Optimal adaptive control of cascading power grid failures
We describe experiments with parallel algorithms for computing adaptive controls for attenuating power grid cascading failures. CitationColumbia University, 2010ArticleDownload View PDF
stochastic optimization
Convex approximations in stochastic programming by semidefinite programming
The following question arises in stochastic programming: how can one approximate a noisy convex function with a convex quadratic function that is optimal in some sense. Using several approaches for constructing convex approximations we present some optimization models yielding convex quadratic regressions that are optimal approximations in $L_1$, $L_\infty$ and $L_2$ norm. Extensive numerical experiments … Read more
On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems
We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully-adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right hand side of the constraints is … Read more
Primal and dual linear decision rules in stochastic and robust optimization
Linear stochastic programming provides a flexible toolbox for analyzing real-life decision situations, but it can become computationally cumbersome when recourse decisions are involved. The latter are usually modelled as decision rules, i.e., functions of the uncertain problem data. It has recently been argued that stochastic programs can quite generally be made tractable by restricting the … Read more
Convergence of stochastic average approximation for stochastic optimization problems with mixed expectation and per-scenario constraints
We present a framework for ensuring convergence of sample average approximations to stochastic optimization problems that include expectation constraints in addition to per-scenario constraints. CitationPreprint ANL/MCS 1562-1108ArticleDownload View PDF
An information-based approximation scheme for stochastic optimization problems in continuous time
Dynamic stochastic optimization problems with a large (possibly infinite) number of decision stages and high-dimensional state vector are inherently difficult to solve. In fact, scenario tree based algorithms are unsuitable for problems with many stages, while dynamic programming type techniques are unsuitable for problems with many state variables. This article proposes a stage aggregation scheme … Read more
Formulation and solution strategies for nonparametric nonlinear stochastic programs, with an application in finance
We consider a class of stochastic programming models where the uncertainty is classically represented using parametric distributions families. The parameters are then usually estimated together with the optimal value of the problem. However, misspecification of the underlying random variables often leads to irrealistic results when little is known about their true distributions. We propose to … Read more
On Rates of Convergence for Stochastic Optimization Problems Under Non-I.I.D. Sampling
In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well-studied problem in case the samples are independent and identically distributed (i.e., when … Read more
Solving the Vehicle Routing Problem with Stochastic Demands using the Cross Entropy Method
An alternate formulation of the classical vehicle routing problem with stochastic demands (VRPSD)is considered. We propose a new heuristic method to solve the problem. The algorithm is a modified version of the so-called Cross-Entropy method, which has been proposed in the literature as a heuristics for deterministic combinatorial optimization problems based upon concepts of rare-event … Read more
Conditional Risk Mappings
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. CitationPreprintArticleDownload View PDF