In this paper, we define a class of linear conic programming (which we call matrix cone programming or MCP) involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone. MCP has recently found many important applications, for example, in nuclear norm relaxations of affine rank minimization problems. In order to make the defined MCP tractable and meaningful, we ...

During the last two decades, major developments in convex optimization were focusing on conic programming, primarily, on linear, conic quadratic and semidefinite optimization. Conic programming allows to reveal rich structure which usually is possessed by a convex program and to exploit this structure in order to process the program efficiently. In the paper, we overview the major components of...

In this paper, minimax design of infinite-impulseresponse (IIR) filters with prescribed stability margin is formulated as a conic quadratic programming (CQP) problem. CQP is known as a class of well-structured convex programming problems for which efficient interior-point solvers are available. By considering factorized denominators, the proposed formulation incorporates a set of linear constra...

We consider coordinate descent methods on convex quadratic problems, in which exact line searches are performed at each iteration. (This algorithm is identical to Gauss-Seidel on the equivalent symmetric positive definite linear system.) We describe a class of convex quadratic problems for which the random-permutations version of cyclic coordinate descent (RPCD) outperforms the standard cyclic ...

We study the trust region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone based approach to derive an exact convex formulation of the TRS, and under slightly stro...

denote the kth elementary symmetric function, and let Γk denote the connected component of {λ ∈ R : σk(λ) > 0} containing the positive cone {λ ∈ R : λ1 > 0, · · · , λn > 0}. It is well known that Γk = {λ ∈ R : σl(λ) > 0, 1 ≤ l ≤ k}. Let S denote the set of n× n real symmetric matrices. For any A ∈ S we denote by λ(A) the eigenvalues of A. Throughout this note we will assume that Γ ⊂ R is an ope...

We introduce a natural definition of Riesz measures and Wishart laws associated to an Ω-positive (virtual) quadratic map, where Ω ⊂ Rn is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone Ω transitively, then the associated Riesz measure and Wish...

In this contribution we present two interior-point path-following algorithms that solve the convex optimisation problem that arises in recentred barrier function model predictive control (MPC), which includes standard MPC as a limiting case. However the optimisation problem that arises in nonlinear MPC may not be convex. In this case we propose sequential convex programming (SCP) as an alternat...