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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.

49K99 Optimality conditions
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