[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

A homotopy method based on penalty function for nonlinear semidefinite programming

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

This paper proposes a homotopy method based on a penalty function for solving nonlinear semidefinite programming problems. The penalty function is the composite function of an exponential penalty function, the eigenvalue function and a nonlinear operator mapping. Representations of its first and second order derivatives are given. Using the penalty function, a new homotopy is constructed. Global convergence of a smooth curve determined by the homotopy is proven under mild conditions. In the process of numerically tracing the curve, the method requires just the solution of a linear system of dimension \(n+2\), whereas a homotopy method proposed by Yang and Yu (Comput Optim Appl 56(1):81–96, 2013) requires a system of dimension \(n+m(m+1)/2+1\) to be solved, where \(n\) is the number of variables while \(m\) is the order of constraint matrix. So, it is expected that the proposed method can improve the efficiency of the method proposed by Yang and Yu. Preliminary numerical experiments are presented and show that the considered algorithm is efficient for some nonlinear semidefinite programming problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R., Robbin, J.: Transversal Mappings and Flows. W. A. Benjamin Inc, New York-Amsterdam (1967)

    MATH  Google Scholar 

  2. Alizadeh, F., Haeberly, J.P.A., Nayakkankuppam, M.V., Overton, M.L., Schmieta, S.: SDPPACK user’s guide. Tech. rep. Courant Institute of Mathematical Sciences, New York University, New York, NY (1997)

  3. Alizadeh, F., Haeberly, J.P.A., Overton, M.L.: Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM J. Optim. 8(3), 746–768 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allgower, E.L., Georg, K.: Numerical Path Following. North-Holland, Amsterdam (1997)

    Book  Google Scholar 

  5. Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Classics in Applied Mathematics, vol. 45. SIAM, Philadelphia, PA (2003)

    Book  Google Scholar 

  6. Auslender, A.: Penalty and barrier methods: a unified framework. SIAM J. Optim. 10(1), 211–230 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-Tal, A., Teboulle, M.: A smoothing technique for nondifferentiable optimization problems. In: Optimization (Varetz, 1988), Lecture Notes in Math., vol. 1405, pp. 1–11. Springer, Berlin (1989)

  8. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)

    Book  Google Scholar 

  9. Chen, X., Qi, H., Qi, L., Teo, K.L.: Smooth convex approximation to the maximum eigenvalue function. J. Glob. Optim. 30(2), 253–270 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeroes of maps: homotopy methods that are constructive with probability one. Math. Comput. 32(143), 887–899 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Correa, R., Ramirez, C.H.: A global algorithm for nonlinear semidefinite programming. SIAM J. Optim. 15(1), 303–318 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. de Klerk, E.: Aspects of Semidefinite Programming. Kluwer Academic Publishers, Dordrecht (2002)

    Book  MATH  Google Scholar 

  13. Fares, B., Noll, D., Apkarian, P.: Robust control via sequential semidefinite programming. SIAM J. Control Optim. 40(6), 1791–1820 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Forsgren, A.: Optimality conditions for nonconvex semidefinite programming. Math. Program. 88(1), 105–128 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Freund, R.W., Jarre, F., Vogelbusch, C.H.: Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling. Math. Program. 109(2–3), 581–611 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gómez, W., Ramírez, H.: A filter algorithm for nonlinear semidefinite programming. Comput. Appl. Math. 29(2), 297–328 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Henrion, D., Löfberg, J., Kočvara, M., Stingl, M.: Solving polynomial static output feedback problems with PENBMI. In: In IEEE (ed.) Proceedings of the 44th IEEE Conference on Decision and Control, Sevilla, Spain, vol. 1, pp. 7581–7586 (2005)

  18. Jarre, F.: An interior method for nonconvex semidefinite programs. Optim. Eng. 1(4), 347–372 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kanzow, C., Nagel, C., Kato, H., Fukushima, M.: Successive linearization methods for nonlinear semidefinite programs. Comput. Optim. Appl. 31(3), 251–273 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kočvara, M., Stingl, M.: Pennon: a code for convex nonlinear and semidefinite programming. Optim. Methods Softw. 18(3), 317–333 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leibfritz, F., Mostafa, E.M.E.: An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems. SIAM J. Optim. 12(4), 1048–1074 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Liqun Qi, P.T.: On almost smooth functions and piecewise smooth functions. Nonlinear Analysis: Theory, Methods & Applications 67(3), 773–794 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Liuzzi, G., Lucidi, S., Sciandrone, M.: A derivative-free algorithm for linearly constrained finite minimax problems. SIAM J. Optim. 16(4), 1054–1075 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Luo, H., Wu, H., Chen, G.: On the convergence of augmented Lagrangian methods for nonlinear semidefinite programming. J. Glob. Optim. 54(3), 599–618 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Noll, D.: Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optim. Methods Softw. 22(5), 777–802 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Noll, D., Apkarian, P.: Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods. Math. Program. 104(2–3), 729–747 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Overton, M.L.: Large-scale optimization of eigenvalues. SIAM J. Optim. 2(1), 88–120 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Overton, M.L., Womersley, R.S.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Program. 62(2, Ser. B), 321–357 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Program. 77(2), 301–320 (1997)

    MATH  Google Scholar 

  30. Stingl, M.: On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods. Ph.D. thesis, Institute of Applied Mathematics II. Friedrich–Alexander University of Erlangen-Nuremberg (2006)

  31. Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12(1–4), 625–653 (1999)

    Article  MathSciNet  Google Scholar 

  32. Sun, D., Sun, J., Zhang, L.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. 114(2), 349–391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95(2), 189–217 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming. International Series in Operations Research & Management Science, 27. Kluwer Academic Publishers, Boston, MA (2000)

    Google Scholar 

  36. Wu, H., Luo, H., Ding, X., Chen, G.: Global convergence of modified augmented lagrangian methods for nonlinear semidefinite programming. Comput. Optim. Appl. 56(3), 531–558 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xiao, Y., Yu, B.: A truncated aggregate smoothing Newton method for minimax problems. Appl. Math. Comput. 216(6), 1868–1879 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xiong, Hj, Yu, B.: An aggregate deformation homotopy method for min–max–min problems with max–min constraints. Comput. Optim. Appl. 47(3), 501–527 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20(3), 267–279 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yamashita, H., Yabe, H.: Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming. Math. Program. 132, 1–30 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Yamashita, H., Yabe, H., Harada, K.: A primal-dual interior point method for nonlinear semidefinite programming. Math. Program. 135, 89–121 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, L., Yu, B.: A homotopy method for nonlinear semidefinite programming. Comput. Optim. Appl. 56(1), 81–96 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhao, X.Y., Sun, D., Toh, K.C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20(4), 1737–1765 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the anonymous referees, whose comments and suggestions led to an improved version of this article.

Author information

Authors and Affiliations

  1. School of Science, Dalian University of Technology, Dalian, 116024, China

    Li Yang

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Bo Yu

  3. Faculty of Management and Economics, Dalian University of Technology, Dalian, 116024, China

    YanXi Li

Authors
  1. Li Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. YanXi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Yang.

Additional information

The work was supported by the National Natural Science Foundation of China (11301050, 11171051, 91230103, 71172136).

Appendix: The gradient and Hessian of \(g_\theta (x,\mu )\)

Appendix: The gradient and Hessian of \(g_\theta (x,\mu )\)

For the convenience of the reader, representations of the gradient and Hessian of \(g_\theta (x,\mu )\) will be derived below. From Theorem 2, we have that

$$\begin{aligned} \frac{\partial \mathrm{exp}G(x)}{\partial x_i}&= Q(x)\left( \left[ \varDelta \varphi \left( \lambda _k(x),\lambda _l(x)\right) \right] _{k,l=1}^m\circledast \left[ Q(x)^TG'_i(x)Q(x)\right] \right) Q(x)^T,\\ \frac{\partial \mathrm{exp}\frac{G(x)}{\theta \mu }}{\partial x_i}&= \frac{1}{\theta \mu } Q(x)\left( \left[ \varDelta \varphi (\lambda _k(x)/\theta \mu ,\lambda _l(x)/\theta \mu )\right] _{k,l=1}^m\right. \\&\left. \circledast [Q(x)^TG'_i(x)Q(x)]\right) Q(x)^T, \end{aligned}$$
$$\begin{aligned} \frac{\partial \mathrm{Tr}\left( \mathrm{exp}G(x)\right) }{\partial x_i}&= \mathrm{Tr}\left( Q(x)\left[ \left( \varDelta \varphi \left( \lambda _k(x),\lambda _l(x)\right) \right) _{k,l=1}^m\circledast \left( Q(x)^TG'_i(x)Q(x)\right) \right] Q(x)^T\right) \\&= \mathrm{Tr}\left( \left( \varDelta \varphi \left( \lambda _k(x),\lambda _l(x)\right) \right) _{k,l=1}^m\circledast \left[ Q(x)^TG'_i(x)Q(x)\right] \right) \\&= \mathrm{Tr}\left( \mathrm{Diag}(\mathrm{exp}\lambda (x)) Q(x)^TG'_i(x)Q(x)\right) \\&= \mathrm{Tr}\left( Q(x)\mathrm{Diag}(\mathrm{exp}{\lambda (x)}) Q(x)^TG'_i(x)\right) \\&= \mathrm{Tr}\left( G'_i(x)\mathrm{exp}G(x)\right) , \end{aligned}$$
$$\begin{aligned} \frac{\partial \mathrm{Tr}\left( \mathrm{exp}\frac{G(x)}{\theta \mu }\right) }{\partial x_i}&= \mathrm{Tr}\left( Q(x)\left( [\varDelta \varphi (\lambda _k(x)/\theta \mu ,\lambda _l(x)/\theta \mu )]_{k,l=1}^m\right. \right. \\&\left. \left. \circledast [Q(x)^TG'_i(x)Q(x)]\right) Q(x)^T\right) {\Big /}\theta \mu \\&= \mathrm{Tr}\left( [\varDelta \varphi (\lambda _k(x)/\theta \mu ,\lambda _l(x)/\theta \mu )]_{k,l=1}^m\circledast [Q(x)^TG'_i(x)Q(x)]\right) {\Big /}\theta \mu \\&= \mathrm{Tr}\left( \mathrm{Diag}\left( \mathrm{exp}{\frac{\lambda (x)}{\theta \mu }}\right) Q(x)^TG'_i(x)Q(x)\right) {\Big /}\theta \mu \\&= \mathrm{Tr}\left( Q(x)\mathrm{Diag}\left( \mathrm{exp}{\frac{\lambda (x)}{\theta \mu }}\right) Q(x)^TG'_i(x)\right) {\Big /}\theta \mu \\&= \mathrm{Tr}\left( G'_i(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) {\Big /}\theta \mu , \end{aligned}$$

where \(\varphi (\alpha )=\mathrm{exp}\,\alpha \), \(\alpha \in \mathbb {R}\). Consequently, the gradient and Hessian of \(g_\theta (x,\mu )\) about \(x\) and \(\mu \) are as follows:

$$\begin{aligned} \nabla _xg_\theta (x,\mu )=\frac{G'(x)^*\mathrm{exp}\frac{G(x)}{\theta \mu }}{\mathrm{Tr}(\mathrm{exp}\frac{G(x)}{\theta \mu })}, \end{aligned}$$
$$\begin{aligned} \nabla _\mu g_\theta (x,\mu )&= \theta \mathrm{ln}\,\mathrm{Tr}\left( \mathrm{exp}\frac{G(x)}{\theta \mu }\right) -\mathrm{Tr}\left( G(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) {\Big /}\mathrm{Tr}\left( \mu \;\mathrm{exp}\frac{G(x)}{\theta \mu }\right) ,\\ \nabla _{x\mu }^2g_\theta (x,\mu )&= \frac{\mathrm{Tr}\left( G(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) G'(x)^*\mathrm{exp}\frac{G(x)}{\theta \mu }}{\theta \mu ^2\left( \mathrm{Tr}\left( \mathrm{exp}\frac{G(x)}{\theta \mu }\right) \right) ^2}- \frac{G'(x)^*\left( G(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) }{\theta \mu ^2\mathrm{Tr}\left( \mathrm{exp}\frac{G(x)}{\theta \mu }\right) },\\ \frac{\partial ^2g_\theta (x,\mu )}{\partial x_i\partial x_j}&= \left[ \mathrm{Tr}\left( G_{ij}''(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) +\mathrm{Tr}\left( G_i'(x)\frac{\partial \mathrm{exp}\frac{G(x)}{\theta \mu }}{\partial x_j}\right) \right] {\Big /}\mathrm{Tr}\left( \mathrm{exp}\frac{G(x)}{\theta \mu }\right) \\&-\mathrm{Tr}\left( G_i'(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) \mathrm{Tr} \left( G_j'(x)\mathrm{exp}\frac{G(x)}{\theta \mu }\right) {\Big /} \left( \!\theta \mu \left[ \mathrm{Tr}\left( \mathrm{exp}\frac{G(x)}{\theta \mu }\!\right) \!\right] ^2\!\right) \!, \end{aligned}$$

where \(G_{ij}''(x)=\frac{\partial ^2 G(x)}{\partial x_ix_j}\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, L., Yu, B. & Li, Y. A homotopy method based on penalty function for nonlinear semidefinite programming. J Glob Optim 63, 61–76 (2015). https://doi.org/10.1007/s10898-015-0276-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-015-0276-5

Keywords

Search

Navigation