Risk-Averse Dynamic Programming for Markov Decision Processes

We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we also develop … Read more

Binary positive semidefinite matrices and associated integer polytopes

We consider the positive semidefinite (psd) matrices with binary entries, along with the corresponding integer polytopes. We begin by establishing some basic properties of these matrices and polytopes. Then, we show that several families of integer polytopes in the literature — the cut, boolean quadric, multicut and clique partitioning polytopes — are faces of binary … Read more

On generalized network design polyhedra

In recent years, there has been an increased literature on so-called Generalized Network Design Problems, such as the Generalized Minimum Spanning Tree Problem and the Generalized Traveling Salesman Problem. In such problems, the node set of a graph is partitioned into clusters, and the feasible solutions must contain one node from each cluster. Up to … Read more

A Sequential Quadratic Programming Algorithm for Nonconvex, Nonsmooth Constrained Optimization

We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on … Read more

Metal Artefact Reduction by Least-Squares Penalized-Likelihood Reconstruction with a Fast Polychromatic Projection Model

We consider penalized-likelihood reconstruction for X-ray computed tomography of objects that contain small metal structures. To reduce the beam hardening artefacts induced by these structures, we derive the reconstruction algorithm from a projection model that takes into account the photon emission spectrum and nonlinear variation of attenuation to photon energy. This algorithm requires excessively long … Read more

Hedge Algorithm and Subgradient Methods

We show that the Hedge Algorithm, a method that is widely used in Machine Learning, can be interpreted as a particular subgradient algorithm for minimizing a well-chosen convex function, namely as a Mirror Descent Scheme. Using this reformulation, we establish three modificitations and extensions of the Hedge Algorithm that are better or at least as … Read more

The minimum spanning tree problem with conflict constraints and its variations

We consider the minimum spanning tree problem with conflict constraints (MSTC). It is observed that computing an $\epsilon$-optimal solution to MSTC is NP-hard for any $\epsilon >0$. For a general conflict graph, computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded … Read more

Approximating the asymmetric profitable tour

We study the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. In \cite{Amico}, the authors … Read more

Alternating Direction Algorithms for $\ell_1hBcProblems in Compressive Sensing

In this paper, we propose and study the use of alternating direction algorithms for several $\ell_1$-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis-pursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from … Read more