Duality theorems and an optimality condition for non-differentiable convex programming

AbstractNecessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.

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A duality theorem for nondifferentiable convex programming with operatorial constraints

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700006419 ◽

1980 ◽

Vol 22 (1) ◽

pp. 145-152 ◽

Cited By ~ 3

Author(s):

P. Kanniappan ◽

Sundaram M.A. Sastry

Keyword(s):

Convex Function ◽

Objective Function ◽

Programming Problem ◽

Duality Theorem ◽

Topological Linear Space ◽

Lower Semi ◽

Convex Objective Function ◽

Continuous Convex Function ◽

Topological Linear ◽

Locally Convex

A duality theorem of Wolfe for non-linear differentiable programming is now extended to minimization of a non-differentiable, convex, objective function defined on a general locally convex topological linear space with a non-differentiable operatorial constraint, which is regularly subdifferentiable. The gradients are replaced by subgradients. This extended duality theorem is then applied to a programming problem where the objective function is the sum of a positively homogeneous, lower semi continuous, convex function and a subdifferentiable, convex function. We obtain another duality theorem which generalizes a result of Schechter.

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Duality theorems for convex programming without constraint qualification

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870002468x ◽

1984 ◽

Vol 36 (2) ◽

pp. 253-266 ◽

Cited By ~ 6

Author(s):

P. Kanniappan

Keyword(s):

Convex Function ◽

Convex Programming ◽

Constraint Qualification ◽

Constraint Qualifications ◽

Convex Programming Problem ◽

Duality Theorems ◽

Finite Dimensional ◽

Lower Semi ◽

Continuous Convex Function

AbstractInvoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

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Optimality and duality in nonsmooth vector optimization with non-convex feasible set

RAIRO - Operations Research ◽

10.1051/ro/2020050 ◽

2020 ◽

Author(s):

Dr. Sunila Sharma ◽

Priyanka Yadav

Keyword(s):

Objective Function ◽

Optimality Conditions ◽

Vector Optimization ◽

Optimization Problem ◽

Vector Optimization Problem ◽

Sufficient Optimality Conditions ◽

Convex Programming Problem ◽

Feasible Set ◽

Nonsmooth Vector Optimization ◽

Necessary And Sufficient

For a convex programming problem, the Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for optimality under suitable constraint qualification. Recently, Suneja et al proved KKT optimality conditions for a differentiable vector optimization problem over cones in which they replaced the cone-convexity of constraint function by convexity of feasible set and assumed the objective function to be cone-pseudoconvex. In this paper, we have considered a nonsmooth vector optimization problem over cones and proved KKT type sufficient optimality conditions by replacing convexity of feasible set with the weaker condition considered by Ho and assuming the objective function to be generalized nonsmooth cone-pseudoconvex. Also, a Mond-Weir type dual is formulated and various duality results are established in the modified setting.

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