Using dual relaxations in multiobjective mixed-integer convex quadratic programming

• Published in: Journal of global optimization Vol. 92; no. 1; pp. 159 - 186
• Main Authors: De Santis, Marianna, Eichfelder, Gabriele, Patria, Daniele, Warnow, Leo
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2025; Springer Nature B.V
• Subjects: Algorithms; Approximation; Branch and bound methods; Branch-and-bound algorithm; Computer Science; Convex quadratic optimization; Enumeration; Integer programming; Mathematics; Mathematics and Statistics; Mixed integer; Mixed-integer quadratic programming; Multiobjective optimization; Nodes; Operations Research/Decision Theory; Optimization; Quadratic programming; Real Functions; Variables; 90C57; 90C25; Multiobjective optimization; 90C11; 90C29; Convex quadratic optimization; Branch-and-bound algorithm; Mixed-integer quadratic programming
• ISSN: 1573-2916; 0925-5001; 1573-2916
• DOI: 10.1007/s10898-024-01440-x

Abstract We present a branch-and-bound method for multiobjective mixed-integer convex quadratic programs that computes a superset of efficient integer assignments and a coverage of the nondominated set. The method relies on outer approximations of the upper image set of continuous relaxations. These outer approximations are obtained addressing the dual formulations of specific subproblems where the values of certain integer variables are fixed. The devised pruning conditions and a tailored preprocessing phase allow a fast enumeration of the nodes. Despite we do not require any boundedness of the feasible set, we are able to prove that the method stops after having explored a finite number of nodes. Numerical experiments on a broad set of instances with two, three, and four objectives are presented.