We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2 = Y$, the matrix analog of binary variables that satisfy $z^2 = z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization … Read more
We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the … Read more
The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. ArticleDownload View PDF
A real symmetric matrix $A$ is copositive if $x^\top Ax\geq 0$ for all $x\geq 0$. As $A$ is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015) … Read more
In this thesis, we settle the computational complexity of some fundamental questions in polynomial optimization. These include the questions of (i) finding a local minimum, (ii) testing local minimality of a candidate point, and (iii) deciding attainment of the optimal value. Our results characterize the complexity of these three questions for all degrees of the … Read more
We utilize association schemes to analyze the quality of semidefinite programming (SDP) based convex relaxations of integral packing and covering polyhedra determined by matchings in hypergraphs. As a by-product of our approach, we obtain bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman. We determine … Read more
We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if … Read more
Second-order necessary optimality conditions for nonlinear conic programming problems that depend on a single Lagrange multiplier are usually built under nondegeneracy and strict complementarity. In this paper we establish a condition of such type for two classes of nonlinear conic problems, namely semidefinite and second-order cone programming, assuming Robinson’s constraint qualification and a generalized form … Read more
In semidefinite programming the dual may fail to attain its optimal value and there could be a duality gap, i.e., the primal and dual optimal values may differ. In a striking paper, Ramana proposed a polynomial size extended dual that does not have these deficiencies and yields a number of fundamental results in complexity theory. … Read more
Distributionally robust optimization (DRO) is a modeling framework in decision making under uncertainty where the probability distribution of a random parameter is unknown while its partial information (e.g., statistical properties) is available. In this framework, the unknown probability distribution is assumed to lie in an ambiguity set consisting of all distributions that are compatible with … Read more