This paper studies exact semidefinite programming relaxations (SDPRs) for separable quadratically constrained quadratic programs (QCQPs). We consider the construction of a larger separable QCQP from multiple QCQPs with exact SDPRs. We show that exactness is preserved when such QCQPs are combined through a separable horizontal connection, where the coupling is induced through the right-hand-side parameters … Read more
We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in {0,±1}. While semidefinite programming (SDP) techniques are well established for {0,1}- and {±1}-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation … Read more
We consider a quadratic minmax problem with coupled inner constraints and propose a method to compute a class of stationary points. To motivate the need to compute such stationary points, we first show that they are meaningful, in the sense that they can be locally optimal for our problem under suitable linear independence and second-order … Read more
In this paper, we propose a new acceleration strategy for gradient-based methods applied to strictly convex Quadratic Programming (QP) problems. The strategy consists in performing, at selected iterations, minimization steps along alternative descent directions or even within low-dimensional affine subspaces. In particular, considering the contribution of the linear and quadratic part of the objective function … Read more
We consider linear and semidefinite programming relaxations of nonconvex quadratic programs given by the reformulation-linearization technique (RLT relaxation), and the Shor relaxation combined with the RLT relaxation (SDP-RLT relaxation). By incorporating the first-order optimality conditions, a quadratic program can be formulated as an optimization problem with complementarity constraints. We investigate the effect of incorporating optimality … Read more
For verifying the safety of neural networks (NNs), Fazlyab et al. (2019) introduced a semidefinite programming (SDP) approach called DeepSDP. This formulation can be viewed as the dual of the SDP relaxation for a problem formulated as a quadratically constrained quadratic program (QCQP). While SDP relaxations of QCQPs generally provide approximate solutions with some gaps, … Read more
We study continuous quadratic submodular minimization with bounds and propose a polynomially sized semidefinite relaxation, which is provably tight for dimension n ≤ 3 and empirically tight for larger n. We apply the relaxation to two moment problems arising in distributionally robust optimization and the computation of covariance bounds. Accordingly, this research advances the ongoing … Read more
Block-structured quadratic programs (QPs) frequently arise in the context of the direct approach to solving optimal control problems. For successful application of direct optimal control algorithms to many real-world problems it is paramount that these QPs can be solved efficiently and reliably. Besides interior-point methods and active-set methods, ADMM-based quadratic programming approaches have gained popularity. … Read more
In this work we propose a generalization of the Spherical Support Vector Machine method, in which the separator is a sphere, applied to Interval-valued data. This type of data belongs to a more general class, known as Symbolic Data, for which features are described by sets, intervals or histograms instead of classic arrays. This paradigm … Read more