TITLE:
Higher Order Strongly Biconvex Functions and Biequilibrium Problems
AUTHORS:
Muhammad Aslam Noor, Khalida Inayat Noor
KEYWORDS:
Biconvex Functions, Convex Functions, kβ-Convex Functions, kβ-Convex Sets, Parallelogram Laws, Biequilibrium Problems, Bivariational Inequalities, Iterative Methods, Convergence Analysis
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.11 No.2,
June
4,
2021
ABSTRACT: In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and k-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.