2013/11/11 by Vsevolod I. Ivanov, Ivanov, Vsevolod Ivanov
Computer Science · Engineering · Mathematics · #26B05 #26B25 #49K10 #90C46 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.1311.2367
openalex publication_date 2013/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is consistent with the classical higher-order Fréchet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order n (n is a positive integer) for a local minimum and isolated local minimum of order n in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A special class \mathcal Fn of functions is defined and optimality conditions for isolated local minimum of order n for a function f∈\mathcal Fn are derived. The derivative of order n does not appear in these characterizations. We prove necessary and sufficient criteria such that every stationary point of order n is a global minimizer. We compare our results with some previous ones.