A Continuous Method for Convex Programming Problems

In this paper, we present a continuous method for convex programming (CP) problems. Our approach converts first the convex problem into a monotone variational inequality (VI) problem. Then, a continuous method, which includes both a merit function and an ordinary differential equation (ODE), is introduced for the resulting variational inequality problem. The convergence of the ODE solution is proved for any starting point. There is no Lipschitz condition required in our proof. We show also that this limit point is an optimal solution for the original convex problem. Promising numerical results are presented.This research was supported in part by Grants FRG/01-02/I-39 and FRG/01-02/II-06 of Hong Kong Baptist University and Grant HKBU2059/02P from the Research Grant Council of Hong Kong.The author thanks Professor Bingsheng He for many helpful suggestions and discussions. The author is also grateful for the comments and suggestions of two anonymous referees. In particular, the author is indebted to one referee who drew his attention to References 15, 17, 18.