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Generalized convex functions and vector variational inequalities. (English) Zbl 0797.90085

Summary: The notion of \((\alpha,\phi,Q)\)-invexity is introduced, where \(\alpha: X\times X\to\text{int }R^ m_ +\), \(\phi: X\times X\to X\), \(X\) is a Banach space, \(Q\) is a convex cone of \(R^ m\). This unifies the properties of many classes of functions, such as \(Q\)-convexity, pseudo- linearity, representation condition, null space condition, and \(V\)- invexity. A generalized vector variational inequality is considered, and its equivalence with a multiobjective programming problem is discussed using \((\alpha,\phi,Q)\)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of \((\alpha,\phi,Q)\)-invexity to multiobjective programming problems and to a special kind of generalized vector variational inequality are given.

MSC:

90C29 Multi-objective and goal programming
26B25 Convexity of real functions of several variables, generalizations
49J40 Variational inequalities
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