The mixing set with a knapsack constraint arises in deterministic equivalent of chance-constrained programming problems with finite discrete distributions. We first consider the case that the chance-constrained program has equal probabilities for each scenario. We study the resulting mixing set with a cardinality constraint and propose facet-defining inequalities that subsume known explicit ine...

In chance-constrained optimization problems, a solution is assumed to be feasible only with certain, sufficiently high probability. For computational and theoretical purposes, the convexity property of the resulting constraint set is treated. It is known, for example, that a suitable combination of a concavity property of the probability distribution and concavity of constraint mappings are suf...

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex se...

To limit the probability of unacceptable worst-case linearization errors that might yield risks for power system operations, this letter proposes a robust data-driven linear flow (RD-LPF) model. It applies to both transmission and distribution systems can achieve better robustness than recent models. The key idea is probabilistically constrain through distributionally chance-constrained program...

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...