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Approximation Methods for Polynomial Optimization

Part of the book series: SpringerBriefs in Optimization ((BRIEFSOPTI))

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Abstract

Polynomial optimization is to optimize a polynomial function subject to polynomial equality and/or inequality constraints, specifically, the following generic optimization model:

$$\begin{array}{lll} (PO)&\min &p(\mathbf{x}) \\ &\text{ s.t.}&{f}_{i}(\mathbf{x}) \leq 0,\,i = 1,2,\ldots, {m}_{1}, \\ & &{g}_{j}(\mathbf{x}) = 0,\,j = 1,2,\ldots, {m}_{2}, \\ & &\mathbf{x} = {({x}_{1},{x}_{2},\ldots, {x}_{n})}^{\text{ T}} \in {\mathbb{R}}^{n}, \end{array}$$

where p(x), f i (x) (i = 1, 2, …, m 1) and g j (x) (j = 1, 2, …, m 2) are some multivariate polynomial functions. This problem is a fundamental model in the field of optimization, and has applications in a wide range of areas. Many algorithms have been proposed for subclasses of (PO), and specialized software packages have been developed.

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  1. Department of Mathematics, Shanghai University, Shanghai, China

    Zhening Li

  2. Department of Management Sciences, City University of Hong Kong, Kowloon Tong, Hong Kong

    Simai He

  3. Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN, USA

    Shuzhong Zhang

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Li, Z., He, S., Zhang, S. (2012). Introduction. In: Approximation Methods for Polynomial Optimization. SpringerBriefs in Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3984-4_1

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