Sufficient Conditions for a Minimum of a Strongly Quasiconvex Function on a Weakly Convex Set

• Published in: Mathematical Notes Vol. 111; no. 1-2; pp. 33 - 46
• Main Authors: Dudov, S. I., Osiptsev, M. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.02.2022; Springer Nature B.V
• Subjects: 14/34; 639/766/189; 639/766/530; 639/766/747; Convexity; Mathematics; Mathematics and Statistics; strongly and weakly convex sets and functions; sufficient conditions for a minimum; subdifferential; radius of the neighborhood of a local minimum; normal cone; strongly quasiconvex function
• ISSN: 0001-4346; 1067-9073; 1573-8876
• DOI: 10.1134/S0001434622010059

We consider a finite-dimensional minimization problem for a strongly quasiconvex function on a weakly convex set. We obtain sufficient conditions for its solution expressed in terms of the strong quasiconvexity constants of the objective function and the weak convexity of the admissible set of arguments, as well as their local characteristics. We separately consider the case of specifying an admissible set by the Lebesgue set of a weakly convex function. For the case of a differentiable objective function, we establish sufficient conditions for a local minimum, including a “strong” stationarity condition and indicate the radius of the corresponding neighborhood.