Chance constrained nonlinear optimization with skewed distributions and dependent rows

This paper discusses chance constrained optimization problems where the constraints are linear to the random variables but nonlinear to the decision variables. For the individual nonlinear chance constraint, we derive tractable reformulation under finite Gaussian mixture distributions and design tight approximation under the generalized hyperbolic distribution. For the joint nonlinear chance constraint, we study several … Read more

An equivalent mathematical program for games with random constraints

This paper shows that there exists a Nash equilibrium of an n-player chance-constrained game for elliptically symmetric distributions. For a certain class of payoff functions, we suitably construct an equivalent mathematical program whose global maximizer is a Nash equilibrium. Article Download View An equivalent mathematical program for games with random constraints

Games with joint chance constraints under mixture distributions

We consider an n-player non-cooperative game where each player has expected value payoff function and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We propose a subset of probability distributions from elliptical family of distributions. We consider the case when the probability distribution of each random constraint … Read more

Equivalent second-order cone programs for distributionally robust zero-sum games

We consider a two player zero-sum game with stochastic linear constraints. The probability distributions of the vectors associated with the constraints are partially known. The available information with respect to the distribution is based mainly on the two first moments. In this vein, we formulate the stochastic linear constraints as distributionally robust chance constraints. We … Read more

Games with distributionally robust joint chance constraints

This paper studies an n-player non-cooperative game with strategy sets defined by stochastic linear constraints. The stochastic constraints of each player are jointly satisfied with a probability exceeding a given threshold. We consider the case where the row vectors defining the constraints are independent random vectors whose probability distributions are not completely known and belong … Read more

Distributionally robust chance constrained geometric optimization

This paper discusses distributionally robust geometric programs with individual and joint chance constraints. Seven groups of uncertainty sets are considered: uncertainty sets with first two order moments information, uncertainty sets constrained by the Kullback-Leibler divergence distance with a normal reference distribution or a discrete reference distribution, uncertainty sets with known first moments or known first … Read more

Bounds for Probabilistic Programming with Application to a Blend Planning Problem

In this paper, we derive deterministic inner approximations for single and joint probabilistic constraints based on classical inequalities from probability theory such as the one-sided Chebyshev inequality, Bernstein inequality, Chernoff inequality and Hoeffding inequality (see Pinter 1989). New assumptions under which the bounds based approximations are convex allowing to solve the problem efficiently are derived. … Read more

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case … Read more

Second-order cone programming formulation for two player zero-sum game with chance constraints

We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium if the row vectors of the random matrices, defining the stochastic constraints of each player, are elliptically symmetric distributed … Read more

Joint rectangular geometric chance constrained programs

This paper discusses joint rectangular geometric chance constrained programs. When the stochastic parameters are elliptically distributed and pairwise independent, we present a reformulation of the joint rectangular geometric chance constrained programs. As the reformulation is not convex, we propose new convex approximations based on variable transformation together with piecewise linear approximation method. Our results show … Read more