Saddle Point Theory and Optimality Conditions

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.

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Lagrangian Duality and Saddle Point Optimality Conditions

Nonlinear Programming ◽

10.1002/0471787779.ch6 ◽

2005 ◽

pp. 257-314

Keyword(s):

Saddle Point ◽

Optimality Conditions ◽

Lagrangian Duality

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

Yugoslav journal of operations research ◽

10.2298/yjor181215014s ◽

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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Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

Abstract and Applied Analysis ◽

10.1155/2012/532430 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Yu-Cheng An ◽

Hong-Min Suo

Keyword(s):

Saddle Point ◽

Point Theorem ◽

Blow Up ◽

Smooth Boundary ◽

Elliptic Systems ◽

Point Theory ◽

Saddle Point Theorem ◽

Multiplicity Results ◽

Near Resonance ◽

Degenerate Elliptic

We study the degenerate semilinear elliptic systems of the form-div(h1(x)∇u)=λ(a(x)u+b(x)v)+Fu(x,u,v),x∈Ω,-div(h2(x)∇v)=λ(d(x)v+b(x)u)+Fv(x,u,v),x∈Ω,u|∂Ω=v|∂Ω=0, whereΩ⊂RN(N≥2)is an open bounded domain with smooth boundary∂Ω, the measurable, nonnegative diffusion coefficientsh1,h2are allowed to vanish inΩ(as well as at the boundary∂Ω) and/or to blow up inΩ¯. Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.

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