Chance constrained problem and its applications

Chance constrained optimization is a natural and widely used approaches to provide profitable and reliable decisions under uncertainty. And the topics around the theory and applications of chance constrained problems are interesting and attractive. However, there are still some important issues requiring non-trivial efforts to solve. In view of this, we will systematically investigate chance constrained problems from the following perspectives. As the basis for chance constrained problems, we first review some main research results about chance constraints in three perspectives: convexity of chance constraints, reformulations and approximations for chance constraints and distributionally robust chance constraints. For stochastic geometric programs, we formulate consider a joint rectangular geometric chance constrained program. With elliptically distributed and pairwise independent assumptions for stochastic parameters, we derive a reformulation of the joint rectangular geometric chance constrained programs. As the reformulation is not convex, we propose new convex approximations based on the variable transformation together with piecewise linear approximation methods. Our numerical results show that our approximations are asymptotically tight. When the probability distributions are not known in advance or the reformulation for chance constraints is hard to obtain, bounds on chance constraints can be very useful. Therefore, we develop four upper bounds for individual and joint chance constraints with independent matrix vector rows. Based on the one-side Chebyshev inequality, Chernoff inequality, Bernstein inequality and Hoeffding inequality, we propose deterministic approximations for chance constraints. In addition, various sufficient conditions under which the aforementioned approximations are convex and tractable are derived. To reduce further computational complexity, we reformulate the approximations as tractable convex optimization problems based on piecewise linear and tangent approximations. Finally, based on randomly generated data, numerical experiments are discussed in order to identify the tight deterministic approximations. In some complex systems, the distribution of the random parameters is only known partially. To deal with the complex uncertainties in terms of the distribution and sample data, we propose a data-driven mixture distribution based uncertainty set. The data-driven mixture distribution based uncertainty set is constructed from the perspective of simultaneously estimating higher order moments. Then, with the mixture distribution based uncertainty set, we derive a reformulation of the data-driven robust chance constrained problem. As the reformulation is not a convex program, we propose new and tight convex approximations based on the piecewise linear approximation method under certain conditions. For the general case, we propose a DC approximation to derive an upper bound and a relaxed convex approximation to derive a lower bound for the optimal value of the original problem, respectively. We also establish the theoretical foundation for these approximations. Finally, simulation experiments are carried out to show that the proposed approximations are practical and efficient. We consider a stochastic n-player non-cooperative game. When the strategy set of each player contains a set of stochastic linear constraints, we model the stochastic linear constraints of each player as a joint chance constraint. For each player, we assume that the row vectors of the matrix defining the stochastic constraints are pairwise independent. Then, we formulate the chance constraints with the viewpoints of normal distribution, elliptical distribution and distributionally robustness, respectively. Under certain conditions, we show the existence of a Nash equilibrium for the stochastic game.