Lagrange duality and saddle point optimality conditions for semi-infinite mathematical programming problems with equilibrium constraints

Abstract

In this paper, we consider a special class of optimization problems which contains infinitely many inequality constraints and finitely many complementarity constraints known as the semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC). We propose Lagrange type dual model for the SIMPEC and obtain their duality results using convexity assumptions. Further, we discuss the saddle point optimality conditions for the SIMPEC. Some examples are given to illustrate the obtained results.

Đối ngẫu lagrange và điều kiện tối ưu dạng điểm yên cho bài toán tối ưu nửa vô hạn với ràng buộc biến mất

“…Tuy nhiên, chưa có bài báo nào đề cập đến dạng đối ngẫu Lagrange và tiêu chuẩn tối ưu dạng điểm yên cho bài toán SIPVC. Lưu ý rằng mô hình đối ngẫu dạng Lagrange có thể dễ xử lý từ quan điểm thuật toán hơn là các mô hình đối ngẫu đã biết khác, xem Mishra et al (2016), Singh et al (2017), Singh et al (2019) và các tài liệu tham khảo trong đó.…”

Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming with equilibrium constraints

“…Suppose by contradiction thatf (x) ≺ L(u, α, λ g , λ h , λ G , λ H ), equivalently, f i (x) < f i (u), ∀i ∈ I.The above inequalities and the pseudoconvexity of f i (i ∈ I) at u tell us that∇f i (u), x − u < 0, ∀i ∈ I,which, along withi∈I α i = 1, lead to m i=1 α i ∇f i (u), x − u < 0,contradicting with(25). (ii) Assume by contradiction that f (x) L(y, α, λ g , λ h , λ G , λ H ).…”

“…By using convexificators, the paper [20] established necessary and sufficient optimality conditions and derived weak and strong duality theorems relating to the semi-infinite mathematical programming problems with equilibrium constraints. The Lagrange type dual model and saddle point optimality criteria of semi-infinite mathematical programming problems with equilibrium constraints were discussed in [25]. However, KKT necessary optimality conditions for multiobjective semi-infinite programming problems with equilibrium constraints have not yet been considered in [21].…”