Every multistage stochastic programming problem with recourse (MSPR) contains a filtration process. In this research, we created a notation that makes the filtration process the central syntactic construction of the MSPR. As a result, we achieve lower redundancy and higher modularity than is possible with the mathematical notation commonly associated with stochastic programming. To experiment … Read more
We present a method to approximate the solution mapping of parametric constrained optimization problems. The approximation, which is of the spectral stochastic finite element type, is represented as a linear combination of orthogonal polynomials. Its coefficients are determined by solving an appropriate finite-dimensional constrained optimization problem. We show that, under certain conditions, the latter problem … Read more
Airlines must dynamically choose how to allocate their flight capacity to incoming travel demand. Because some passengers take connecting flights, the decisions for all network flights must be made simultaneously. To simplify the decision making process, most practitioners assume demand is deterministic and equal to average demand. We propose a multistage stochastic programming approach that … Read more
In the paper, we consider the chance constrained version $$ \Prob\{A_0[x]+\sum_{i=1}^d\zeta_i A_i[x]\succeq0\}\geq1-\epsilon, $$ of an affinely perturbed Linear Matrix Inequality constraint; here $A_i[x]$ are symmetric matrices affinely depending on the decision vector $x$, and $\zeta_1,…,\zeta_d$ are independent of each other random perturbations with light tail distributions (e.g., bounded or Gaussian). Constraints of this type, playing … Read more
We discuss in this paper asymptotics of the sample average approximation (SAA) of the optimal value of a minimax stochastic programming problem. The main tool of our analysis is a specific version of the infinite dimensional Delta Method. As an example, we discuss asymptotics of SAA of risk averse stochastic programs involving the absolute semideviation … Read more
We study in this paper a Markov Decision Problem (MDP) with continuous state space and discrete decision variables. We propose an extension of the Q-learning algorithm introduced to solve this problem by Watkins in 1989 for completely discrete MDPs. Our algorithm relies on stochastic approximation and functional estimation, and uses kernels to locally update the … Read more
This paper addresses the problem of finding cutting planes for multi-stage stochastic integer programs. We give a general method for generating cutting planes for multi-stage stochastic integer programs based on combining inequalities that are valid for the individual scenarios. We apply the method to generate cuts for a stochastic version of a dynamic knapsack problem … Read more
Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but-bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of {\sl robust counterpart} of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links … Read more
We describe a method of generating a warm-start point for interior point methods in the context of stochastic programming. Our approach exploits the structural information of the stochastic problem so that it can be seen as a structure-exploiting initial point generator. We solve a small-scale version of the problem corresponding to a reduced event tree … Read more
This paper proposes a new stochastic algorithm, Search via Probability (SP) algorithm, for single-objective optimization problems. The SP algorithm uses probabilities to control the process of searching for optimal solutions. We calculate probabilities of the appearance of a better solution than the current one on each iteration, and on the performance of SP algorithm we … Read more