A property of bicriteria affine vector variational inequalities

Abstract

By a scalarization method, it is proved that both the Pareto solution set and the weak Pareto solution set of a bicriteria affine vector variational inequality have finitely many connected components provided that a regularity condition is satisfied. An explicit upper bound for the numbers of connected components of the Pareto solution set and the weak Pareto solution set is obtained. Consequences of the results for bicriteria quadratic vector optimization problems and linear fractional vector optimization problems are discussed in detail. Under an additional assumption on the data set, Theorems 3.1 and 3.2 in this article solve in the affirmative Question 1 in Yen and Yao [N.D. Yen and J.-C. Yao, Monotone affine vector variational inequalities, Optimization 60 (2011), pp. 53–68] and Question 9.3 in Yen [N.D. Yen, Linear fractional and convex quadratic vector optimization problems, in Recent Developments in Vector Optimization, Q.H. Ansari and J.-C. Yao, eds, Springer-Verlag, New York, 2012, pp. 297–328] for the case of bicriteria problems without requiring the monotonicity. Besides, the theorems also give a partial solution to Question 2 in Yen and Yao (2011) about finding an upper bound for the numbers of connected components of the solution sets under investigation.

Keywords:

• bicriteria affine vector variational inequality
• scalarization
• solution set
• connectedness
• number of connected components

AMS Subject Classifications::

• 49J40
• 49J53
• 90C29

Acknowledgements

The authors were supported by the National Foundation for Science and Technology Development (NAFOSTED), Vietnam. They are grateful to the referees for their careful reading and valuable suggestions which have helped to improve this article substantially.