In this paper we show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce "uncertainty structures" for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second-order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs. Moreover, we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures used to estimate the market parameters. We demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence.

Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or ‘instance ’ parameter. A recently emerged successful paradigm for attacking these problems is robust optimization, where one seeks a solution which simultaneously satisfies all possible constraint instances. In practice, however, the robust approach is effective only for problem families with rather simple dependence on the instance parameter (such as affine or polynomial), and leads in general to conservative answers, since the solution is usually computed by transforming the original semi-infinite problem into a standard one, by means of relaxation techniques. In this paper, we take an alternative ‘randomized ’ or ‘scenario ’ approach: by randomly sampling the uncertainty parameter, we substitute the original infinite constraint set with a finite set of N constraints. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

Inspired by a recently proposed constructive framework for the distributed source coding problem, 1 we propose a powerful constructive approach to the watermarking problem, emphasizing the dual roles of "source codes" and "channel codes." In our framework, we explore various source and channel codes to achieve watermarks that are robust to attackers in terms of maximizing the distortion between the corrupted coded-source signal and the original signal while holding the distortion between the coded-source signal and the original signal constant. We solve the resulting combinatorial optimization problem using an original technique based on robust optimization and convex programming. Keywords: Data Hiding, Digital Watermarking, Multimedia, Convex Optimization, Robustness 1. INTRODUCTION Digital watermarking (data hiding) is an emerging research area that has received a considerable amount of attention in recent years. The basic idea behind digital watermarking is to embed information...

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We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to single-stage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than singe-stage optimization. For network flow and design under demand uncertainty we give a characterization of the first-stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is N P-hard even for a network flow problem on a bipartite graph. We show, however, that if the second-stage network topology is totally ordered or an arborescence, then the separation problem is tractable. Unlike single-stage robust optimization under demand uncertainty, two-stage robust optimization allows one to control conservatism of the solutions by means of an allowed “budget for demand uncertainty.” Using a budget of uncertainty we provide an upper

This paper presents a new approach to finite-horizon guaranteed state prediction for discrete-time systems affected by bounded noise and unknown-but-bounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the state-space matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization. With n states, l uncertain parameters appearing linearly in the state-space matrices, with rank-one matrix coefficients, the worst-case complexity grows as O(l(n + l) 3:5 ). With unstructured uncertainty in all system matrices, the worst-case complexity reduces to O(n 3:5 ).

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We primarily focus on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : # p (Q, Q 0 ) # #}, where # p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q 0 . The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with high probability. We also show that the robust sampled problem can be solved e#ciently both in theory and in practice. 1

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call “robust-optimization equilibrium, ” to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results.

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In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows.