scholarly journals Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

2020 ◽  
Vol 25 (2) ◽  
pp. 24
Author(s):  
Yue Wang ◽  
Jinchuan Zhou ◽  
Jingyong Tang
Keyword(s):  
Perturbation Analysis ◽  
Lagrange Multipliers ◽  
Duality Theory ◽  
Optimization Problems ◽  
Saddle Points ◽  
Perturbation Function ◽  
Zero Duality Gap ◽  
Constrained Optimization Problems ◽  
Augmented Lagrange Multiplier ◽  
Nonlinear Support

The augmented Lagrange multiplier as an important concept in duality theory for optimization problems is extended in this paper to generalized augmented Lagrange multipliers by allowing a nonlinear support for the augmented perturbation function. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis. Meanwhile, the relations among generalized augmented Lagrange multipliers, saddle points, and zero duality gap property are developed.

Characterizations of ɛ-duality gap statements for constrained optimization problems

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Horaţiu-Vasile Boncea ◽  
Sorin-Mihai Grad
Keyword(s):  
Constrained Optimization ◽  
Optimization Problems ◽  
Duality Gap ◽  
Regularity Conditions ◽  
Locally Convex Spaces ◽  
Zero Duality Gap ◽  
Constrained Optimization Problems ◽  
Locally Convex ◽  
Convex Spaces

AbstractIn this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.

Duality Theory

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Duality Theory ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Perturbation Function ◽  
Primal Problem ◽  
Theory Section ◽  
Existence Conditions ◽  
Definition Of ◽  
Optimal Multiplier

Nonlinear optimization problems have two different representations, the primal problem and the dual problem. The relation between the primal and the dual problem is provided by an elegant duality theory. This chapter presents the basics of duality theory. Section 4.1 discusses the primal problem and the perturbation function. Section 4.2 presents the dual problem. Section 4.3 discusses the weak and strong duality theorems, while section 4.4 discusses the duality gap. This section presents the formulation of the primal problem, the definition and properties of the perturbation function, the definition of stable primal problem, and the existence conditions of optimal multiplier vectors.

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