Abstract. In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set. The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems. 1.

Abstract. We propose a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. The attractive features of the proposed approach are: (a) It incorporates a wide variety of phenomena, including demands that are not identically distributed over time and capacity on the echelons and links; (b) it uses very little information on the demand distributions; (c) it leads to qualitatively similar optimal policies (basestock policies) as in dynamic programming; (d) it is numerically tractable for large scale supply chain problems even in networks, where dynamic programming methods face serious dimensionality problems; (e) in preliminary computational experiments, it often outperforms dynamic programming based solutions for a wide range of parameters. 1

. A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. However, in contrast to LMIs, parameterized LMI (PLMIs) feasibility problems involve infinitely many LMIs hence are very hard to solve. In this paper, we propose several effective relaxation techniques to replace PLMIs by a finite set of LMIs. The resulting relaxed feasibility problems thus become convex and hence can be solved by very efficient interior point methods. Applications of these techniques to different problems such as robustness analysis, or Linear Parameter-Varying (LPV) control are then thoroughly discussed and illustrated by examples. 1 Introduction Linear matrix inequalities (LMIs) have emerged as a very powerful tool in the analysis and synthesis for robust control problems (see e.g. [6, 8, 12] and references therein). From a computational point of view,...

Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-butbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control. Keywords optimization under uncertainty · robust optimization · convex programming · chance constraints · robust linear control

informs doi 10.1287/moor.1100.0444

We study semi-infinite systems of Linear Matrix Inequalities which are generically NP-hard. For these systems, we introduce computationally tractable approximations and derive quantitative guarantees of their quality. As applications, we discuss the problem of maximizing a Hermitian quadratic form over the complex unit cube and the problem of bounding the complex structured singular value. With the help of our complex Matrix Cube Theorem we demonstrate that the standard scaling upper bound on µ(M) is a tight upper bound on the largest level of structured perturbations of the matrix M for which all perturbed matrices share a common Lyapunov certificate for the (discrete time) stability. 1

Traditional models of decision-making under uncertainty assume perfect information, i.e., ac-curate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inputs might be infeasible or exhibit poor performance when implemented. The purpose of this tutorial is to present a mathematical framework that is well-suited to the limited information available in real-life problems and captures the decision-maker’s attitude towards uncertainty; the proposed approach builds upon recent developments in robust and data-driven optimization. In robust optimization, random variables are modeled as uncertain parameters be-longing to a convex uncertainty set and the decision-maker protects the system against the worst case within that set. Data-driven optimization uses observations of the random variables as direct inputs to the mathematical programming problems. The first part of the tutorial describes the robust optimization paradigm in detail in single-stage and multi-stage problems. In the second part, we address the issue of constructing uncertainty sets using historical realizations of the random variables and investigate the connection between convex sets, in particular polyhedra, and a specific class of risk measures.

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We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministically-bounded model for the channel uncertainty of each user, and our goal is to design a robust precoder that minimizes the total transmission power required to satisfy the users ’ QoS constraints for all channels within a specified uncertainty region around the transmitter’s estimate of each user’s channel. Since this problem is not known to be computationally tractable, we will derive three conservative design approaches that yield convex and computationally-efficient restrictions of the original design problem. The three approaches yield semidefinite program (SDP) formulations that offer different trade-offs between the degree of conservatism and the size of the SDP. We will also show how these conservative approaches can be used to derive efficiently-solvable quasi-convex restrictions of some related design problems, including the robust counterpart to the problem of maximizing the minimum signal-to-interference-plus-noise-ratio (SINR) subject to a given power constraint. Our simulation results indicate that in the presence of uncertain CSI the proposed approaches can satisfy the users ’ QoS requirements for a significantly larger set of uncertainties than existing methods, and require less transmission power to do so.

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