Abstract
Yuan’s theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan’s theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially nonpositive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.
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Acknowledgments
The authors would like to express their sincere thanks to the referees for their constructive comments and valuable suggestions, which have contributed to the revision of this paper. Moreover, the second author would like to thank Prof. J. B. Lasserre and Prof. T. S. Pham for pointing out the related references [50, 51] during their visit in UNSW. Research was partially supported by the Australian Research Council Future Fellowship (FT130100038) and the Hong Kong Research Grant Council (Grant Nos. PolyU 502510, 502111, 501212, and 501913), and the National Natural Science Foundation of China (Grant No. 11401428 and 11101303).
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Appendix
Appendix
Proof of Proposition 2.1
Proof
As any SOS polynomial takes non-negative value, \(\mathrm{SOS}_{m,n} \cap E_{m,n} \subseteq \mathrm{PSD}_{m,n} \cap E_{m,n}\) always holds. We only need to show the converse inclusion. To establish this, let \(\mathcal {A} \in \mathrm{PSD}_{m,n} \cap E_{m,n}\), and let us consider the associated homogeneous polynomial:
Then, \(f\) is a polynomial which takes non-negative value. Note that
where \(I:=\{(i,i,\ldots ,i) \in \mathbb {N}^m: 1 \le i \le n\}.\) As \(\mathcal {A}\) is essentially nonpositive, \(\mathcal {A}_{i_1i_2\cdots i_m} \le 0\) for all \((i_1,\ldots ,i_m) \notin I\). Now, let \(f(x)=\sum _{i=1}^n f_{m,i} x_i^{m}+\sum _{\alpha \in \Omega _f}f_{\alpha }x^{\alpha }\). Then, \(f_{m,i}=\mathcal {A}_{ii\cdots i}\) and \(f_{\alpha } < 0\) for all \(\alpha \in \Omega _f\) where \(\Omega _f=\{\alpha =(\alpha _1,\ldots ,\alpha _n) \in (\mathbb {N}\cup \{0\})^n: f_{\alpha } \ne 0 \text { and } \alpha \ne m e_i, \ i=1,\ldots ,n\},\) and \(e_i\) is the vector where its \(i\)th component is one and all the other components are zero. Recall that \(\Delta _f = \{\alpha =(\alpha _1,\ldots ,\alpha _n) \in \Omega _f: f_{\alpha } < 0 \text { or } \alpha \notin (2\mathbb {N}\cup \{0\})^n\}.\) Note that \(f_{\alpha }<0\) for all \(\alpha \in \Omega _f\), and so, \(\Delta _f=\Omega _f\). It follows that
So, \(\hat{f}\) is also a polynomial which takes non-negative value. Thus, the conclusion follows by Lemma 2.1. \(\square \)
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Hu, S., Li, G. & Qi, L. A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure . J Optim Theory Appl 168, 446–474 (2016). https://doi.org/10.1007/s10957-014-0652-1
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DOI: https://doi.org/10.1007/s10957-014-0652-1
Keywords
- Alternative theorem
- Symmetric tensors
- Nonconvex polynomial optimization
- Sum-of-squares relaxation
- Semidefinite programming