Randomized First-order Methods for Saddle Point Optimization

In this paper, we present novel randomized algorithms for solving saddle point problems whose dual feasible region is a direct product of many convex sets. Our algorithms can achieve ${\cal O}(1/N)$ rate of convergence by solving only one dual subproblem at each iteration. Our algorithms can also achieve ${\cal O}(1/N^2)$ rate of convergence if a … Read more

A Proximal Multiplier Method for Convex Separable Symmetric Cone Optimization

This work is devoted to the study of a proximal decomposition algorithm for solving convex symmetric cone optimization with separable structures. The algorithm considered is based on the decomposition method proposed by Chen and Teboulle (1994), and the proximal generalized distance defined by Auslender and Teboulle (2006). Under suitable assumptions, first a class of proximal … Read more

Adaptive Robust Optimization with Dynamic Uncertainty Sets for Multi-Period Economic Dispatch under Significant Wind

The exceptional benefits of wind power as an environmentally responsible renewable energy resource have led to an increasing penetration of wind energy in today’s power systems. This trend has started to reshape the paradigms of power system operations, as dealing with uncertainty caused by the highly intermittent and uncertain wind power becomes a significant issue. … Read more

The Descriptor Continuous-Time Algebraic Riccati Equation. Numerical Solutions and Some Direct Applications

We investigate here the numerical solution of a special type of descriptor continuous-time Riccati equation which is involved in solving several key problems in robust control, formulated under very general hypotheses. We also give necessary and sufficient existence conditions together with computable formulas for both stabilizing and antistabilizing solutions in terms of the associated matrix … Read more

Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs

In view of solving nonsmooth and nonconvex problems involving complex constraints (like standard NLP problems), we study general maximization-minimization procedures produced by families of strongly convex sub-problems. Using techniques from semi-algebraic geometry and variational analysis –in particular Lojasiewicz inequality– we establish the convergence of sequences generated by this type of schemes to critical points. The … Read more

Efficient First-Order Methods for Linear Programming and Semidefinite Programming

We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making virtually all subgradient methods be immediately applicable. We observe, moreover, that the objective function is naturally “smoothed,” thereby allowing most first-order … Read more

Relative Entropy Relaxations for Signomial Optimization

Signomial programs (SPs) are optimization problems specified in terms of signomials, which are weighted sums of exponentials composed with linear functionals of a decision variable. SPs are non convex optimization problems in general, and families of NP-hard problems can be reduced to SPs. In this paper we describe a hierarchy of convex relaxations to obtain … Read more

Integer programming formulations for the elementary shortest path problem

Given a directed graph G = (V, A) with arbitrary arc costs, the Elementary Shortest Path Problem (ESPP) consists of finding a minimum-cost path between two nodes s and t such that each node of G is visited at most once. If the costs induce negative cycles on G, the problem is NP-hard. In this … Read more

Approximation algorithms for the Transportation Problem with Market Choice and related models

Given facilities with capacities and clients with penalties and demands, the transportation problem with market choice consists in finding the minimum-cost way to partition the clients into unserved clients, paying the penalties, and into served clients, paying the transportation cost to serve them. We give polynomial-time reductions from this problem and variants to the (un)capacitated … Read more

A Gentle, Geometric Introduction to Copositive Optimization

This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We intend the paper for readers new to the area, and hence the exposition is largely self-contained. We focus on examples having just a few variables … Read more