© IJARW | ISSN (O) - 2582-1008 April 2023 | Vol. 4 Issue. 10 www.ijarw.com IJARW1830 International Journal of All Research Writings 37 SUFFICIENT CONDITIONS FOR THE EXISTENCE OF SOLUTIONS TO TWO – LEVEL LEADERSHIP OPTIMIZATION PROBLEM Nguyen Quynh Hoa 1 , Tran Thi Mai 2 , Nguyen Thi Thu Hang 3 1,2,3 Thai Nguyen University of Economics and Business Administration, Thai Nguyen, Viet Nam. Corresponding Author: Tran Thi Mai ABSTRACT Optimal theory is derived from the ideas of economic equilibrium, the value theory of Edgeworth and Pareto from the late 19th and early 20th centuries. Vector optimization theory is an important part of theory. optimization theory. Vector optimization is really an independent branch of mathematics and has many practical applications, especially in the field of economics. In this paper, we will consider a sufficient condition for the existence of a solution to two-level leadership optimization problem when the utility function is the sum of the lower and upper semicontinuous mappings. Keyword: Applied Mathematics; economical model; quasi-equilibrium problem;sufficient conditions; optimal theory; quasi-variational relation; etc 1. MAKE A PROBLEM The problem that plays an important role in optimization theory is the problem: Finding x D  such that ( ) ( ) f x f x  for all x D  , or ( ) () min . 1 xD f x  Where, D is a non-empty subset in space R and mapping : f D → . In 1994, Blum and Oettli introduced the following equilibrium problem: Given the mapping ( ) : , , 0, . D D xx x D    → =  , Finding x D  such that ( ) , 0 xx   for all . x D  As we know, the equilibrium problem is a generalization of the variable inequality problem, the fixed point problem, the Nash equilibrium problem, etc. Therefore, the study of the closed equilibrium point problem. an important role in optimization theory. The above problems are stated for the vector case and the multivalued vector case in different directions, respectively. But in general, we can see that these problems are all reduced to the multivalued equilibrium problem: Given Hausdorff linear topological spaces X, Y, D X  and multivalued mapping : 2 Y F D → . Finding x D  such that: ( ) () 0 2 Fx  Problem (2) is also known by many as a multivalued equation. The study of this problem is very important and meaningful in practice. But in practice, many times the constraint domain D changes, depends on a mapping, : 2 D PD → . Then, we consider the problem: Finding x D  such that: () () () 3 0 x Px Fx        Problem (3) is called the general quasi- equilibrium problem. This problem is also known as the two-level optimization problem, that is, solving one problem on the solution set of the other problem. Sufficient conditions for the existence of a solution of this problem have been studied in the case that P is a continuous map, F is a u.s.c mapping (see [3]). In order to mitigate the condition of continuity of the mappings P, F in finding a sufficient condition for the existence of a solution of a general quasi-equilibrium problem, one often considers this problem in the form of a product as follows: X, Y, Z are Hausdorff local convex topological spaces, , D XK Z   , multivalued mappings : 2, : 2, : 2 D K Y PD K QD K F D K  →  →  → .