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Vector Equilibrium Problems

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Vector Variational Inequalities and Vector Optimization

Part of the book series: Vector Optimization ((VECTOROPT))

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Abstract

Motivated by various applications of multi-criteria decision making, extensions of scalar equilibrium problems, discussed in Chap. 1, for the vector case were proposed. Among them, the most investigated problems are vector optimization and vector saddle point. In 1956, Blackwell (Pacific Journal of Mathematics 6(1):1–8, 1956) considered matrix games with vector payoff’s and proved the existence theorem for such problems. Since then, many researchers studied vector non-cooperative games in finite- and infinite-dimensional spaces. In 1980, F. Giannessi extended the variational inequality problem for vector-valued functions known as vector variational inequality problem” (in short, VVIP) with further applications. Vector equilibrium problems (in short, VEPs) can be viewed as further and natural extension of the previous concepts. It is a unified model of several known problems, namely, vector variational inequality problems, vector optimization problems, vector saddle point problems and Nash equilibrium problems for vector-valued functions. The theory of VVIPs and VEPs has been developing extensively since the early ninety’s. In particular, a number of various kinds of these problems were proposed and the corresponding existence results both on bounded and on unbounded sets were established. The mathematical theory of VEPs is presented in this chapter.

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Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, India

    Qamrul Hasan Ansari

  2. Institute of Mathematics, Martin Luther University Halle-Wittenberg, Halle, Germany

    Elisabeth Köbis

  3. Center for General Education, China Medical University, Taichung, Taiwan

    Jen-Chih Yao

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  1. Qamrul Hasan Ansari
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  2. Elisabeth Köbis
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  3. Jen-Chih Yao
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Ansari, Q.H., Köbis, E., Yao, JC. (2018). Vector Equilibrium Problems. In: Vector Variational Inequalities and Vector Optimization. Vector Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-63049-6_9

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