One limit theorem, optimal criterion and existence theorem of the dual nonlinear program
With definitions and terminology consistent with those of \textit{R. T. Rockafellar} [``Convex analysis'' (1970; Zbl 0193.184)], the authors study two programs, primal (P) and dual (DP) of the type \[ (P):\quad \min \{g_ 0(x):\quad x\in D\},\quad D=\{x:\quad g_ i(x)\leq 0,\quad 1\leq i\leq m\}\subset E^ n \] where all \(g_ i's\), \(0\leq i\leq m\), are real functions and \(g_ i(x)>-\infty\) for each \(0\leq i\leq m\) and all \(x\in E^ n\). Now if the real function F(u) is defined for all \(u\in E_{01}^{m+1}=\{u:\) \((u_ 0,u_ 1,u_ 2,...,u_ m)\in E^{m+1}\), \(u_ 0=1\}\) as \(F(u)=\inf \{u^ Tg(x):\) \(x\in D\}\) with \(g=(g_ 0,g_ 1,...,g_ m)\) the dual program is \[ (DP):\quad \max \{F(u):\quad u\in E_{01}^{m+1}\}. \] Under certain conditions a ``characteristic function'' f(\(\alpha\),u,\(\delta)\) with domain \(E^{m+1}\times E^{m+1}\times E^ 1\) defined which approaches the directional derivative of F(u) in the direction \(\alpha\) as the ``perturbation parameter'' \(\delta\) approaches F(u). This property of f is used to develop an optimality criterium and an existence theorem for solutions of (DP) as well as, in some cases, a relationship between the existence of an optimal solution for (DP) and that of a feasible solution for (P). There is a number of ancillary definitions and lemmas and the style is somewhat confusing at times.
scientific article; zbMATH DE number 4110474