We derive a \emph{lower bound} for the \emph{sample complexity} of the Sample Average Approximation method for a certain class of multistage stochastic optimization problems. In previous works, \emph{upper bounds} for such problems were derived. We show that the dependence of the \emph{lower bound} with respect to the complexity parameters and the problem’s data are comparable … Read more
We study the Lov\'{a}sz-Schrijver lift-and-project operator ($\LS_+$) based on the cone of symmetric, positive semidefinite matrices, applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the $\LS_+$-operator generates the stable set polytope in one step has been open since 1990. We call these graphs … Read more
To facilitate fast solution of deterministic dynamic programming problems, we present a parameter-free variation of the Sampled Fictitious Play (SFP) algorithm. Its random tie-braking procedure imparts a natural randomness to the algorithm which prevents it from “getting stuck” at a local optimal solution and allows the discovery of an optimal path in a finite number … Read more
The worst case complexity of direct-search methods has been recently analyzed when they use positive spanning sets and impose a sufficient decrease condition to accept new iterates. Assuming that the objective function is smooth, it is now known that such methods require at most O(n^2 epsilon^{-2}) function evaluations to compute a gradient of norm below … Read more
In this paper, we propose a filter active-set algorithm for the minimization problem over a product of multiple ball/sphere constraints. By making effective use of the special structure of the ball/sphere constraints, a new limited memory BFGS (L-BFGS) scheme is presented. The new L-BFGS implementation takes advantage of the sparse structure of the Jacobian of … Read more
In this paper, we study properties of the solution of the pseudomonotone second-order cone linear complementarity problems (SOCLCP). Based upon Tao’s recent work [Tao, J. Optim. Theory Appl., 159(2013), pp. 41–56] on pseudomonotone LCP on Euclidean Jordan algebras, we made two noticeable contributions on the solutions of the pseudomonotone SOCLCP: First, we introduce the concept … Read more
The alternating direction method of multipliers (ADMM) is a benchmark for solving a two-block linearly constrained convex minimization model whose objective function is the sum of two functions without coupled variables. Meanwhile, it is known that the convergence is not guaranteed if the ADMM is directly extended to a multiple-block convex minimization model whose objective … Read more
We consider discounted Markov Decision Processes (MDPs) with countably-infinite state spaces, finite action spaces, and unbounded rewards. Typical examples of such MDPs are inventory management and queueing control problems in which there is no specific limit on the size of inventory or queue. Existing solution methods obtain a sequence of policies that converges to optimality … Read more
In this note we announce the availability of an electronic compendium of extreme functions for Gomory–Johnson’s infinite group problem. These functions serve as the strongest cut-generating functions for integer linear optimization problems. We also close several gaps in the literature. ArticleDownload View PDF
In this paper we present a variant of the proximal forward-backward splitting method for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be call the Proximal Subgradient Splitting Method, extends the classical projected subgradient iteration for important classes of … Read more