In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.

For small cell technology to significantly increase the capacity of tower-based cellular networks, mobile users will need to be actively pushed onto the more lightly loaded tiers (corresponding to, e.g., pico and femtocells), even if they offer a lower instantaneous SINR than the macrocell base station (BS). Optimizing a function of the long-term rates for each user requires (in general) a massive utility maximization problem over all the SINRs and BS loads. On the other hand, an actual implementation will likely resort to a simple biasing approach where a BS in tier j is treated as having its SINR multiplied by a factor A j ≥ 1, which makes it appear more attractive than the heavily-loaded macrocell. This paper bridges the gap between these approaches through several physical relaxations of the network-wide association problem, whose solution is NP hard. We provide a low-complexity distributed algorithm that converges to a near-optimal solution with a theoretical performance guarantee, and we observe that simple per-tier biasing loses surprisingly little, if the bias values A j are chosen carefully. Numerical results show a large (3.5x) throughput gain for cell-edge users and a 2x rate gain for median users relative to a maximizing received power association.

Singular Value Decomposition (and Principal Component Analysis) is one of the most widely used techniques for dimensionality reduction: successful and efficiently computable, it is nevertheless plagued by a well-known, well-documented sensitivity to outliers. Recent work has considered the setting where each point has a few arbitrarily corrupted components. Yet, in applications of SVD or PCA such as robust collaborative filtering or bioinformatics, malicious agents, defective genes, or simply corrupted or contaminated experiments may effectively yield entire points that are completely corrupted.We present an efficient convex optimization-based algorithm we call Outlier Pursuit, that under some mild assumptions on the uncorrupted points (satisfied, e.g., by the standard generative assumption in PCA problems) recovers the exact optimal low-dimensional subspace, and identifies the corrupted points. Such identification of corrupted points that do not conform to the low-dimensional approximation, is of paramount interest in bioinformatics and financial applications, and beyond. Our techniques involve matrix decomposition using nuclear norm minimization, however, our results, setup, and approach, necessarily differ considerably from the existing line of work in matrix completion and matrix decomposition, since we develop an approach to recover the correct column space of the uncorrupted matrix, rather than the exact matrix itself. In any problem where one seeks to recover a structure rather than the exact initial matrices, techniques developed thus far relying on certificates of optimality, will fail. We present an important extension of these methods, that allows the treatment of such problems.The authors are with the A preliminary version appeared in the proceedings of NIPS, 2010 [1].Our results: We consider a novel but natural convex optimization approach to the recovery problem above. The main result of this paper is to establish that, under certain natural conditions, the optimum of this convex program will yield the column space of L 0 and the identities of the outliers (i.e., the non-zero columns of C 0 ). Our conditions depend on the fraction of points that are outliers (which can otherwise be completely arbitrary), and incoherence of the row space of L 0 . The latter condition essentially requires that each direction in the column space of L 0 be represented in a sufficient number of non-outlier points; we discuss in more detail below. We note that our results do not require incoherence of the column space, as is done, e.g., in the papers [5], [6]. This is due to to our alternative convex formulation, and our analytical approach that focuses only on recovery of the column space, instead of "exact recovery" of the entire L 0 matrix. This also means our method's performance is rotation invariant -in particular, applying the same rotation to all given points (i.e., columns) will not change its performance. This is again not true for the method in [5], [6]. Finally, we extend our analysis to the noisy case when al...

Lasso, or ℓ 1 regularized least squares, has been explored extensively for its remarkable sparsity properties. It is shown in this paper that the solution to Lasso, in addition to its sparsity, has robustness properties: it is the solution to a robust optimization problem. This has two important consequences. First, robustness provides a connection of the regularizer to a physical property, namely, protection from noise. This allows a principled selection of the regularizer, and in particular, generalizations of Lasso that also yield convex optimization problems are obtained by considering different uncertainty sets.Secondly, robustness can itself be used as an avenue to exploring different properties of the solution. In particular, it is shown that robustness of the solution explains why the solution is sparse. The analysis as well as the specific results obtained differ from standard sparsity results, providing different geometric intuition. Furthermore, it is shown that the robust optimization formulation is related to kernel density estimation, and based on this approach, a proof that Lasso is consistent is given using robustness directly. Finally, a theorem saying that sparsity and algorithmic stability contradict each other, and hence Lasso is not stable, is presented.

This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when a (natural) recently proposed method, based on convex optimization, succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. On the one hand, corollaries obtained by specializing this one single result in different ways recovers (upto poly-log factors) all the existing works in matrix completion, and sparse and low-rank matrix recovery. On the other hand, our results also provide the first guarantees for (a) deterministic matrix completion, and (b) recovery when we observe a vanishing fraction of entries of a corrupted matrix.

Abstract-In multistage problems, decisions are implemented sequentially, and thus may depend on past realizations of the uncertainty. Examples of such problems abound in applications of stochastic control and operations research; yet, where robust optimization has made great progress in providing a tractable formulation for a broad class of single-stage optimization problems with uncertainty, multistage problems present significant tractability challenges. In this paper we consider an adaptability model designed with discrete second stage variables in mind. We propose a hierarchy of increasing adaptability that bridges the gap between the static robust formulation, and the fully adaptable formulation. We study the geometry, complexity, formulations, algorithms, examples and computational results for finite adaptability. In contrast to the model of affine adaptability proposed in [2], our proposed framework can accommodate discrete variables. In terms of performance for continuous linear optimization, the two frameworks are complementary, in the sense that we provide examples that the proposed framework provides stronger solutions and vice versa. We prove a positive tractability result in the regime where we expect finite adaptability to perform well, and illustrate this claim with an application to Air Traffic Control.

Principal Component Analysis plays a central role in statistics, engineering and science. Because of the prevalence of corrupted data in real-world applications, much research has focused on developing robust algorithms. Perhaps surprisingly, these algorithms are unequipped -indeed, unable -to deal with outliers in the high dimensional setting where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a High-dimensional Robust Principal Component Analysis (HR-PCA) algorithm that is as efficient as PCA, robust to contaminated points, and easily kernelizable. In particular, our algorithm achieves maximal robustness -it has a breakdown point of 50% (the best possible) while all existing algorithms have a breakdown point of zero. Moreover, our algorithm recovers the optimal solution exactly in the case where the number of corrupted points grows sub linearly in the dimension.

This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both (a) erasures: most entries are not observed, and (b) errors: values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when the natural convex relaxation of minimizing rank plus support succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. On the one hand, corollaries obtained by specializing this one single result in different ways recover (up to poly-log factors) all the existing works in matrix completion, and sparse and low-rank matrix recovery. On the other hand, our results also provide the first guarantees for (a) recovery when we observe a vanishing fraction of entries of a corrupted matrix, and (b) deterministic matrix completion.