On subdifferential calculus of regular functions
Continuing previous work [e.g. \textit{S. K. Zavriev} and the author, Zh. Vychisl. Mat. Mat. Fiz. 30, No. 4, 491-500 (1990; Zbl 0708.90061)] the author studies differentiability properties of marginal functions in parametrized mathematical programming of the form: \(q(x)=\max\{F^ 0(x,y);\;y\in Y(x)\}\), \(Y(x)=\{y\in Y:\;F^ j(x,y)\leq 0\), \(j=1,2,\dots,s^ 1\}\), \(Y=\{y\in R^ m;\;F^ j(y)\leq 0\), \(j=s^ 1+1,\dots,s^ 2\}\). Under the usual constraint qualifications it is proved that if, in addition, the multifunction \(Y(.)\) is Hausdorff continuous and regular and if \(F^ 0\) is weakly convex in the sense that the set \(G(x,y)\) of its quasigradients \((g,h)\) satisfying \(F^ 0(x',y')- F^ 0(x,y)\geq\langle g,x'-x\rangle+\langle h,y'-y\rangle+r(x',y',x,y)\), \(r(x',y',x,y)/(\| x'-x\|+\| y'-y\|)\to 0\) as \((x',y')\to(x,y)\), is nonempty, then \(q(\cdot)\) is also weakly convex (hence regular) and a formula for its subdifferential is provided. Moreover, some approximating properties of the subdifferential of \(q(\cdot)\) are obtained.
differentiability properties of marginal functions
scientific article; zbMATH DE number 20598