On the worst case performance of the steepest descent algorithm for quadratic functions

\begin{abstract} The existing choices for the step lengths used by the classical steepest descent algorithm for minimizing a convex quadratic function require in the worst case $ \Or(C\log(1/\eps)) $ iterations to achieve a precision $ \eps $, where $ C $ is the Hessian condition number. We show how to construct a sequence of step … Read more

Inertial primal-dual algorithms for structured convex optimization

The primal-dual algorithm recently proposed by Chambolle \& Pock (abbreviated as CPA) for structured convex optimization is very efficient and popular. It was shown by Chambolle \& Pock in \cite{CP11} and also by Shefi \& Teboulle in \cite{ST14} that CPA and variants are closely related to preconditioned versions of the popular alternating direction method of … Read more

On the ergodic convergence rates of a first-order primal-dual algorithm

We revisit the proofs of convergence for a first order primal-dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. Article Download View On the ergodic convergence rates of a first-order primal-dual algorithm

Optimality gap of constant-order policies decays exponentially in the lead time for lost sales models

Inventory models with lost sales and large lead times have traditionally been considered intractable due to the curse of dimensionality. Recently, Goldberg and co-authors laid the foundations for a new approach to solving these models, by proving that as the lead time grows large, a simple constant-order policy is asymptotically optimal. However, the bounds proven … Read more

A primal-simplex based Tardos’ algorithm

In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin’s primal-based modification and Mizuno’s use of the simplex method, we introduce a modification of Tardos’ algorithm considering only the primal problem and using simplex method … Read more

Information Relaxation Bounds for Infinite Horizon Markov Decision Processes

We consider the information relaxation approach for calculating performance bounds for stochastic dynamic programs (DPs), following Brown, Smith, and Sun (2010). This approach generates performance bounds by solving problems with relaxed nonanticipativity constraints and a penalty that punishes violations of these constraints. In this paper, we study infinite horizon DPs with discounted costs and consider … Read more

On a nonconvex MINLP formulation of the Euclidean Steiner tree problems in n-space

The Euclidean Steiner Tree Problem in dimension greater than two is notoriously difficult. The successful methods for exact solution are not based on mathematical-optimization methods — rather, they involve very sophisticated enumeration. There are two types of mathematical-optimization formulations in the literature, and it is an understatement to say that neither scales well enough to … Read more

Multistage Adaptive Robust Optimization for the Unit Commitment Problem

The growing uncertainty associated with the increasing penetration of wind and solar power generation has presented new challenges to the operation of large-scale electric power systems. Motivated by these challenges, we present a multistage adaptive robust optimization model for the most critical daily operational problem of power systems, namely the unit commitment (UC) problem, in … Read more

Dynamic Generation of Scenario Trees

We present new algorithms for the dynamic generation of scenario trees for multistage stochastic optimization. The different methods described are based on random vectors, which are drawn from conditional distributions given the past and on sample trajectories. The structure of the tree is not determined beforehand, but dynamically adapted to meet a distance criterion, which … Read more

Normally admissible stratifications and calculation of normal cones to a finite union of polyhedral sets

This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more