Search

This paper discusses distributionally robust geometric programs with individual or joint chance constraints. Several groups of uncertainty sets are considered: uncertainty sets with first two order moments information; uncertainty sets with known first order or first two order moments information under nonnegative support; uncertainty sets constrained by the Kullback–Leibler divergence with a normal or discrete reference distribution; uncertainty sets constrained by the Wasserstein distance under discrete, full, or nonnegative real-space support; and joint uncertainty sets for the product of random variables. Under each group of uncertainty sets, we find deterministic reformulations of the distributionally robust geometric programs with individual or joint chance constraints. Convexity, solution methods, and relationships of the reformulation programs are discussed. Finally, numerical tests are carried out on a shape optimization problem. [ABSTRACT FROM AUTHOR]

In the paper we consider the chance-constrained version of an affinely perturbed linear matrix inequality (LMI) constraint, assuming the primitive perturbations to be independent with light-tail distributions (e.g., bounded or Gaussian). Constraints of this type, playing a central role in chance-constrained linear/conic quadratic/semidefinite programming, are typically computationally intractable. The goal of this paper is to develop a tractable approximation to these chance constraints. Our approximation is based on measure concentration results and is given by an explicit system of LMIs. Thus, the approximation is computationally tractable; moreover, it is also safe, meaning that a feasible solution of the approximation is feasible for the chance constraint. [ABSTRACT FROM AUTHOR]

In the multidimensional 0-1 knapsack problem, we are given a set of items, each with a value and multiple attributes, and we want to select a subset in such a way that the total value is maximized while the total quantity of each attribute satisfies a capacity constraint. In this paper, we assume that quantities of the item attributes are independent random variables such that those of the same attribute across different items follow the same type of probability distribution, not necessarily with the same parameters. A joint probabilistic constraint is imposed on the capacity constraints, and the objective function is the same as that of the underlying deterministic problem. We prove that the problem is convex, under some condition on the parameters, for special continuous and discrete distributions: gamma, normal, Poisson, and binomial, in which the latter two discrete distribution functions are extended to log-concave continuous distribution functions. We present computational experiments to demonstrate the tractability of our approach. [ABSTRACT FROM AUTHOR]

We provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for, e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at the same time. Moreover, the precision of gradients can be controlled by that of function values, which is a great advantage over using finite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a transportation network with stochastic demands. [ABSTRACT FROM AUTHOR]