On safe tractable approximations of chance-constrained linear matrix inequalities with partly dependent perturbations.

Cheung, Sin Shuen.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (leaves 55-59).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 2 --- Preliminaries --- p.5Chapter 2.1 --- Heuristics by Hoeffding and Janson --- p.5Chapter 2.2 --- Facts in Matrix Theory --- p.10Chapter 2.3 --- Facts in Probability --- p.11Chapter 3 --- General Chance-Constrained LMIs --- p.18Chapter 3.1 --- Probability Inequalities --- p.18Chapter 3.2 --- Safe Tractable Approximations --- p.22Chapter 4 --- Chance-Constrained Quadratically Perturbed LMIs --- p.27Chapter 4.1 --- Exact Proper Fractional Covers --- p.27Chapter 4.2 --- Bounding the Matrix Moment Generating Functions --- p.32Chapter 5 --- Computational Study --- p.39Chapter 5.1 --- Update Procedures for Safe Tractable Approximations --- p.39Chapter 5.2 --- A Numerical Example and Comparisons --- p.44Chapter 6 --- Conclusion --- p.54Bibliography --- p.5

A distributionally robust perspective on uncertainty quantification and chance constrained programming

The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

Safe Approximations of Chance Constraints Using Historical Data

This paper proposes a new way to construct uncertainty sets for robust optimization. Our approach uses the available historical data for the uncertain parameters and is based on goodness-of-fit statistics. It guarantees that the probability that the uncertain constraint holds is at least the prescribed value. Compared to existing safe approximation methods for chance constraints, our approach directly uses the historical-data information and leads to tighter uncertainty sets and therefore to better objective values. This improvement is significant especially when the number of uncertain parameters is low. Other advantages of our approach are that it can handle joint chance constraints easily, it can deal with uncertain parameters that are dependent, and it can be extended to nonlinear inequalities. Several numerical examples illustrate the validity of our approach.robust optimization;chance constraint;phi-divergence;goodness-of-fit statistics

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

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 (SLP) for the downlink of multiuser multiple-input single-output wireless channels, when imperfect channel state information (CSI) is available at the transmitter. In particular, we consider a well known model for the CSI imperfection, namely, stochastic Gaussian-distributed uncertainty. Our design objective is to minimize the total (per-symbol) transmission power subject to constructive interference (CI) constraints as well as the users’ quality-of-service requirements in terms of signal-to-interference-plus-noise ratio. Assuming stochastic channel uncertainties, we first define probabilistic CI constraints in order to achieve robustness to statistically known CSI errors. Since these probabilistic constraints are difficult to handle, we resort to their convex approximations in the form of tractable (deterministic) robust constraints. Three convex approximations are obtained based on different conservatism levels, 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 settings, while both of them always outperform the benchmark. Using the proposed CI constraints, we formulate the robust SLP optimization problem as a second-order cone program. Extensive simulation results are provided to validate our analytic discussions and to make comparisons with conventional block-level robust precoding schemes. We show that the robust design of symbol-level precoder leads to an improved performance in terms of energy efficiency at the cost of increasing the computational complexity by an order of the number of users in the large system limit, compared to its non-robust counterpart

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization