New Sequential Lagrange Multiplier Conditions Characterizing Optimality without Constraint Qualification for Convex Programs

2003 ◽  
Vol 14 (2) ◽  
pp. 534-547 ◽  
Author(s):  
V. Jeyakumar ◽  
G. M. Lee ◽  
N. Dinh
Keyword(s):  
Lagrange Multiplier ◽  
Constraint Qualification ◽  
Convex Programs

SEQUENTIAL LAGRANGE MULTIPLIER CONDITIONS FOR MINIMAX PROGRAMMING PROBLEMS

2008 ◽  
Vol 25 (02) ◽  
pp. 113-133 ◽  
Author(s):  
ANULEKHA DHARA ◽  
APARNA MEHRA
Keyword(s):  
Lagrange Multiplier ◽  
Constraint Qualification ◽  
Minimax Programming ◽  
Duality Results ◽  
Multiplier Rules ◽  
Cone Constraint

In this article, we study nonsmooth convex minimax programming problems with cone constraint and abstract constraint. Our aim is to develop sequential Lagrange multiplier rules for this class of problems in the absence of any constraint qualification. These rules are obtained in terms of ∊-subdifferentials of the functions. As an application of these rules, a sequential dual is proposed and sequential duality results are presented.

scholarly journals Lagrange multiplier characterizations of constrained best approximation with nonsmooth nonconvex constraints

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4669-4684
Author(s):  
H. Mohebi
Keyword(s):  
Lagrange Multiplier ◽  
Best Approximation ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Dual Cone ◽  
The Best Approximation ◽  
Constraint Set ◽  
Nonconvex Constraint ◽  
Constrained Best Approximation

In this paper, we consider the constraint set K := {x ? Rn : gj(x)? 0,? j = 1,2,...,m} of inequalities with nonsmooth nonconvex constraint functions gj : Rn ? R (j = 1,2,...,m).We show that under Abadie?s constraint qualification the ?perturbation property? of the best approximation to any x in Rn from a convex set ?K := C ? K is characterized by the strong conical hull intersection property (strong CHIP) of C and K, where C is an arbitrary non-empty closed convex subset of Rn: By using the idea of tangential subdifferential and a non-smooth version of Abadie?s constraint qualification, we do this by first proving a dual cone characterization of the constraint set K. Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set ?K is closed and convex, we show that the Lagrange multiplier characterizations of constrained best approximation holds under a non-smooth version of Abadie?s constraint qualification. The obtained results extend many corresponding results in the context of constrained best approximation. Several examples are provided to clarify the results.

Generalized Lagrange multiplier rule for non-convex vector optimization problems

2016 ◽  
Vol 146 (2) ◽  
pp. 297-308
Author(s):  
Maria Bernadette Donato
Keyword(s):  
Vector Optimization ◽  
Lagrange Multiplier ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Condition ◽  
Lagrange Multiplier Rule ◽  
Convex Vector Optimization ◽  
Multiplier Rule ◽  
Necessary And Sufficient

In this paper a non-convex vector optimization problem among infinite-dimensional spaces is presented. In particular, a generalized Lagrange multiplier rule is formulated as a necessary and sufficient optimality condition for weakly minimal solutions of a constrained vector optimization problem, without requiring that the ordering cone that defines the inequality constraints has non-empty interior. This paper extends the result of Donato (J. Funct. Analysis261 (2011), 2083–2093) to the general setting of vector optimization by introducing a constraint qualification assumption that involves the Fréchet differentiability of the maps and the tangent cone to the image set. Moreover, the constraint qualification is a necessary and sufficient condition for the Lagrange multiplier rule to hold.

Further Development in the Global Resolution of Convex Programs with Complementarity Constraints

2014 ◽  
Author(s):  
John E. Mitchell ◽  
Jong-Shi Pang ◽  
Yu-Ching Lee ◽  
Bin Yu ◽  
Lijie Bai
Keyword(s):  
Complementarity Constraints ◽  
Convex Programs ◽  
Further Development

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

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