Askar Hussain

This paper introduces the concepts of strongly geodesic preinvexity, strongly η-invexity of order m, and strongly invariant η-monotonicity of order m on Riemannian manifolds. Additionally, it discusses an important characterization of these functions under a condition, known as Condition C (The Condition C is defined in Remark 1 of this article), defined by Barani and Pouryayevali [J. Math. Anal. Appl. 328 (2007) 767–779]. The paper provides various non-trivial examples to support these definitions. Furthermore, it presents a significant characterization of strict η-minimizers (or η-minimizers) of order m for multi-objective optimization problems and a solution to the vector variational-like inequality problem.

In this paper, we present strongly geodesic preinvexity on Riemannian manifolds (RM) and strongly η-invexity of order m on RM. Furthermore, we define strongly invariant η-monotonicity of order m on RM. Under Condition C, an important characterization of these functions are studied. We construct several non-trivial examples in support of these definitions. Afterwords, an important and significant characterization of a strict η-minimizers (η-minimizers)of order m for MOP and a solution to the variational like-inequality problem (VVLIP) has been derived.

In this article, we present semi strongly $E$-preinvexity and semi strongly $E$-invexity. To demonstrate the existence of these functions, certain nontrivial examples have been developed. Several significant relationships and characterizations of these functions on strongly $E$-invex sets are discussed. Furthermore, we consider a non-linear programming problem for semi strongly $E$-preinvex functions and investigate relationships between the set of optimal solutions and these functions.

In this paper, we extend the class of strongly $E$-preinvex and strongly $E$-invex functions to quasi strongly $E$-preinvex, quasi strongly $E$-invex and pseudo strongly $E$-invex functions. Some nontrivial suitable examples have been constructed in support of our definitions. Several interesting properties and relationships of these functions are discussed. Furthermore, to show the application of our results, we consider a nonlinear programming problem and show that the local minimum point is also a strictly global minimum.

The concepts of stronglyE-invex sets, stronglyE-invex, stronglyE-preinvex, and strongly pseudoE-preinvex functions are introduced in this paper. We have included several non-trivial examples to support our definitions. The family of stronglyE-invex sets has been shown to form a vector space overR, and other interesting properties have been addressed. The epigraph of stronglyE-preinvex function has been derived, as well as the relationship between the stronglyE-preinvex function and the strongly pseudoE-preinvex function has been established. To show an important relationship between stronglyE-invex and stronglyE-preinvex functions, a new Condition A has been introduced. A nonlinear programming problem for stronglyE-preinvex functions is explored as an application. Under a few conditions, it has been proved that the local minimum point is the global minimum for a nonlinear programming problem.