Linear Matrix Inequalities with Stochastically Dependent Perturbations and Applications to Chance-Constrained Semidefinite Optimization

Abstract

The wide applicability of chance-constrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so-called safe tractable approximations of chance-constrained programs, where a chance constraint is replaced by a deterministic and efficiently computable inner approximation. Currently, such approach applies mainly to chance-constrained linear inequalities, in which the data perturbations are either independent or define a known covariance matrix. However, its applicability to chanceconstrained conic inequalities with dependent perturbations-which arises in finance, control and signal processing applications-remains largely unexplored. In this paper, we develop safe tractable approximations of chance-constrained affinely perturbed linear matrix inequalities, in which the perturbations are not necessarily independent, and the only information available about the dependence structure is a list of independence relations. To achieve this, we establish new large deviation bounds for sums of dependent matrix-valued random variables, which are of independent interest. A nice feature of our approximations is that they can be expressed as systems of linear matrix inequalities, thus allowing them to be solved easily and efficiently by off-the-shelf solvers. We also provide a numerical illustration of our constructions through a problem in control theory.

Distributionally Robust Convex Optimization

“…Among the first proponents of this idea are Ghaoui et al (2003), who study distributionally robust quantile optimization problems. Their methods have later been extended to linear and conic chance constraints where only the mean, covariance matrix, and support of the underlying probability distribution are specified, see, e.g., Calafiore and Ghaoui (2006), Chen et al (2010), Cheung et al (2012), andZymler et al (2013).…”

Partial sample average approximation method for chance constrained problems

“…The parameters are set as follows: n = 50, η ∈ {0.05, 0.1, 0.15, 0.2}, and c is uniformly generated on the interval [10,100]. The upper and lower bounds of ξ 1 are uniformly generated on the intervals [10,20] and [5,10], respectively, whilst the upper and lower bounds of ξ 2 are uniformly generated on the intervals [50, 100] and [0, 50], respectively.…”

“…Nemirovski and Shapiro [18] developed a large deviation-type approximation, referred to as Bernstein approximation. Cheung et al [10] established a new large deviation bounds for the stochastic linear matrix inequalities. On the other hand, probability discretization based approaches consider discrete probability distribution or representation of empirical distribution obtained from Monte Carlo samples.…”

Probabilistically Robust SWIPT for Secrecy MISOME Systems

“…The merit of Lemma 5 is that it helps decomposing a sum of dependent random variables into sums of independent random variables. This idea has been used extensively in the literature of probability theory; see, e.g., [28], [29].…”