxation of a non-convex program are used in the lower bound calculation phase. To achieve this phase, most deterministic global optimization codes use reformulation-linearization techniques. However, there exist also two interval-based polyhedral relaxation techniques which produce reliable bounds without adding new auxiliary variables, and which can take into account mathematical operations and most transcendental functions: (i) the affine relaxation technique, used in the IBBA code, based on affine forms and affine arithmetic, and (ii) the extremal Taylor technique, used in the Ibex-Opt code, which is based on a specific interval-based Taylor form. In this paper, we describe how these two interval-based linear relaxation techniques can be hybridized. These two approaches appear to be complementary, and such a hybrid method performs well on a representative sample of constrained global optimization instances.