We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: “given an x ∈ V , does x belong to X?” can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, ...

The quadratic assignment problem (QAP) is arguably one of the hardest NP-hard discrete optimization problems. Problems of dimension greater than 25 are still considered to be large scale. Current successful solution techniques use branch-and-bound methods, which rely on obtaining strong and inexpensive bounds. In this paper, we introduce a new semidefinite programming (SDP) relaxation for gener...

General QOPs (quadratic optimization problems) have a linear objective function cTx to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. Difficulties in solving a QOP arise from the nonconvexity in its quadratic terms. We propose a branch and bound algorithm for QOPs where branching operations have been designed to effectively reduce...

The dual form of convex quadratic semidefinite programming (CQSDP) problem, with nonnegative constraints on the matrix variable, is a 4-block convex optimization problem. It is known that, the directly extended 4-block alternating direction method of multipliers (ADMM) is very efficient to solve this dual, but its convergence are not guaranteed. In this paper, we reformulate it as a 3-block con...

We propose a randomized second-order method for optimization known as the Newton sketch: it is based on performing an approximate Newton step using a randomly projected Hessian. For self-concordant functions, we prove that the algorithm has superlinear convergence with exponentially high probability, with convergence and complexity guarantees that are independent of condition numbers and relate...

We propose a randomized second-order method for optimization known as the Newton Sketch: it is based on performing an approximate Newton step using a randomly projected or sub-sampled Hessian. For self-concordant functions, we prove that the algorithm has super-linear convergence with exponentially high probability, with convergence and complexity guarantees that are independent of condition nu...

Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binarity constraint on the variables, a minimizer of the separable...

A convex semidefinite programming problem is a convex constrained optimization problem, where the constraints are linear matrix inequalities, for which the standard Lagrangian condition is sufficient for optimality. However, this condition requires constraint qualifications to completely characterize optimality. We present a necessary and sufficient condition for optimality without a constraint...

We present a coordinate ascent method for a class of semidefinite programming problems that arise in non-convex quadratic integer optimization. These semidefinite programs are characterized by a small total number of active constraints and by low-rank constraint matrices. We exploit this special structure by solving the dual problem, using a barrier method in combination with a coordinate-wise ...

This manuscript develops a new framework to analyze and design iterative optimization algorithms built on the notion of Integral Quadratic Constraints (IQC) from robust control theory. IQCs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. We discuss how to adapt IQC theory to study optimizatio...