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

• Published in: SIAM journal on control and optimization Vol. 35; no. 2; pp. 532 - 551
• Main Authors: Sun, Jie, Zhu, Jishan, Zhao, Gongyun
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.03.1997
• Subjects: Algorithms; Applied sciences; Computer science; control theory; systems; Control theory. Systems; Convex analysis; Exact sciences and technology; Linear programming; Mathematical programming; Methods; Operational research and scientific management; Operational research. Management science; Optimal control; Optimization; Predictor corrector method; Optimal control; Interior point method; Complementarity problem; Non linear system; Non linear complementarity problem; Saddle point; Convergence; Stochastic programming
• ISSN: 0363-0129; 1095-7138
• DOI: 10.1137/S0363012994276111

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.