This paper studies the asymptotic behavior of the central path (X(ν), S(ν), y(ν)) as ν ↓ 0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose “degenerate diagonal blocks” XT (ν) and ST (ν) of the central path are assumed to satisfy max{‖XT (ν)‖, ‖ST (ν)‖} = O(√ν). We establish the conver...

In this paper we consider the problem of minimizing a nonconvex quadratic function, subject to two quadratic inequality constraints. As an application, such quadratic program plays an important role in the trust region method for nonlinear optimization; such problem is known as the CDT subproblem in the literature. The Lagrangian dual of the CDT subproblem is a Semidefinite Program (SDP), hence...

Inference in probabilistic graphical models can be represented as a constrained optimization problem of a free-energy functional. Substantial research has been focused on the approximation of the constraint set, also known as the marginal polytope. This paper presents a novel inference algorithm that tightens and solves the optimization problem by intersecting the popular local polytope and the...

The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov’s accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the conv...

This paper considers the problem of minimizing a quadratic cost subject to purely quadratic equality constraints. This problem is tackled by first relating it to a standard semidefinite programming problem. The approach taken leads to a dynamical systems analysis of semidefinite programming and the formulation of a gradient descent flow which can be used to solve semidefinite programming proble...

In this paper, we consider partial Lagrangian relaxations of continuous quadratic formulations of the Quadratic Assignment Problem (QAP) where the assignment constraints are not relaxed. These relaxations are a theoretical limit for semidefinite relaxations of the QAP using any linearized quadratic equalities made from the assignment constraints. Using this framework, we survey and compare stan...

ABSTRACT We describe efficient interior-point methods for the design of filters with constraints on the magnitude spectrum, for example, piecewise-constant upper and lower bounds, and arbitrary phase. Several researchers have observed that problems of this type can be solved via convex optimization and spectral factorization. The associated optimization problems are usually solved via linear pr...

A discriminative method is proposed for learning monotonic transformations of the training data while jointly estimating a large-margin classifier. In many domains such as document classification, image histogram classification and gene microarray experiments, fixed monotonic transformations can be useful as a preprocessing step. However, most classifiers only explore these transformations thro...

In this paper we propose the Graduated NonConvexity and Graduated Concavity Procedure (GNCGCP) as a general optimization framework to approximately solve the combinatorial optimization problems on the set of partial permutation matrices. GNCGCP comprises two sub-procedures, graduated nonconvexity (GNC) which realizes a convex relaxation and graduated concavity (GC) which realizes a concave rela...