Lagrangian Duality and Saddle Point Optimality Conditions

2005 ◽  
pp. 257-314
Keyword(s):  
Saddle Point ◽  
Optimality Conditions ◽  
Lagrangian Duality

scholarly journals Multiobjective Convex Optimization in Real Banach Space

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2148
Author(s):  
Kin Keung Lai ◽  
Mohd Hassan ◽  
Jitendra Kumar Maurya ◽  
Sanjeev Kumar Singh ◽  
Shashi Kant Mishra
Keyword(s):  
Banach Space ◽  
Multiobjective Optimization ◽  
Saddle Point ◽  
Optimality Conditions ◽  
Pareto Optimality ◽  
Real Banach Space ◽  
Optimization Problems ◽  
Mixed Integer ◽  
Multiobjective Optimization Problems ◽  
Necessary And Sufficient

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn–Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019.

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

2019 ◽  
Vol 29 (4) ◽  
pp. 433-448
Author(s):  
Kunwar Singh ◽  
J.K. Maurya ◽  
S.K. Mishra
Keyword(s):  
Mathematical Programming ◽  
Saddle Point ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Mathematical Programming Problem ◽  
Dual Model ◽  
Equilibrium Constraints ◽  
Convexity Assumptions ◽  
Mathematical Programming Problems

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.

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