Robust Conjugate Duality for Convex Optimization under Uncertainty with Application to Data Classification∗

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also is necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences. ∗The authors are grateful to the referee and the Editor for their helpful comments and valuable suggestions which have contributed to the final preparation of the paper. Research was partially supported by a grant from the Australian Research Council and the Korea Science and Engineering Foundation NRL program grant of the Korea government (MEST)(No. ROA-2008-000-20010-0). †Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. Email: [email protected] ‡Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. Email: [email protected] §Department of Applied Mathematics, Pukyong National University, Busan 608-737, Korea.