Related papers: Invariants of SDP exactness in quadratic programmi…
Relying on the co-area formula, an exact relaxation framework for minimizing objectives involving the total variation of a binary valued function (of bounded variation) is presented. The underlying problem class covers many important…
In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x…
Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is…
We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation…
The moment-SOS (sum of squares) hierarchy is a powerful approach for solving globally non-convex polynomial optimization problems (POPs) at the price of solving a family of convex semidefinite optimization problems (called moment-SOS…
We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…
Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP)…
We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix $X$ of size $n$. Following the Burer--Monteiro approach, we optimize a factor $Y$ of size $n \times p$…
We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1…
Disjointly constrained multilinear programming concerns the problem of maximizing a multilinear function on the product of finitely many disjoint polyhedra. While maximizing a linear function on a polytope (linear programming) is known to…
In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…
We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a…
Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science.…
We introduce a sublevel Moment-SOS hierarchy where each SDP relaxation can be viewed as an intermediate (or interpolation) between the d-th and (d+1)-th order SDP relaxations of the Moment-SOS hierarchy (dense or sparse version). With the…
It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are…
In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions.…
Globally optimizing a nonconvex quadratic over the intersection of $m$ balls in $\mathbb{R}^n$ is known to be polynomial-time solvable for fixed $m$. Moreover, when $m=1$, the standard semidefinite relaxation is exact. When $m=2$, it has…
A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only…
In this article we develop a max-strategy improvement algorithm for computing least fixpoints of operators on on the reals that are point-wise maxima of finitely many monotone and order-concave operators. Computing the uniquely determined…
In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the…