Solving Indefinite Quadratic Programs by Dynamical Systems: Preliminary Investigations
Massimo Pappalardo, Nguyen Nang Thieu, Nguyen Dong Yen
TL;DR
This work addresses solving indefinite quadratic programs under a convex constraint by casting them as affine variational inequalities and solving them with dynamical systems derived from DC decompositions. The authors formulate two continuous-time dynamics, analyze their well-posedness, and prove that trajectories remain in the constraint set and converge to KKT points in a parametric subproblem under specific conditions. A key contribution is the demonstration that projected dynamical systems can solve AVIs without requiring symmetry of the quadratic term, along with a detailed study of strong pseudomonotonicity for affine operators on convex sets and its implications for convergence. The results provide a foundational, theory-grounded path toward understanding dynamical-systems solvers for indefinite QPs with geometric constraints, setting the stage for further development and open questions.
Abstract
Preliminary results of our investigations on solving indefinite qua\-dra\-tic programs by dynamical systems are given. First, dynamical systems corresponding to two fundamental DC programming algorithms to deal with indefinite quadratic programs are considered. Second, the existence and the uniqueness of the global solution of the dynamical system are proved by using some theorems from nonsmooth analysis and the theory of ordinary differential equations. Third, the strong pseudomonotonicity of the restriction of an affine operator on a closed convex set is analyzed in a special case. Finally, for a parametric indefinite quadratic program related to that special case, convergence of the trajectories of the dynamical system to the Karush-Kuhn-Tucker points is established. The elementary direct proofs in the third and fourth topics would be useful for understanding the meaning and significance of several open problems proposed in this paper.