Convergence theory for nonconvex stochastic programming with an application to mixed logit

Monte Carlo methods have extensively been used and studied in the area of stochastic programming. Their convergence properties typically consider global minimizers or first-order critical points of the sample average approximation (SAA) problems and minimizers of the true problem, and show that the former converge to the latter for increasing sample size. However, the assumption of global minimization essentially restricts the scope of these results to convex problems. We review and extend these results in two directions: we allow for local SAA minimizers of possibly nonconvex problems and prove, under suitable conditions, almost sure convergence of local second-order solutions of the SAA problem to second-order critical points of the true problem. We also apply this new theory to the estimation of mixed logit models for discrete choice analysis. New useful convergence properties are derived in this context, both for the constrained and unconstrained cases, and associated estimates of the simulation bias and variance are proposed. Research Fellow of the Belgian National Fund for Scientific Research     

Ground-state energy of the Hubbard model in the BCS approximation

In the paper it is shown that it is possible to apply the BCS theory to the Hubbard model with positiveU for a half-filled band. A calculation of the ground-state energy of the model is presented.     

On uniform convergence of measures

Summary A new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsØe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.Most of the work on the paper was done while the author was with the Institute for Information Theory and Automation of Czechoslovak Academy of Sciences. After August 21, 1968 the author worked on it while a guest of the Forschungsinstitut für Mathematik, Eidgenössische Technische Hochschule, Zürich, and the paper was completed at Michigan State University.     

Approaches for interpreting π electronic states and π electronic transitions

Scope and limitations of different approaches which enable π electronic systems of topologically equivalent structures to be compared have been examined. Particular attention has been paid to analyses in terms of molecular subsystems.     

Unified Treatment of EPR and Bell Arguments in Algebraic Quantum Field Theory

A conjecture concerning vacuum correlations in axiomatic quantum field theory is proved. It is shown that this result can be applied both in the context of EPR-type experiments and Bell-type experiments.