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A homotopy method based on penalty function for nonlinear semidefinite programming

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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.

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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
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  2. Bo Yu
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  3. YanXi Li
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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 \(G_{ij}''(x)=\frac{\partial ^2 G(x)}{\partial x_ix_j}\).

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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

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