An Approximate Point Algorithm for a Time-Varying Ill-Posed Quadratic Programming Problem

This paper addresses the numerical instability in time-varying quadratic programming caused by ill-conditioned matrices, which often leads to error amplification and significant deviation from the true optimum. We propose a timevarying proximal point algorithm that reformulates the illposed problem into a sequence of well-conditioned subproblems, thereby improving the Hessian condition number while enforcing temporal consistency along the solution trajectory. In the static-parameter case, the method reduces to a proximal point algorithm with linear convergence, for which we establish a refined convergence rate. In the time-varying setting, we further prove convergence of the resulting solution sequence. Numerical experiments demonstrate that, relative to conventional approaches, the proposed framework significantly suppresses solution oscillations in high-condition-number regimes, yielding substantial improvements in numerical stability and decision smoothness.