In this paper a simple derivation of duality is presented for convex quadratic programs with a convex quadratic constraint. This problem arises in a number of applications including trust region subproblems of nonlinear programming, regularized solution of ill-posed least squares problems, and ridge regression problems in statistical analysis. In general, the dual problem is a concave maximizat...

A symmetric matrix S is copositive if yT S y≥0 for all y≥0, and the set of all copositive matrices, denoted C∗, is a closed, pointed, convex cone; see [25] for a recent survey. Researchers have realized how to model many NP-hard optimization problems as copositive programs, that is, programs over C∗ for which the objective and all other constraints are linear [7, 9, 13, 16, 32–34]. This makes c...

Symmetry is the essential element of lifted inference that has recently demonstrated the possibility to perform very efficient inference in highly-connected, but symmetric probabilistic models models. This raises the question, whether this holds for optimisation problems in general. Here we show that for a large class of optimisation methods this is actually the case. More precisely, we introdu...

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programm...

Problems involving estimation and inference under linear inequality constraints arise often in statistical modeling. In this paper we propose an algorithm to solve the quadratic programming problem of minimizing ψ(θ) = θ′Qθ−2c′θ for positive-definite Q, where θ is constrained to be in a closed polyhedral convex cone C = {θ : Aθ ≥ d}, and the m×n matrix A is not necessarily full row-rank. The th...

In this paper we study the problem of parametric minimization of convex piecewise quadratic functions. Our study provides a unifying framework for convex parametric quadratic and linear programs. Furthermore, it extends parametric programming algorithms to problems with piecewise quadratic cost functions, paving the way for new applications of parametric programming in dynamic programming and o...