On the Numbers of Connected Components in the Solution Sets of Polynomial Vector Variational Inequalities
- 25 Downloads
In this paper, we establish explicit upper bounds for the number of connected components in the proper Pareto solution sets and the weak Pareto solution sets of polynomial vector variational inequalities. Consequently, upper bounds for the numbers of connected components in the stationary point sets, the proper stationary point sets, and the weak Pareto solution sets of polynomial vector optimization problems are obtained.
KeywordsPolynomial vector variational inequality Polynomial vector optimization Solution set Semi-algebraic set Number of connected components
Mathematics Subject Classification90C29 90C33 49J40 14P10
The author is indebted to Professor Nguyen Dong Yen for many stimulating conversations. The author would like to thank the referees for their valuable comments which helped to improve the manuscript.
- 2.Hieu, V.T.: The Tarski–Seidenberg theorem with quantifiers and polynomial vector variational inequalities. https://arxiv.org/abs/1803.00201 (2018)
- 3.Coste, M.: An Introduction to Semialgebraic Geometry. Institut de Recherche Mathématique de Rennes, Université de Rennes, Rennes (2002)Google Scholar
- 4.Giannessi, F.: Theorems of alternative, quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.-L. (eds.) Variational Inequality and Complementarity Problems, pp. 151–186. Wiley, New York (1980)Google Scholar