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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.

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