Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T19:38:18.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Semidefinite Optimization Estimating Bounds on Linear Functionals Defined on Solutions of Linear ODEs

Part of: Smooth dynamical systems: general theory Partial differential equations

Published online by Cambridge University Press:  27 May 2016

Guangming Zhou ,
Chao Deng  and
Kun Wu
Guangming Zhou*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Hunan 411105, China
Chao Deng*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
Kun Wu*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
*
*Corresponding author. Email:[email protected] (G. Zhou), [email protected] (C. Deng), [email protected] (K. Wu)
*Corresponding author. Email:[email protected] (G. Zhou), [email protected] (C. Deng), [email protected] (K. Wu)
*Corresponding author. Email:[email protected] (G. Zhou), [email protected] (C. Deng), [email protected] (K. Wu)
Get access

Abstract

In this paper, semidefinite optimization method is proposed to estimate bounds on linear functionals defined on solutions of linear ordinary differential equations (ODEs) with smooth coefficients. The method can get upper and lower bounds by solving two semidefinite programs, not solving ODEs directly. Its convergence theorem is proved. The theorem shows that the upper and lower bounds series of linear functionals discussed can approach their exact values infinitely. Numerical results show that the method is effective for the estimation problems discussed. In addition, in order to reduce calculation amount, Cheybeshev polynomials are applied to replace Taylor polynomials of smooth coefficients in computing process.

Keywords

Semidefinite optimizationboundlinear functionalordinary differential equation

MSC classification

Secondary: 35A24: Methods of ordinary differential equations 37C10: Vector fields, flows, ordinary differential equations 90C22: Semidefinite programming
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016 , pp. 599 - 615
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bellman, R. and Fan, K., On systems of linear inequalities in Hermitian matrix variables, in: Convexity, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, 7 (1963), pp. 1–11.Google Scholar
[2]Alizzdeh, F., Combinatorial Optimization with Interior Point Methods and Semidefinite Matrices, Ph.D. thesis, University of Minnesota, Minneapolis, 1991.Google Scholar
[3]Alizzdeh, F., Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13–51.Google Scholar
[4]Wu, S. P., Boyd, S. and Vandenberghe, L., FIR filter design via semidefinite programming and spectral factorization, Proceeding 35th Conference Decision and Control, (1996), pp. 271–276.Google Scholar
[5]Overton, M. L., On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl., 9 (1998), pp. 256–268.Google Scholar
[6]Peng, H. J. and Rasmussen, L. K., The application of semidefinite programming for detection in CDMA, IEEE Selected Areas in Communication, 19 (2001), pp. 1442–1449.Google Scholar
[7]Lasserre, J. B., Glabal optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), pp. 796–817.CrossRefGoogle Scholar
[8]Lasserre, J. B., Bounds on measures satisfying moment conditions, Ann. Appl. Prob., 12 (2002), pp. 1114–1137.Google Scholar
[9]Lasserre, J. B., Moments, Positive Polynomials and Their Applications, London: Imperial College Press, 2010.Google Scholar
[10]Lasserre, J. B., Robust global optimization with polynomials, Math. Program. Ser. A, 107 (2006), pp. 275–293.Google Scholar
[11]Blekherman, G., There are significantly more nonnegative polynomials than sums of squares, Israel J. Math., 153 (2006), pp. 355–380.Google Scholar
[12]Kuhlmann, S. and Putinar, M., Positive polynomials on fibre products, C. R. Acad. Sci. Paris, Ser. 1, 1344 (2007), pp. 681–684.Google Scholar
[13]Laurent, M., Sums of squares, moment matrices and optimization over polynomials, in Putinar, M. and Sullivant, S. (eds), Emerging Applications of Algebraic Geometry, 149 (2008), pp. 157–271.Google Scholar
[14]Scheiderer, C., Positivity and sums of squares: a guide to some recent results, in Putinar, M. and Sullivant, S. (eds), Emerging Applications of Algebraic Geometry, 149 (2008), pp. 272–324.Google Scholar
[15]Marshall, M., Representation of non-negative polynomials, degree bounds and applications to optimization, Canada J. Math., 61 (2009), pp. 205–221.CrossRefGoogle Scholar
[16]Helton, J. W. and Nie, J. W., Semidefinite representation of convex sets, Math. Program. Ser. A, 122 (2010), pp. 21–64.Google Scholar
[17]Nie, J. W., An exact Jacobian SDP relaxation for polynomial optimization, Math. Program. Ser. A, 137 (2013), pp. 225–255.Google Scholar
[18]Bertsimas, D. and Caramanis, C., Bounds on linear PDEs via semidefinite optimization, Math. Program. Ser. A, 108 (2006), pp. 135–158.Google Scholar
[19]Sturm, J. F. and the Advanced Optimization Laboratory at McMaster University, Canada, SeDuMi version 1.1R3, 2006.Google Scholar