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A solution method for combined semi-infinite and semi-definite programming

Published online by Cambridge University Press:  17 February 2009

S. J. Li ,
X. Q. Yang ,
K. L. Teo  and
S. Y. Wu
S. J. Li
Affiliation:
Department of Information and Computer Sciences, College of Sciences, Chongqing University, Chongqing, 400044, China; e-mail: lisj@cqu.edu.cn.
X. Q. Yang
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong; e-mail: mayangxq@polyu.edu.hk and mateokl@polyu.edu.hk.
K. L. Teo
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong; e-mail: mayangxq@polyu.edu.hk and mateokl@polyu.edu.hk.
S. Y. Wu
Affiliation:
Institute of Applied Mathematics, National Cheng-Kung University, Tainan 700, Taiwan; e-mail: soonyi@mail.ncku.edu.tw.
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Abstract

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In this paper, we develop a discretisation algorithm with an adaptive scheme for solving a class of combined semi-infinite and semi-definite programming problems. We show that any sequence of points generated by the algorithm contains a convergent subsequence; and furthermore, each accumulation point is a local optimal solution of the combined semi-infinite and semi-definite programming problem. To illustrate the effectiveness of the algorithm, two specific classes of problems are solved. They are relaxations of quadratically constrained semi-infinite quadratic programming problems and semi-infinite eigenvalue problems.

Type
Research Article
Information
The ANZIAM Journal , Volume 45 , Issue 4 , April 2004 , pp. 477 - 494
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Alizadeh, F., “Interior point methods in semidefinite programming with applications to combinatorial optimization”, SIAM J. Optim. 5 (1995) 13–51.CrossRefGoogle Scholar
[2]Anderson, E. J. and Lewis, A. S., “An extension of the simplex method for semi-infinite programming”, Math. Program. 44 (1989) 247–269.CrossRefGoogle Scholar
[3]Anderson, E. J. and Nash, P., Linear programming infinite-dimensional spaces (John Wiley & Sons, Chichester, 1987).Google Scholar
[4]Duffin, R. J., Jeroslow, R. G. and Karlovitz, L. A., “Duality in semi-infinite linear programming”, in Semi-infinite programming and applications (eds. Fiacco, A. V. and Kortanek, K. O.), Lecture Notes in Economics and Mathematical Systems 215, (Springer, Berlin, 1983) 50–62.CrossRefGoogle Scholar
[5]Goberna, M. A. and Lopez, M. A., Linear semi-infinite optimization (John Wiley & Sons, Chichester, 1998).Google Scholar
[6]Helmberg, C., Rendl, F., Vanderbei, R. J. and Wolkowicz, H., “An interior-point method for semidefinite programming”, SIAM J. Optim. 6 (1996) 342–361.CrossRefGoogle Scholar
[7]Hettich, R. and Kortanek, K. O., “Semi-infinite programming: theory, methods, and applications”, SIAM Rev. 35 (1993) 380–429.CrossRefGoogle Scholar
[8]Jarre, F., “An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices”, SIAM J. Control Optim. 31 (1993) 1360–1377.CrossRefGoogle Scholar
[9]Jeyakumar, V. and Gwinner, J., “Inequality systems and optimization”, J. Math. Anal. Appl. 159 (1991) 51–71.CrossRefGoogle Scholar
[10]Jeyakumar, V. and Wolkowicz, H., “Zero duality gaps in infinite-dimensional programming”, J. Optim. Theory Appl. 67 (1990) 87–108.CrossRefGoogle Scholar
[11]Kaplan, A. and Tichatschke, R., “Proximal interior point methods for convex semi-infinite programming”, Optim. Methods Softw. 15 (2001) 87–119.Google Scholar
[12]Kelley, J. L. and Namtoka, I., Linear topological spaces (Springer, New York, 1963).CrossRefGoogle Scholar
[13]Kortanek, K. O. and Zhang, Q., “Perfect duality in semi-infinite and semidefinite programming”, Math. Program. 91 (2001) 127–144.CrossRefGoogle Scholar
[14]Li, S. J., Teo, K. L., Yang, X. Q. and Wu, S. Y., “Robust envelope-constrained filter with orthonorinal bases and semi-definite and semi-infinite programming”, IEEE Trans. Circuits Sys.-1, submitted.Google Scholar
[15]Nayakkankuppam, M. V. and Overton, M. L., “Conditioning of semidefinite programs”, Math. Program. Series A 85 (1999) 525–540.CrossRefGoogle Scholar
[16]Nesterov, Y. and Nemirovskii, A., Interior-point polynomial algorithms in convex programming (SIAM, Philadelphia, 1994).CrossRefGoogle Scholar
[17]Pardalos, P. M., “Quadratic programming with one negative eigenvalue is NP-hard”, J. Global Optim. 1 (1991) 15–22.CrossRefGoogle Scholar
[18]Pataki, G., “Cone-LP's and semidefinite programs: Geometry and a simplex-type method”, in Integer programming and combinational optimization (Vancouver BC, 1996), (Springer, 1996) 162–174.CrossRefGoogle Scholar
[19]Polak, E., Optimization: algorithm and consistent applications (Springer, New York, 1997).CrossRefGoogle Scholar
[20]Potra, F. A. and Sheng, R., “A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming”, SIAM J. Optim. 8 (1998) 1007–1028.Google Scholar
[21]Ramana, M., Tuncel, L. and Wolkowicz, H., “Strong duality for semidefinite programming”, SIAM J. Optim. 7 (1997) 641–662.CrossRefGoogle Scholar
[22]Reemtsen, R., “Some outer approximation methods for semi-infinite optimization problems”, J. Comput. Appl. Math. 53 (1994) 87–108.CrossRefGoogle Scholar
[23]Reemtsen, R. and Ruckmann, J. J., Semi-infinite programming (Kluwer Academic Publisher, Drodrecht, 1998).CrossRefGoogle Scholar
[24]Rendl, F., Vanderbei, R. J. and Wolkowicz, H., “Max-min eigenvalue problems, primal-dual interior point algorithms, and trust region subproblems”, Optim. Methods Softw. 5 (1995) 1–16.CrossRefGoogle Scholar
[25]Teo, K. L., Yang, X. Q. and Jennings, L. S., “Computational discretization algorithms for functional inequality constrained optimization”, Ann. Oper. Res. 98 (2000) 215–234.CrossRefGoogle Scholar
[26]Todd, M. J., “Semidefinite optimization”, Acta Numer. 10 (2001) 515–560.Google Scholar
[27]Vandenberghe, L. and Boyd, S., “Semidefinite programming”, SIAM Rev. 38 (1996) 49–95.CrossRefGoogle Scholar
[28]Vavasis, S., Nonlinear optimization (Oxford University Press, New York, 1991).Google Scholar
[29]Wolkowicz, H., Saigal, R. and Vandenberghe, L., Handbook of semidefinite programming theory, algorithms, and applications (Kluwer Academic Publishers, Dordrecht, 2000).Google Scholar
[30]Wu, S. Y., Fang, S. C. and Lin, C. J., “Relaxed cutting plane method for solving linear semi-infinite programming problems”, J. Optim. Theory Appl. 99 (1998) 759–779.CrossRefGoogle Scholar
[31]Wu, S. Y., Fang, S. C. and Lin, C. J., “Solving quadratic semi-infinite programming problems by using relaxed cutting plane scheme”, J. Comput. Appl. Math. 129 (2001) 89–104.Google Scholar
[32]Yang, X. Q. and Teo, K. L., “Nonlinear Lagrangian functions and applications to semi-infinite programs”, Ann. Oper. Res. 103 (2001) 235–250.CrossRefGoogle Scholar