Abstract
This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivatives, second-order pseudoinvexity functions, and the second-order Karush–Kuhn–Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization, respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of “Higgs Boson like” potentials, among others.
Highlights
This article concerns itself with several questions, the first of which is why we study the optimality conditions of scalar mathematical programming problems
Optimality conditions are a crucial asset to solve optimization problems, which constitute some of the most ubiquitous types of problems across many scientific disciplines. Optimality conditions and their associated optimal points play a vital role in activities as interdisciplinary as finding the best-fit parameters of a model given a set of data
In Ruiz-Garzón et al [16], we proved the existence of optimality conditions from KKT to constrained vector optimization problems on Hadamard manifolds as a particular case of equilibrium vector problems with constraints
Summary
This article concerns itself with several questions, the first of which is why we study the optimality conditions of scalar mathematical programming problems. More closely related to our study, Ginchev and Ivanov [6] obtain second-order optimality conditions of the KKT type for a problem with inequality constraints using pseudoconvex functions and Euclidean spaces. We present necessary and sufficient optimality conditions for both unconstrained and constrained scalar optimization problems on Hadamard manifolds, looking for the function types for which the second-order critical points and the global minimum points coincide.
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