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$$r$$-convex transformability in nonlinear programming problems. (English) Zbl 1121.49026
The $$r$$-convexity is defined in M. Avriel [Math. Program. 2, 309–323 (1972; Zbl 0249.90063)]. A locally Lipschitz function $$f\: \mathbb R^n \to \overline {\mathbb R}$$ is said to be $$r$$-convex transformable or $$r$$-convexifiable with respect to $$\varphi$$, if there exists a $$C^1$$ diffeomorphism $$\varphi \: \mathbb R^n \to \mathbb R^n$$ with $$C^1$$-inverse $$\varphi ^{-1}$$ such that the composed function $$f\circ \varphi ^{-1}$$ is $$r$$-convex. The authors consider minimization problems with $$r$$-convexifiable objective functions and constraints $$f_i(x) \leq 0$$, $$i=1,\dots ,m$$, where $$f_i\: \mathbb R^n \to \mathbb R$$ are $$r$$-convexifiable functions. It is shown that for problems of this kind the Karush-Kuhn-Tucker conditions are not only necessary but also sufficient for optimality. The authors provide a method for solving such problems and illustrate the theoretical results by a small numerical example. Some applications and connections with $$r$$-invexity are briefly discussed.

MSC:
 49K99 Optimality conditions
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