On generalized surrogate duality in mixed-integer nonlinear programming

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global $$\epsilon $$ ϵ -optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.

An Algorithm of Schatten ρ-Norm Regularized Least Squares Problems for Video Restoration

Minimizing the nuclear norm is recently considered as the convex relaxation of the rank minimization problem and arises in many applications as Netflix challenge. A closest nonconvex relaxation - Schatten norm minimization has been proposed to replace the NP hard rank minimization. In this paper, an algorithm based on Majorization Minimization has be proposed to solve Schatten norm minimization. The numerical experiments show that Schatten norm with recovers low rank matrix from fewer measurements than nuclear norm minimization. The numerical results also indicate that our algorithm give a more accurate reconstruction