Optimization and variational inequalities with pseudoconvex functions
The author defines a class of pseudoconvex functions in an extended sense and a notion of stationarity in an extended sense, which is called stationarity of order infinity. The inf-stationary points of infinite order are called extended inf-stationary points. Then it is proved that each global minimizer of a function \(f\) over a convex set \(X\) is an extended inf-stationary point. By defining a new variational inequality of Stampacchia type whose solution set is denoted by \(S(f,X)\) it is proved that \(S(f,X)\) gives an approximation of the set of global minimizers \(GM(f,X)\) of \(f\) over \(X\). Then, it is shown that the pseudoconvex functions in the extended sense are the largest class such that \(S(f,Y)\) coincides with \(GM(f,Y)\) for all convex subsets \(Y\) of \(X\). In the last part of the paper the author introduces a notion of pseudomonotone lower Dini directional derivatives in an extended sense using higher-order Dini derivatives and proves that a radially lower semicontinuous function is pseudoconvex in the extended sense if and only if its Dini derivatives are pseudomonotone in the extended sense. It is proved that the radially lower semicontinuous pseudoconvex functions in the extended sense are the largest class such that \(S(f,Y)=M(f,Y)\) for all convex subsets \(Y\) of \(X\), where \(M(f,Y)\) denotes the solution set of the Minty variational inequality of differential type over \(Y\). Finally, the paper deals with the characterizations of the solution set of nonlinear programming problems, whose objective functions are pseudoconvex in the extended sense.
variational inequalities of differential type
characterizations of the solution set of the nonlinear programming problem
scientific article; zbMATH DE number 5828739
scientific article