We consider an optimization problem of minimization of a linear function subject to chance constraints. In the multidimensional case this problem is, generically, a problem of minimizing under a nonconvex and difficult to compute constraints and as such is computationally intractable. We investigate the potential of conceptually simple scenario approximation of the chance constraints. The … Read more
The main focus of this paper is discussion of complexity of stochastic programming problems. We argue that two-stage (linear) stochastic programming problems with recourse can be solved with a reasonable accuracy by using Monte Carlo sampling techniques, while multi-stage stochastic programs, in general, are intractable. We also discuss complexity of chance constrained problems and multi-stage … Read more
We show that for even quasi-concave objective functions the worst-case distribution, with respect to a family of unimodal distributions, of a stochastic programming problem is a uniform distribution. This extends the so-called “Uniformity Principle” of Barmish and Lagoa (1997) where the objective function is the indicator function of a convex symmetric set. Article Download View … Read more
We consider probabilistic constrained linear programs with general distributions for the uncertain parameters. These problems generally involve non-convex feasible sets. We develop a branch and bound algorithm that searches for a global solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partitions. … Read more
In this paper we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of convergence of sample average approximations of such stochastic problems. Citation School of Industrial … Read more
The vast size of real world stochastic programming instances requires sampling to make them practically solvable. In this paper we extend the understanding of how sampling affects the solution quality of multistage stochastic programming problems. We present a new heuristic for determining good feasible solutions for a multistage decision problem. For power and log-utility functions … Read more
Computing the optimal portfolio policy of an investor facing capital gains tax is a challenging problem: because the tax to be paid depends on the price at which the security was purchased (the tax basis), the optimal policy is path dependent and the size of the problem grows exponentially with the number of time periods. … Read more
Traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criteria. A common approach to addressing risk in decision making problems is to consider a weighted mean-risk criterion, where some dispersion statistic is used as a measure of risk. We investigate the computational suitability of various … Read more
Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear … Read more
We introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings. Citation Preprint Article Download View Conditional Risk Mappings