Optimization and equilibrium are two important concepts in system science. In this paper we show that the two concepts are equivalent in some sence. The variational inequality is an important tool in many applications. A nonlinear programming can be transformed into a variational inequality under convexity and differentiability. Indeed, in some practical cases, if we do not know the accurate expression of the objective function in an optimization problem, but we know its variable speed of values of objective function, then an optimization problem can be formulated as a variational inequality. Except for the optimization field, there are other foundations to introduce variational inequalities. In the sixties, Stampacchia transformed a partial differential equation to a variational inequality problem. A network equilibrium problem and a variational inequality is equivalent under some conditions by Wardrop equilibrium principle. Besides, some problems in the network economics can be solved by variational inequalities. In this paper we introduce a vector variational inequality and we show that an multiobjective optimization problem is equivalent with a vector variational inequality in some conditions. For this end we will introduce nonlinear scalarization function to a variable preference structure, and we obtain some properties of the nonlinear scalarization function. For approximation analysis of multiobjective optimization problems, we will introduce vector variational principles for vector-valued functions and some equivalence among the vector variational principles and some important theorems in nonlinear analysis.
CHEN Guangya. , {{custom_author.name_en}}.
Equivalency Between Optima and Equilibria. Journal of Systems Science and Mathematical Sciences, 2009, 29(11): 1441-1446 https://doi.org/10.12341/jssms08474
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