A predictor-corrector algorithm for a class of nonlinear saddle point problems

SIAM Journal on Control and Optimization

Volumn 35, Issue 2, 1997, Pages 532-551

  Sun, Jie ^a   Zhu, Jishan ^b   Zhao, Gongyun ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b Newbury College   (United States)

Author keywords

Interior point methods; Nonlinear complementarity problem; Optimal control; Saddle point problem; Stochastic programming

Indexed keywords

ALGORITHMS; MATHEMATICAL PROGRAMMING; NONLINEAR CONTROL SYSTEMS; NUMERICAL METHODS; OPTIMAL CONTROL SYSTEMS; OPTIMIZATION; PROBLEM SOLVING;

CENTRAL PATH; ERROR BOUND; INTERIOR POINT METHODS; NONLINEAR COMPLEMENTARY PROBLEM; NONLINEAR SADDLE POINT PROBLEMS; PREDICTOR CORRECTOR ALGORITHMS; SADDLE POINT PROBLEM; STOCHASTIC PROGRAMMING;

CONTROL THEORY;

EID: 0031094383     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994276111     Document Type: Article

References (38)

1. J. R. BIRGE, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res., 33 (1985), pp. 989-1007.
2. J. R. BIRGE AND D. F. HOLMES, Efficient solution of two-stage stochastic linear programs using interior point methods, Comput. Optim. Appl., 1 (1992), pp. 245-276.
3. J. R. BIRGE AND L. QI, Computing block-angular Karmarkar projections with application to stochastic programming, Management Sci., 34 (1988), pp. 1472-1479.
4. D. DEN HERTOG, F. JARRE, C. ROOS, AND T. TERLAKY, A sufficient condition for self-concordance, with application to some classes of structured convex programming problems, Math. Programming, 69 (1995), pp. 75-88.
5. G. DANTZIG AND A. MADANSKY, On the solution of two-stage linear programs under uncertainty, in Proc. Fourth Berkeley Symposium on Math. and Probability, Vol. 1, University of California Press, Berkeley, CA, 1961.
6. C. C. GONZAGA, Interior point path following algorithms, in Mathematical Programming: State of the Art 1994, J. Berge and K. Murty, eds., University of Michigan Press, Ann Arbor, MI, 1994.
7. O. GÜLER, Existence of interior point and interior paths in nonlinear monotone complementarity problems, Math. Oper. Res., 18 (1993), pp. 128-147.
8. F. JARRE AND M. A. SAUNDERS, A practical interior-point method for convex programming, SIAM J. Optim., 5 (1995), pp. 129-148.
9. J. JI, F. POTRA, AND S. HUANG, A predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence, J. Optim. Theory Appl., 85 (1995), pp. 187-199.
10. M. KOJIMA, S. MIZUNO, AND A. YOSHISE, A primal-dual algorithm for a class of linear complementarity problems, Math. Programming, 44 (1989), pp. 1-26.
11. M. KOJIMA, M. MEGIDDO, T. NOMA, AND A. YOSHISE, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1991.
12. M. KOJIMA, T. NOMA, AND A. YOSHISE, Global convergence in infeasible-interior-point algorithms, Math. Programming, 65 (1994), pp. 43-72.
13. S. MIZUNO, A superlinearly convergent infeasible-interior-point algorithm for geometrical LCPs without a strictly complementary condition, Math. Oper. Res., 21 (1996), pp. 382-400.
14. S. MIZUNO, M. TODD, AND Y. YE, On adaptive step primal-dual interior-point algorithm for linear programming, Math. Oper. Res., 18 (1993), pp. 964-981.
15. Y. NESTEROV and A. NEMIROVSKII, Interior Point Polynomial Algorithms in Convex Programming, SIAM, Philadelphia, PA, 1994.
16. L. QI AND R. S. WOMERSLEY, An SQP algorithm for extended linear-quadratic problems in stochastic programming, Ann. Oper. Res., 56 (1995), pp. 251-285.
17. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
18. R. T. ROCKAFELLAR, Linear-quadratic programming and optimal control, SIAM J. Control Optim., 25 (1987), pp. 781-814.
19. R. T. ROCKAFELLAR, Computational schemes for large-scale problems in extended linear-quadratic programming, Math. Programming, 48 (1990), pp. 447-474.
20. R. T. ROCKAFELLAR, Large-scale extended linear-quadratic programming and multistage optimization, in Proc. Fifth Mexico-U.S. Workshop on Numerical Analysis, S. Gomez, J. .P. Hennat, and R. Tapia, eds., SIAM, Philadelphia, PA, 1990.
21. R. T. ROCKAFELLAR, Nonsmooth optimization, in Mathematical Programming: State of the Art 1994, J. Berge and K. Murty, eds., University of Michigan Press, Ann Arbor, MI, 1994.
22. R. T. ROCKAFELLAR AND J. SUN, A finite simplex-active-set method for monotropic piecewise quadratic programming, in Advances in Optimization and Approximation, D. Du and J. Sun, eds., Kluwer Academic Publishers, Dordrecht, the Netherlands, 1994.
23. R. T. ROCKAFELLAR AND R. J.-B. WETS, A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming, Math. Programming Studies, 28 (1986), pp. 63-93.
24. R. T. ROCKAFELLAR AND R. J.-B. WETS, Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time, SIAM J. Control Optim., 28 (1990), pp. 810-822.
25. G. SONNEVEND, J. STOER, AND G. ZHAO, On the complexity of following the central path of linear programs by linear extrapolation II, Math. Programming, 52 (1991), pp. 527-553.
26. J. SUN, K. WEE, AND J. ZHU, An interior point method for solving a class of linear-quadratic stochastic programming problems, in Recent Advances in Nonsmooth Optimization, D. Du, L. Qi, and R. S. Womersley, eds., World Scientific, Singapore, 1995.
27. J. SUN AND J. ZHU, A predictor-corrector method for extended linear-quadratic programming, Comput. Oper. Res., 23 (1996), pp. 755-767.
28. P. TSENG, Global linear convergence of a path-following algorithm for some monotone variational inequality problems, J. Optim. Theory Appl., 75 (1992), pp. 265-279.
29. P. TSENG, An Infeasible Path-Following Method for Monotone Complementarity Problem, Department of Mathematics, University of Washington, Seattle, WA, 1994, preprint.
30. R. VAN SLYKE AND R. J.-B. WETS, L-shaped linear programs with applications to optimal control and stochastic linear programs, SIAM J. Appl. Math., 17 (1969), pp. 638-663.
31. R. J.-B. WETS, Solving stochastic programs with simple recourse, Stochastics, 10 (1983), pp. 219-242.
32. S. J. WEIGHT, Interior point methods for optimal control of discrete-time systems, J. Optim. Theory Appl., 77 (1993), pp. 161-187.
33. S. J. WRIGHT AND D. RALPH, A superlinear infeasible-interior-point algorithm for monotone complementarity problems, Math. Oper. Res., 21 (1996).
34. Y. YE AND K. ANSTREICHER, On quadratic and O(√nL) convergence of a predictor-corrector algorithm for LCP, Math. Programming, 62 (1993), pp. 537-552.
35. Y. ZHANG, On the convergence of a class of infeasible-interior-point methods for the horizontal linear complementarity problem, SIAM J. Optim., 4 (1994), pp. 208-227.
36. C. ZHU, On the primal-dual steepest descent algorithm for extended linear-quadratic programming, SIAM J. Optim., 5 (1995), pp. 114-128.
37. C. ZHU AND R. T. ROCKAFELLAR, Primal-dual projected gradient algorithms for extended linear-quadratic programming, SIAM J. Optim., 3 (1993), pp. 751-783.
38. J. ZHU, A path-following algorithm for a class of convex programming problems, Z. Oper. Res., 36 (1992), pp. 359-377.