Are Weaker Stationarity Concepts of Stochastic MPCC Problems Significant in Absence of SMPCC-LICQ?

In this article, we study weak stationarity conditions (A- and C-) for a particular class of degenerate stochastic mathematical programming problems with complementarity constraints (SMPCC, for short). Importance of the weak stationarity concepts in absence of SMPCC-LICQ are presented through toy problems in which the point of local or global minimum are weak stationary points … Read more

Large Deviation Bounds for Markov Chain Sample Average Approximation via Weak Convergence

A common approach to solve stochastic optimization problems with expectations is to replace the expectations by its sample averages. Large sample asymptotic properties of this approximation are well studied when the sample is i.i.d. In many cases, however, i.i.d. samples are not readily available. On the contrary, one can generate a Harris recurrent Markov chain … Read more

Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks

Bayesian Networks (BNs) represent conditional probability relations among a set of random variables (nodes) in the form of a directed acyclic graph (DAG), and have found diverse applications in knowledge discovery. We study the problem of learning the sparse DAG structure of a BN from continuous observational data. The central problem can be modeled as … Read more

Epi-convergence of Sample Averages of a Random Lower Semi-continuous Functional Generated by a Markov Chain and Application to Stochastic Optimization

The purpose of this article is to establish epigraphical convergence of the sample averages of a random lower semi-continuous functional associated with a Harris recurrent Markov chain with stationary distribution $\pi$. Sample averages associated with an ergodic Markov chain with stationary probability distribution will epigraphically converge from $\pi$-almost all starting points. The property of Harris … Read more

Achieving Consistency with Cutting Planes

Cutting planes accelerate branch-and-bound search primarily by cutting off fractional solutions of the linear programming (LP) relaxation, resulting in tighter bounds for pruning the search tree. Yet cutting planes can also reduce backtracking by excluding inconsistent partial assignments that occur in the course of branching. A partial assignment is inconsistent with a constraint set when … Read more

A Robust Multi-Batch L-BFGS Method for Machine Learning

This paper describes an implementation of the L-BFGS method designed to deal with two adversarial situations. The first occurs in distributed computing environments where some of the computational nodes devoted to the evaluation of the function and gradient are unable to return results on time. A similar challenge occurs in a multi-batch approach in which … Read more

An Enhanced Spatial Branch-and-Bound Method in Global Optimization with Nonconvex Constraints

We discuss some difficulties in determining valid upper bounds in spatial branch-and-bound methods for global minimization in the presence of nonconvex constraints. In fact, two examples illustrate that standard techniques for the construction of upper bounds may fail in this setting. Instead, we propose to perturb infeasible iterates along Mangasarian-Fromovitz directions to feasible points whose … Read more