![The feasible sets K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{K}$$\end{document} (enclosed by the green curves) and their SDr approximations Λ6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _6$$\end{document} (the gray areas enclosed by the red curves) in Problem (19)–(22) (color figure online)](https://www.researchgate.net/publication/367497800/figure/fig1/AS:11431281216904158@1704943225735/The-feasible-sets-Kdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
January 2023
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1 Citation
Optimization Letters
In this paper, we study a class of fractional semi-infinite polynomial programming problems involving sos-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre’s measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the considered problem. This method enables us to compute convergent lower bounds of the optimum and extract approximate minimizers. Moreover, for a set defined by infinitely many sos-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations (SDr). The convergence rate of the lower bounds and outer SDr approximations are also discussed.
