Beyond local optimality conditions: the case of maximizing a convex function

In this paper, we design an algorithm for maximizing a convex function over a convex feasible set. The algorithm consists of two phases: in phase 1 a feasible solution is obtained that is used as an initial starting point in phase 2. In the latter, a biconvex problem equivalent to the original problem is solved … Read more

Convex Maximization via Adjustable Robust Optimization

Maximizing a convex function over convex constraints is an NP-hard problem in general. We prove that such a problem can be reformulated as an adjustable robust optimization (ARO) problem where each adjustable variable corresponds to a unique constraint of the original problem. We use ARO techniques to obtain approximate solutions to the convex maximization problem. … Read more

Tractable approximation of hard uncertain optimization problems

Robust Optimization is a widespread approach to treat uncertainty in optimization problems. Finding a computationally tractable formulation of the robust counterpart of an uncertain optimization problem is a key step in applying this approach. Techniques for finding a computationally tractable robust counterpart are available for constraints concave in the uncertain parameters. In many problems, however, … Read more

Computing the channel capacity of a communication system affected by uncertain transition probabilities

We study the problem of computing the capacity of a discrete memoryless channel under uncertainty affecting the channel law matrix, and possibly with a constraint on the average cost of the input distribution. The problem has been formulated in the literature as a max-min problem. We use the robust optimization methodology to convert the max-min … Read more

Extending the Scope of Robust Quadratic Optimization

We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. In particular, we show how to reformulate the support functions of uncertainty sets represented in terms of matrix norms and cones. Our … Read more

A Tractable Approach for designing Piecewise Affine Policies in Two-stage Adjustable Robust Optimization

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. A significant challenge in designing a practical piecewise affine policy is constructing good pieces of … Read more

Robust optimization with ambiguous stochastic constraints under mean and dispersion information

In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the old result of Ben-Tal … Read more

Adjustable robust optimization with decision rules based on inexact revealed data

Adjustable robust optimization (ARO) is a technique to solve dynamic (multistage) optimization problems. In ARO, the decision in each stage is a function of the information accumulated from the previous periods on the values of the uncertain parameters. This information, however, is often inaccurate; there is much evidence in the information management literature that even … Read more

Solving large scale polynomial convex problems on \ell_1/nuclear norm balls by randomized first-order algorithms

One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformulation of the problem of interest and solving the resulting problem by a fast First Order saddle point method utilizing smoothness of the saddle point cost function. In this paper, we demonstrate that when … Read more

Deriving robust and globalized robust solutions of uncertain linear programs with general convex uncertainty sets

We propose a new way to derive tractable robust counterparts of a linear program by using the theory of Beck and Ben-Tal (2009) on the duality between the robust (“pessimistic”) primal problem and its “optimistic” dual. First, we obtain a new {\it convex} reformulation of the dual problem of a robust linear program, and then … Read more