Mathematical Programming

, Volume 147, Issue 1–2, pp 519–538 | Cite as

A Frank–Wolfe type theorem for nondegenerate polynomial programs

  • Si Tiep Dinh
  • Huy Vui Ha
  • Tien Son PhamEmail author
Full Length Paper Series A


In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let \(f_0\) and \(f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be convenient polynomial functions, and let \(S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .\) Under the assumption that the map \((f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}\) is non-degenerate at infinity, we show that if \(f_0\) is bounded from below on \(S,\) then \(f_0\) attains its infimum on \(S.\)


Existence of optimal solutions Frank–Wolfe type theorem  Newton polyhedron Nondegenerate polynomial programs 

Mathematics Subject Classification (1991)

90C26 90C30 



The authors would like to thank the referees, whose helpful comments and suggestions much improved the original manuscript. This research was performed while the authors had been visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the Institute for hospitality and support.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of MathematicsUniversity of DalatDalatVietnam

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