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Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions

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Abstract

This paper presents a canonical dual approach for solving a nonconvex global optimization problem governed by a sum of 4th-order polynomial and a log-sum-exp function. Such a problem arises extensively in engineering and sciences. Based on the canonical duality–triality theory, this nonconvex problem is transformed to an equivalent dual problem, which can be solved easily under certain conditions. We proved that both global minimizer and the biggest local extrema of the primal problem can be obtained analytically from the canonical dual solutions. As two special cases, a quartic polynomial minimization and a minimax problem are discussed. Existence conditions are derived, which can be used to classify easy and relative hard instances. Applications are illustrated by several nonconvex and nonsmooth examples.

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Notes

  1. We use the same definition of the neighborhood as defined in [15] (Note 1 on page 306), i.e., a subset \(\mathcal{X}_0\) is said to be the neighborhood of the critical point \(\bar{\varvec{x}}\) if \(\bar{\varvec{x}}\) is the only critical point in \(\mathcal{X}_0\).

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Authors and Affiliations

  1. School of Applied and Biomedical Sciences, Federation University Australia, Mount Helen, Ballarat, VIC, 3350, Australia

    Yi Chen & David Y. Gao

Authors
  1. Yi Chen
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  2. David Y. Gao
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Correspondence to Yi Chen.

Appendix

Appendix

The following lemma is a generalization of Lemma 6 in [23].

Lemma 7

Suppose that \( \varvec{P}\in \mathbb {R}^{n\times n}\), \( \varvec{U}\in \mathbb {R}^{m\times m}\) and \( \varvec{D}\in \mathbb {R}^{n\times m}\) are given symmetric matrices with

$$\begin{aligned} \varvec{P}=\begin{bmatrix} \varvec{P}_{11}&\quad \! \varvec{P}_{12}\\ \varvec{P}_{21}&\quad \! \varvec{P}_{22} \end{bmatrix} \prec 0, ~~ \varvec{U}=\begin{bmatrix} \varvec{U}_{11}&\quad \! \varvec{0}\\ \varvec{0}&\quad \! \varvec{U}_{22} \end{bmatrix}\succ 0, \,\mathrm{and }\,\varvec{D}=\begin{bmatrix} \varvec{D}_{11}&\quad \! \varvec{0}\\ \varvec{0}&\quad \! \varvec{0} \end{bmatrix}, \end{aligned}$$

where \( \varvec{P}_{11}\), \( \varvec{U}_{11}\) and \( \varvec{D}_{11}\) are \(r\times r\)-dimensional matrices, and \( \varvec{D}_{11}\) is nonsingular. Then,

$$\begin{aligned} \varvec{P}+ \varvec{D}\varvec{U}\varvec{D}^T\preceq 0 \Leftrightarrow - \varvec{D}^T \varvec{P}^{-1} \varvec{D}- \varvec{U}^{-1}\preceq 0. \end{aligned}$$
(46)

Proof

Obviously, \( \varvec{P}+ \varvec{D}\varvec{U}\varvec{D}^T\preceq 0\) is equivalent to

$$\begin{aligned} - \varvec{P}- \varvec{D}\varvec{U}\varvec{D}^T= \begin{bmatrix} - \varvec{P}_{11}- \varvec{D}_{11} \varvec{U}_{11} \varvec{D}_{11}^T&\quad \! - \varvec{P}_{12}\\ - \varvec{P}_{21}&\quad \! - \varvec{P}_{22} \end{bmatrix}\succeq 0. \end{aligned}$$
(47)

By Schur lemma, Eq. (47) is equivalent to

$$\begin{aligned} - \varvec{P}_{11}- \varvec{D}_{11} \varvec{U}_{11} \varvec{D}_{11}^T+ \varvec{P}_{12} \varvec{P}_{22}^{-1} \varvec{P}_{21}\succeq 0 \,\mathrm{and }\,\varvec{P}_{22}\prec 0. \end{aligned}$$
(48)

The inverse of matrix \( \varvec{P}\) is

$$\begin{aligned} \varvec{P}^{-1}= \begin{bmatrix} ( \varvec{P}_{11}- \varvec{P}_{12} \varvec{P}_{22}^{-1} \varvec{P}_{21})^{-1}&\quad \! - \varvec{P}_{11}^{-1} \varvec{P}_{12}( \varvec{P}_{22}- \varvec{P}_{21} \varvec{P}_{11}^{-1} \varvec{P}_{12})^{-1}\\ -( \varvec{P}_{22}- \varvec{P}_{21} \varvec{P}_{11}^{-1} \varvec{P}_{12})^{-1} \varvec{P}_{21} \varvec{P}_{11}^{-1}&\quad \! ( \varvec{P}_{22}- \varvec{P}_{21} \varvec{P}_{11}^{-1} \varvec{P}_{12})^{-1} \end{bmatrix}. \end{aligned}$$

Then, it is easy to prove that \(- \varvec{P}_{11}+ \varvec{P}_{12} \varvec{P}_{22}^{-1} \varvec{P}_{21}\succ 0\). Since \( \varvec{D}_{11}\) is nonsingular and \( \varvec{U}_{11}\succ 0\), we have \( \varvec{D}_{11} \varvec{U}_{11} \varvec{D}_{11}^T\succ 0\). Thus the Eq. (48) is equivalent to

$$\begin{aligned} (- \varvec{P}_{11}+ \varvec{P}_{12} \varvec{P}_{22}^{-1} \varvec{P}_{21})^{-1}\preceq ( \varvec{D}_{11} \varvec{U}_{11} \varvec{D}_{11}^T)^{-1}, \end{aligned}$$
(49)

which is further equivalent to

$$\begin{aligned} \varvec{D}_{11}^T(- \varvec{P}_{11}+ \varvec{P}_{12} \varvec{P}_{22}^{-1} \varvec{P}_{21})^{-1} \varvec{D}_{11}\preceq \varvec{U}_{11}^{-1}. \end{aligned}$$
(50)

Since \( \varvec{D}_{11}^T(- \varvec{P}_{11}+ \varvec{P}_{12} \varvec{P}_{22}^{-1} \varvec{P}_{21})^{-1} \varvec{D}_{11}=- \varvec{D}^T \varvec{P}^{-1} \varvec{D}\) and \( \varvec{U}_{22}\succ 0\), the Eq. (50) is equivalent to

$$\begin{aligned} - \varvec{D}^T \varvec{P}^{-1} \varvec{D}- \varvec{U}^{-1}\preceq 0. \end{aligned}$$
(51)

The lemma is proved. \(\square \)

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Chen, Y., Gao, D.Y. Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions. J Glob Optim 64, 417–431 (2016). https://doi.org/10.1007/s10898-014-0244-5

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