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

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.