Existence of solutions for extended generalized complementarity problems
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Positivity
Existence of solutions for extended generalized complementarity problems Bijaya Kumar Sahu1 · Ouayl Chadli2 Sabyasachi Pani1
· Ram N. Mohapatra3 ·
Received: 24 November 2019 / Accepted: 11 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we introduce extended generalized complementarity problems in Hausdorff topological vector spaces in duality and study the existence of their solutions. We use a different method than those in literature on the existence of solutions of complementarity problems, which are usually based on arguments from generalized monotonicity. This leads us to obtain new results and improve many existing results in literature. We also prove some existence results for extended generalized complementarity problems in reflexive Banach spaces by means of a Tikhonov regularization procedure under a copositivity assumption and arguments from the recession analysis. Keywords Complementarity problem · Variational inequality · Multi-valued mapping · Copositive mapping · Recession cone · Recession function Mathematics Subject Classification 47H04 · 47H10 · 46N10
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Ouayl Chadli [email protected] Bijaya Kumar Sahu [email protected] Ram N. Mohapatra [email protected] Sabyasachi Pani [email protected]
1
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar 752050, India
2
Department of Economics, Faculty of Economics and Social Sciences, Ibn Zohr University, B.P. 8658, Poste Dakhla, Agadir, Morocco
3
Department of Mathematics, University of Central Florida, Orlando, FL, USA
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B. K. Sahu et al.
1 Introduction The theory of complementarity problems, has been introduced by Lemke [34], Cottle and Dantzig [17] in the early 1960s, it occurs in many applied problems; see for instance [14–16,20,25,29] and the references therein. Many investigators have been concerned with both the computational and the theoretical aspects of the above problem; see e.g. [3,28,31,33,43]. Complementarity problems have been generalized and extended in several directions using novel and innovative techniques. Habetler and Price [26] generalized complementarity problems by replacing the usual nonnegative partial ordering of an n-dimensional Euclidean space E n with partial orderings generated by a given cone and its polar. The problem they consider is described as the following: Given a closed, convex cone K in En , its polar K ∗ and a mapping F from K into En , find a vector x ∈ En such that x ∈ K , F(x) ∈ K ∗ and x T F(x) = 0.
(1)
There are several modifications done in this line; see [5,30,40,43,44,46]. Robinson [41] pointed out that problem (1) can be presented in the form of a generalized equation 0 ∈ F(x) + Q(x),
(2)
where Q(x) := N (x; K ) is the normal cone mapping generated by the convex set K ⊂ Rn . A general formulation of (1) has been introduced by Outrata [39] when he considered equilibria governed by a parameter dependent implicit complementarity problem (ICP) defined as the following: For a given x ∈ Rn , find y ∈ Rm such that F(x,
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