On the Sample Size of Random Convex Programs with Structured Dependence on the Uncertainty (Extended Version)

The "scenario approach" provides an intuitive method to address chance constrained problems arising in control design for uncertain systems. It addresses these problems by replacing the chance constraint with a finite number of sampled constraints (scenarios). The sample size critically depends on Helly's dimension, a quantity always upper bounded by the number of decision variables. However, this standard bound can lead to computationally expensive programs whose solutions are conservative in terms of cost and violation probability. We derive improved bounds of Helly's dimension for problems where the chance constraint has certain structural properties. The improved bounds lower the number of scenarios required for these problems, leading both to improved objective value and reduced computational complexity. Our results are generally applicable to Randomized Model Predictive Control of chance constrained linear systems with additive uncertainty and affine disturbance feedback. The efficacy of the proposed bound is demonstrated on an inventory management example.Comment: Accepted for publication at Automatic

Sample Approximation-Based Deflation Approaches for Chance SINR Constrained Joint Power and Admission Control

Consider the joint power and admission control (JPAC) problem for a multi-user single-input single-output (SISO) interference channel. Most existing works on JPAC assume the perfect instantaneous channel state information (CSI). In this paper, we consider the JPAC problem with the imperfect CSI, that is, we assume that only the channel distribution information (CDI) is available. We formulate the JPAC problem into a chance (probabilistic) constrained program, where each link's SINR outage probability is enforced to be less than or equal to a specified tolerance. To circumvent the computational difficulty of the chance SINR constraints, we propose to use the sample (scenario) approximation scheme to convert them into finitely many simple linear constraints. Furthermore, we reformulate the sample approximation of the chance SINR constrained JPAC problem as a composite group sparse minimization problem and then approximate it by a second-order cone program (SOCP). The solution of the SOCP approximation can be used to check the simultaneous supportability of all links in the network and to guide an iterative link removal procedure (the deflation approach). We exploit the special structure of the SOCP approximation and custom-design an efficient algorithm for solving it. Finally, we illustrate the effectiveness and efficiency of the proposed sample approximation-based deflation approaches by simulations.Comment: The paper has been accepted for publication in IEEE Transactions on Wireless Communication

Convex Relaxations and Approximations of Chance-Constrained AC-OPF Problems

This paper deals with the impact of linear approximations for the unknown nonconvex confidence region of chance-constrained AC optimal power flow problems. Such approximations are required for the formulation of tractable chance constraints. In this context, we introduce the first formulation of a chance-constrained second-order cone (SOC) OPF. The proposed formulation provides convergence guarantees due to its convexity, while it demonstrates high computational efficiency. Combined with an AC feasibility recovery, it is able to identify better solutions than chance-constrained nonconvex AC-OPF formulations. To the best of our knowledge, this paper is the first to perform a rigorous analysis of the AC feasibility recovery procedures for robust SOC-OPF problems. We identify the issues that arise from the linear approximations, and by using a reformulation of the quadratic chance constraints, we introduce new parameters able to reshape the approximation of the confidence region. We demonstrate our method on the IEEE 118-bus system

Two-Stage Subspace Constrained Precoding in Massive MIMO Cellular Systems

We propose a subspace constrained precoding scheme that exploits the spatial channel correlation structure in massive MIMO cellular systems to fully unleash the tremendous gain provided by massive antenna array with reduced channel state information (CSI) signaling overhead. The MIMO precoder at each base station (BS) is partitioned into an inner precoder and a Transmit (Tx) subspace control matrix. The inner precoder is adaptive to the local CSI at each BS for spatial multiplexing gain. The Tx subspace control is adaptive to the channel statistics for inter-cell interference mitigation and Quality of Service (QoS) optimization. Specifically, the Tx subspace control is formulated as a QoS optimization problem which involves an SINR chance constraint where the probability of each user's SINR not satisfying a service requirement must not exceed a given outage probability. Such chance constraint cannot be handled by the existing methods due to the two stage precoding structure. To tackle this, we propose a bi-convex approximation approach, which consists of three key ingredients: random matrix theory, chance constrained optimization and semidefinite relaxation. Then we propose an efficient algorithm to find the optimal solution of the resulting bi-convex approximation problem. Simulations show that the proposed design has significant gain over various baselines.Comment: 13 pages, accepted by IEEE Transactions on Wireless Communication