Adaptive Global Algorithm for Solving Box-Constrained Non-convex Quadratic Minimization Problems

• Published in: Journal of optimization theory and applications Vol. 192; no. 1; pp. 360 - 378
• Main Authors: Andjouh, Amar, Bibi, Mohand Ouamer
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2022; Springer Nature B.V
• Subjects: Adaptive algorithms; Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Constraints; Convexity; Eigenvalues; Engineering; Global optimization; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Quadratic equations; Theory of Computation; Abstract convexity; Optimality conditions; Global optimization; Non-convex quadratic minimization; Box constraints; Convex support; Adaptive global algorithm (AGA)
• DOI: 10.1007/s10957-021-01980-2
• ISSN: 0022-3239

In this paper, we propose a new adaptive method for solving the non-convex quadratic minimization problem subject to box constraints, where the associated matrix is indefinite, in particular with one negative eigenvalue. We investigate the derived sufficient global optimality conditions by exploiting the particular form of the Moreau envelope ( L -subdifferential) of the quadratic function and abstract convexity, also to develop a new algorithm for solving the original problem without transforming it, that we call adaptive global algorithm, which can effectively find one global minimizer of the problem. Furthermore, the research of the convex support of the objective function allows us to characterize the global optimum and reduce the complexity of the big size problems. We give some theoretical aspects of global optimization and present numerical examples with test problems for illustrating our approach.