Journal of Convex Analysis 09 (2002), No. 1, 237--244
Copyright Heldermann Verlag 2002
Convex Stochastic Duality and the "Biting Lemma"
Igor V. Evstigneev
School of Economic Studies, University of Manchester, Oxford Road, Manchester M13 9PL, Great Britain
igor.evstigneev@man.ac.uk
Sjur D. Flåm
Dept. of Economics, University of Bergen, Fosswickels gate 6, 5007 Bergen, Norway
sjur.flaam@econ.uib.no
[Abstract-pdf]
A standard approach to duality in stochastic optimization problems with constraints in $L_{\infty}$ relies upon the Yosida - Hewitt theorem. We develop an alternative technique which employs only "elementary" means. The technique is based on an $\varepsilon$-regularization of the original problem and on passing to the limit as $\varepsilon \to 0$ with the help of a simple measure-theoretic fact -- the biting lemma.
Keywords: Stochastic optimization, convex duality, constraints in L-infinity, stochastic Lagrange multipliers, bounded sets in L-1, biting lemma, Gale's economic model.
MSC: 90C15, 51A41; 90C19, 90A16
[ Fulltext-pdf (262 KB)]