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Strict minimizers of order \(m\) in nonsmooth optimization problems. (English) Zbl 1150.90007
The concept of differentiable \((F,\rho )\)-convex function of order \(m\) is introduced. This concept generalizes the concept of \((F,\rho )\)-convex function known from the literature. The \((F,\rho )\)-convex functions are in general non-differentiable and for their description Clarke’s generalized gradient is used. Further the concept of the strict local minimizer of order \(m\) is introduced. A point \(x^0 \in D\) is called strict local minimizer of order \(m\) of a given function \(f\) on \(D\), if there exists a neighborhood \(U\) of \(x^0\) and a positive real number \(\beta \) such that the inequality \[ f(x)\geq f(x^0)+\beta \, \| x-x^0 \| \] is fulfilled for all \(x\in D \cap U\). Sufficient optimality conditions for strict minimizers of order \(m\) in constraint nonlinear mathematical programming problems involving \((F,\rho )\)-convex function of order \(m\) are presented. Duality results for such optimization problems are proved.

90C26 Nonconvex programming, global optimization
90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
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