In this paper we consider the general cone programming problem, and propose primaldual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method [9, 11], Nesterov’s smooth approximation scheme [11], and Nemirovski’s prox-method [8], and propose a variant of Nesterov’s opti...

This paper is devoted to the study of a broad class of problems in conic programming modeled via parameter-dependent generalized equations. In this framework we develop a secondorder generalized differential approach of variational analysis to calculate appropriate derivatives and coderivatives of the corresponding solution maps. These developments allow us to resolve some important issues rela...

Given a 0-1 integer programming problem, several authors have introduced sequential relaxation techniques — based on linear and/or semidefinite programming — that generate the convex hull of integer points in at most n steps. In this paper, we introduce a sequential relaxation technique, which is based on p-order cone programming (1 ≤ p ≤ ∞). We prove that our technique generates the convex hul...

In this paper we consider the general cone programming problem, and propose primaldual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method [10, 11], Nesterov’s smooth approximation scheme [11], and Nemirovski’s prox-method [9], and propose a variant of Nesterov’s opt...

This paper addresses the globally optimal solution of the network-constrained unit commitment (NCUC) problem incorporating a nonlinear alternating current (AC) model of the transmission network. We formulate the NCUC as a mixed-integer quadratically constrained quadratic programming (MIQCQP) problem. A global optimization algorithm is developed based on a multi-tree approach that iterates betwe...

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We pro...