Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

A scenario approach for non-convex control design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro

Sparse and Constrained Stochastic Predictive Control for Networked Systems

This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. The control channel is assumed to have i.i.d. Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. We further consider a regularization term in a quadratic performance index to promote sparsity in control. We demonstrate how to augment the underlying optimization problem with a constant negative drift constraint to ensure mean-square boundedness of the closed-loop states, yielding a convex quadratic program to be solved periodically online. The states of the closed-loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control and any non-zero probability of successful transmission

Robust SINR-Constrained Symbol-Level Multiuser Precoding with Imperfect Channel Knowledge

In this paper, we address robust design of symbol-level precoding for the downlink of multiuser multiple-input multiple-output wireless channels, in the presence of imperfect channel state information (CSI) at the transmitter. In particular, we consider two common uncertainty models for the CSI imperfection, namely, spherical (bounded) and stochastic (Gaussian). Our design objective is to minimize the total (per-symbol) transmission power subject to constructive interference (CI) constraints as well as users' quality-of-service requirements in terms of signal-to-interference-plus-noise ratio. Assuming bounded channel uncertainties, we obtain a convex CI constraint based on the worst-case robust analysis, whereas in the case of Gaussian uncertainties, we define probabilistic CI constraints in order to achieve robustness to statistically-known CSI errors. Since the probabilistic constraints of actual interest are difficult to handle, we resort to their convex approximations, yielding tractable (deterministic) robust constraints. Three convex approximations are developed based on different robust conservatism approaches, among which one is introduced as a benchmark for comparison. We show that each of our proposed approximations is tighter than the other under specific robustness conditions, while both always outperform the benchmark. Using the developed CI constraints, we formulate the robust precoding optimization as a convex conic quadratic program. Extensive simulation results are provided to validate our analytic discussions and to make comparisons with existing robust precoding schemes. We also show that the robust design increases the computational complexity by an order of the number of users in the large system limit, compared to its non-robust counterpart.Comment: 19 pages, 9 figures, Submitted to IEEE Transactions in Signal Processin