An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions

SIAM Journal on Optimization

Volumn 6, Issue 2, 1996, Pages 461-487

  Zhou, Jian L ^a   Tits, André L ^a  

^a UNIVERSITY OF MARYLAND   (United States)

Author keywords

Continuous minimax; Discretization; Global convergence; Many constraints; Semi infinite programming; Sequential quadratic programming

Indexed keywords

EID: 0030303851     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/0806025     Document Type: Article

References (46)

1. M. C. BIGGS, Constrained minimization using recursive equality quadratic programming, in Numerical Methods for Non-Linear Optimization, F. A. Lootsma, ed., Academic Press, London, New York, 1972, pp. 411-428.
2. S. P. BOYD AND C. H. BARRATT, Linear Controller Design: Limits of Performance, Prentice-Hall, Englewood Cliffs, NJ, 1991.
3. R. K. BRAYTON, G. D. HACHTEL, AND A. L. SANGIOVANNI-VINCENTELLI, A survey of optimization techniques for integrated circuit design, IEEE Proc., 69 (1981), pp. 1334-1362.
4. A. R. CONN AND Y. LI, A structure-exploiting algorithm for nonlinear minimax problems, SIAM J. Optim., 2 (1992), pp. 242-263.
5. I. D. COOPE AND G. A. WATSON, A projected Lagrangian algorithm for semi-infinite programming, Math. Programming, 32 (1985), pp. 337-356.
6. H. A. ESCHENAUER, C. MATTHECK, AND N. OLHOFF, EDS., Engineering Optimization in Design Processes, Springer-Verlag, Berlin, 1991.
7. M. K. H. FAN, A. L. TITS, J. L. ZHOU, L. S. WANG, AND J. KONINCKX, CONSOL-OPTCAD User's Manual (Version c1.2-.s1.7, Released 8/1992), Tech. report 87-212r4, Institute for Systems Research, University of Maryland, College Park, MD, 1992.
8. A. V. FIACCO, Introduction to sensitivity and stability analysis in nonlinear programming, in Academic Press, New York, London, R. Bellman, ed., Mathematics in Science and Engineering 165, 1983.
9. P. E. GILL, W. MURRAY, M. A. SAUNDERS, AND M. H. WRIGHT, User's Guide for QPSOL (Version 3.2): A Fortran Package for Quadratic Programming, Tech. report SOL 84-6, Systems Optimization Laboratory, Stanford University, Stanford, CA, 1984.
10. C. GONZAGA AND E. POLAK, On constraint dropping schemes and optimality functions for a class of outer approximations algorithms, SIAM J. Control Optim., 17 (1979), pp. 477-493.
11. C. GONZAGA, E. POLAK, AND R. TRAHAN, An improved algorithm for optimization problems with functional inequality constraints, IEEE Trans. Automat. Control, AC-25 (1980), pp. 49-54.
12. G. GRAMLICH, R. HETTICH, AND E. W. SACHS, Local convergence of SQP methods in semi-infinite programming, SIAM J. Optim., 5 (1995), pp. 641-658.
13. S. A. GUSTAFSON, A three-phase algorithm for semi-infinite programs, in Semi-Infinite Programming and Applications, A. V. Fiacco and K. O. Kortanek, eds., Lecture Notes in Economics and Mathematical Systems 215, Springer-Verlag, New York, Heidelberg, Berlin, 1983, pp. 138-157.
14. S. P. HAN, Superlinear Convergence of a Minimax Method, Tech. report 78-336, Department of Computer Science, Cornell University, Ithaca, NY, 1978.
15. S. P. HAN, Variable metric methods for minimizing a class of nondifferentiable functions, Math. Programming, 20 (1981), pp. 1-13.
16. R. HETTICH, An implementation of a discretization method for semi-infinite programming, Math. Programming, 34 (1986), pp. 354-361.
17. R. HETTICH AND W. VAN HONSTEDE, On quadratically convergent methods for semi-infinite programming, in Semi-Infinite Programming, R. Hettich, ed., Lecture Notes in Control and Information Sciences 15, Springer-Verlag, Berlin, New York, 1979, pp. 97-111.
18. R. HETTICH AND K. O. KORTANEK, Semi-infinite programming: Theory, methods, and applications, SIAM Rev., 35 (1993), pp. 380-429.
19. R. HETTCIH AND P. ZENCKE, Numerische Methoden der Approximation und Semi-Infiniten Optimierung, Teubner Studienbücher, Leipzig, Stuttgart, 1982.
20. W. VAN HONSTEDE, An approximation method for semi-infinite problems, in Semi-Infinite Programming, R. Hettich, ed., Lecture Notes in Control and Information Sciences 15, Springer-Verlag, Berlin, New York, 1979, pp. 126-136.
21. K. C. KIWIEL, Methods of Descent in Nondifferentiable Optimization, in Lecture Notes in Mathematics 1133, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.
22. C. T. LAWRENCE, J. L. ZHOU AND A. L. TITS, User's Guide for CFSQP Version 2.0: A C Code for Solving Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality and Linear Constraints, Tech. report 94-16, Institute for Systems Research, University of Maryland, College Park, MD, 1994.
23. C. LEMARÉCHAL, Nondifferentiable optimization, in Optimization, G. Nemhauser, A. Rinnooy-Kan, and M. Todd, eds., Handbooks in Operations Research and Management Science, North-Holland, Amsterdam, 1989.
24. H. MINE, M. FUKUSHIMA, AND Y. TANAKA, On the use of e-most-active constraints in an exact penalty function method for nonlinear optimization, IEEE Trans. Automat. Control, AC-29 (1984), pp. 1040-1042.
25. C. A. MOTA SCARES, ED., Computer-Aided Optimal Design: Structural and Mechanical Systems, NATO ASI Series, Series F, 27, Springer-Verlag, Berlin, 1987.
26. K. OETTERSHAGEN, Ein Superlinear Konvergenter Algorithmus zur Losung Semi-Infiniter Optimierungsprobleme, Ph.D. thesis, Bonn University, Bonn, Germany, 1982.
27. E. R. PANIER AND A. L. TITS, A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design, IEEE Trans. Automat. Control, AC-34 (1989), pp. 903-908.
28. _, A superlinearly convergent method of feasible directions for optimization problems arising in the design of engineering systems, in Proc. 7th International Conf. on Analysis and Optimization of Systems, Antibes, 1986, A. Bensoussan and J. L. Lions, eds., Lecture Notes in Control and Information Sciences 83, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, June 1986, pp. 65-73.
29. E. POLAK, Computational Methods in Optimization, Academic Press, New York, 1971.
30. _, On the mathematical foundations of nondifferentiable optimization, SIAM Rev., 29 (1987), pp. 21-89.
31. E. POLAK AND L. HE, Rate preserving discretization strategies for semi-infinite programming and optimal control, in Proc. 29th IEEE Conf. on Decision and Control, 1990, pp. 2444-2449.
32. E. POLAK AND D. Q. MAYNE, An algorithm for optimization problems with functional inequality constraints, IEEE Trans. Automat. Control, 21 (1976), pp. 184-193.
33. E. POLAK, D. Q. MAYNE, AND D. M. STIMLER, Control system design via semi-infinite optimization: A review, IEEE Proc., 72 (1984), pp. 1777-1794.
34. E. POLAK AND A. L. TITS, A recursive quadratic programming algorithm for semi-infinite optimization problems, Appl. Math. Optim., 8 (1982), pp. 325-349.
35. M. J. D. POWELL, A fast algorithm for nonlinearly constrained optimization calculations, in Numerical Analysis, Dundee, 1977, Lecture Notes in Mathematics 630, G. A. Watson, ed., Springer-Verlag, Berlin, New York, 1978, pp. 144-157.
36. _, A tolerant algorithm for linearly constrained optimization calculations, Math. Programming, 45 (1989), pp. 547-566.
37. J. RAKOWSKA, R. T. HAFTKA, AND L. T. WATSON, Multi-objective control-structure optimization via homotopy methods, SIAM J. Optim., 3 (1993), pp. 654-667.
38. R. REEMTSEN, Discretization methods for the solution of semi-infinite programming problems, J. Optim. Theory Appl., 71 (1991), pp. 85-103.
39. S. M. ROBINSON, Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms, Math. Programming, 7 (1974), pp. 1-16.
40. K. SCHITTKOWSKI, QLD: A FORTRAN Code for Quadratic Programming, User's Guide, Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany, 1986.
41. _, Solving Nonlinear Programming Problems with Very Many Constraints, Report No. 294, Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany, 1991.
42. G. A. WATSON, Globally convergent methods for semi-infinite programming, BIT, 21 (1981), pp. 362-373.
43. W. ZHAO AND S. AZARM, A cross-sectional shape multiplier method for two-level optimum design of frames, in Advances in Design Automation, Volume II-Optimal Design and Mechanical Systems Analysis, ASME Publication DE, 23-2, September 1990, pp. 197-205.
44. J. L. ZHOU, Fast, Globally Convergent Optimization Algorithms, with Application to Engineering System Design, Ph.D. thesis, University of Maryland, College Park, MD, 1992.
45. J. L. ZHOU AND A. L. TITS, Nonmonotone Line Search for Minimax Problems, J. Optim. Theory Appl., 76 (1993), pp. 455-476.
46. G. ZOUTENDIJK, Methods of Feasible Directions, Elsevier Scientific Publishing Company, Amsterdam, The Netherlands, 1960.