A Predictor-Corrector Algorithm for a Class of Nonlinear Saddle Point Problems

Abstract

An interior path-following algorithm is proposed for solving the nonlinear saddle point problem $$ {\rm minimax}\ c^Tx+\ph(x)+b^Ty-\psi(y)-y^TAx $$ \vspace*{-18pt} $$ {\rm subject\ to\ }(x,y)\in \X\ti \Y\su R^n\ti R^m, $$ \noindent where $\ph(x)$ and $\ps(y)$ are smooth convex functions and $\X$ and $\Y$ are boxes (hyperrectangles). This problem is closely related to the models in stochastic programming and optimal control studied by Rockafellar and Wets (Math. Programming Studies, 28 (1986), pp. 63--93; SIAM J. Control Optim., 28 (1990), pp. 810--822). Existence and error-bound results on a central path are derived. Starting from an initial solution near the central path with duality gap $O(\mu)$, the algorithm finds an $\ep$-optimal solution of the problem in $O(\sqrt{m+n}\,|\log\mu/\ep|)$ iterations if both $\ph(x)$ and $\ps(y)$ satisfy a scaled Lipschitz condition.