Article

Journal of Optimization Theory and Applications

, Volume 122, Issue 1, pp 1-17

Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities

  • M. BianchiAffiliated withIstituto di Econometria e Matematica per le Applicazioni Economiche, Finanziarie e Attuariali, Università
  • , N. HadjisavvasAffiliated withDepartment of Product and Systems Design Engineering, University of the Aegean
  • , S. SchaibleAffiliated withA.G. Anderson Graduate School of Management, University of California

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Abstract

A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.

Variational inequalities quasimonotone maps pseudomonotone maps coercivity conditions exceptional families of elements