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Regularization of nonmonotone variational inequalities. (English) Zbl 1115.49010

When a Tikhonov-Brouwder regularization is used for solving a multivalued variational inequality problem, it is advisable that the two following properties are satisfied: (i) each auxiliary variational inequality has a unique solution \(x^{\epsilon}\) depending of a parameter \(\epsilon >0\). (ii) the sequence of solutions \(\{x^{\epsilon}\}\) converges to the minimal norm solution of the initial problem when \(\epsilon \to 0\).
In this paper, Konnov et al. present such a regularization for solving a nonmonotone multivalued VI. First they obtain property (ii) for VIs whose dual problem is solvable and has the same solution set. This condition is weaker than monotonicity, but allows unbounded solution sets. Next, they present parametric weakened coercivity conditions which also enable them to consider unbounded problems via their reduction to a bounded VI. In a second part they utilize the \(P_0\) properties of the cost mapping to provide property (i). Finally they describe two rather broad classes of perfectly and nonperfectly competitive economic equilibrium models that satisfy these conditions.

MSC:

49J40 Variational inequalities
47H14 Perturbations of nonlinear operators
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
65K10 Numerical optimization and variational techniques
91B50 General equilibrium theory
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