Activity Identification and Local Linear Convergence of Douglas-Rachford/ADMM under Partial Smoothness

Proximal splitting algorithms are becoming popular to solve convex optimization problems in variational image processing. Within this class, Douglas-Rachford (DR) and ADMM are designed to minimize the sum of two proper lower semicontinuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local convergence behaviour of … Read more

A Filter SQP Method: Local Convergence and Numerical Results

The work by Gould, Loh, and Robinson [“A filter method with unified step computation for nonlinear optimization”, SIAM J. Optim., 24 (2014), pp. 175–209] established global convergence of a new filter line search method for finding local first-order solutions to nonlinear and nonconvex constrained optimization problems. A key contribution of that work was that the … Read more

A Cycle-Based Formulation and Valid Inequalities for DC Power Transmission Problems with Switching

It is well-known that optimizing network topology by switching on and off transmission lines improves the efficiency of power delivery in electrical networks. In fact, the USA Energy Policy Act of 2005 (Section 1223) states that the U.S. should “encourage, as appropriate, the deployment of advanced transmission technologies” including “optimized transmission line configurations”. As such, … Read more

Set approach for set optimization with variable ordering structures

This paper aims at combining variable ordering structures with set relations in set optimization, which have been defined using the constant ordering cone before. Since the purpose is to connect these two important approaches in set optimization, we do not restrict our considerations to one certain relation. Conversely, we provide the reader with many new … Read more

Constrained trace-optimization of polynomials in freely noncommuting variables

The study of matrix inequalities in a dimension-free setting is in the realm of free real algebraic geometry (RAG). In this paper we investigate constrained trace and eigenvalue optimization of noncommutative polynomials. We present Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrate its convergence properties. Finite convergence of this relaxation … Read more

An Axiomatic Duality Framework for the Theta Body and Related Convex Corners

Lovász theta function and the related theta body of graphs have been in the center of the intersection of four research areas: combinatorial optimization, graph theory, information theory, and semidefinite optimization. In this paper, utilizing a modern convex optimization viewpoint, we provide a set of minimal conditions (axioms) under which certain key, desired properties are … Read more

Clustering-Based Preconditioning for Stochastic Programs

We present a clustering-based preconditioning strategy for KKT systems arising in stochastic programming within an interior-point framework. The key idea is to perform adaptive clustering of scenarios (inside-the-solver) based on their influence on the problem as opposed to cluster scenarios based on problem data alone, as is done in existing (outside-thesolver) approaches. We derive spectral … Read more

Some lower bounds on sparse outer approximations of polytopes

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [1] studied the idealized problem of how well a polytope is approximated by the use of sparse valid inequalities. As an extension to this work, we study the following “less idealized” questions in this pa- … Read more

On an open question about the complexity of a dynamic spectrum management problem

In this paper we discuss the complexity of a dynamic spectrum management problem within a multi-user communication system with K users and N available tones. In this problem a common utility function is optimized. In particular, so called min-rate, harmonic mean and geometric mean utility functions are considered. The complexity of the optimization problems with … Read more

Error estimates for the Euler discretization of an optimal control problem with first-order state constraints

We study the error introduced in the solution of an optimal control problem with first order state constraints, for which the trajectories are approximated with a classical Euler scheme. We obtain order one approximation results in the $L^\infty$ norm (as opposed to the order 2/3 obtained in the literature). We assume either a strong second … Read more