Related papers: Semidefinite programming in matrix unknowns which …
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…
We propose a new homotopy-based conditional gradient method for solving convex optimization problems with a large number of simple conic constraints. Instances of this template naturally appear in semidefinite programming problems arising…
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…
In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on…
Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of…
In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming…
In this note, we use a natural desingularization of the conormal variety of the variety of n x n symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.
Matrix representations are a powerful tool for designing efficient algorithms for combinatorial optimization problems such as matching, and linear matroid intersection and parity. In this paper, we initiate the study of matrix…
When faced with a mathematical model, often the first step is to reduce the complexity of the model by turning variables and parameters into dimensionless quantities. This process is often performed by hand, relying on a skill practiced…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent…
Studying structural properties of linear dynamical systems through invariant subspaces is one of the key contributions of the geometric approach to system theory. In general, a model of the dynamics is required in order to compute the…
Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we…
Matrix configurations define noncommutative spaces endowed with extra structure including a generalized Laplace operator, and hence a metric structure. Made dynamical via matrix models, they describe rich physical systems including…
We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite…
We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we…
We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality…
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite…
We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular we study…
In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting…