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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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