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.
In this paper we propose the Online Prize-Collecting Facility Location problem (OPCFL), which is an online version of the Prize-Collecting Facility Location problem (PCFL). The PCFL is a generalization of the Uncapacitated Facility Location problem (FL) in which some clients may be left unconnected by paying a penalty. Another way to think about it is that every client has a prize that can only be collected if it is connected. We give a primal-dual O(log n)-competitive algorithm for the OPCFL based on previous algorithms for Online Facility Location (OFL) due to Fotakis [3] and Nagarajan and Williamson [9].
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