Related papers: A note on the Lovasz-Schrijver Semidefinite Progra…
Convex relaxations have emerged as a promising approach for verifying desirable properties of neural networks like robustness to adversarial perturbations. Widely used Linear Programming (LP) relaxations only work well when networks are…
This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous…
We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more…
Random projection, a dimensionality reduction technique, has been found useful in recent years for reducing the size of optimization problems. In this paper, we explore the use of sparse sub-gaussian random projections to approximate…
Covering and elimination inequalities are central to combinatorial optimization, yet their role has largely been studied in problem-specific settings or via no-good cuts. This paper introduces a unified perspective that treats these…
Semidefinite programming (SDP) provides a powerful relaxation for the maximum cut problem. For a graph with rational weights, the decision problem of whether the SDP relaxation for the maximum cut problem is exact is known to be $NP$-hard;…
The unconstrained binary quadratic programming (UBQP) problem is a class of problems of significant importance in many practical applications, such as in combinatorial optimization, circuit design, and other fields. The positive…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and are valid for a set described by a separable bilinear constraint together with…
We point out that Chubanov's oracle-based algorithm for linear programming [5] can be applied almost as it is to linear semi-infinite programming (LSIP). In this note, we describe the details and prove the polynomial complexity of the…
For a large class of optimization problems, namely those that can be expressed as finite-valued constraint satisfaction problems (VCSPs), we establish a dichotomy on the number of levels of the Lasserre hierarchy of semi-definite programs…
Semidefinite programming (SDP) is widely acknowledged as one of the most effective methods for deriving the tightest lower bounds of the optimal power flow (OPF) problems. In this paper, an enhanced semidefinite relaxation model that…
We consider approximation algorithms for covering integer programs of the form min $\langle c, x \rangle $ over $x \in \mathbb{N}^n $ subject to $A x \geq b $ and $x \leq d$; where $A \in \mathbb{R}_{\geq 0}^{m \times n}$, $b \in…
This two-part paper is concerned with the problem of minimizing a linear objective function subject to a bilinear matrix inequality (BMI) constraint. In this part, we first consider a family of convex relaxations which transform BMI…
Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques which is based on Mathematical Programming with…
We study the lift-and-project procedures of Lov\'asz and Schrijver for 0-1 integer programming problems. We prove that the procedure using the positive semidefiniteness constraint is not better than the one without it, in the worst case.…
In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the…
Many probabilistic inference tasks involve summations over exponentially large sets. Recently, it has been shown that these problems can be reduced to solving a polynomial number of MAP inference queries for a model augmented with randomly…
Two-stage stochastic mixed-integer programming (SMIP) problems with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing the subset of the…
Lagrangian Relaxation (LR) is a powerful technique for solving large-scale Mixed Integer Linear Programming (MILP), particularly those with decomposable structures, such as vehicle routing or unit commitment problems. By relaxing the…