scholarly journals Duality theorems for convex programming without constraint qualification

1984 ◽  
Vol 36 (2) ◽  
pp. 253-266 ◽  
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.

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

1982 ◽  
Vol 32 (3) ◽  
pp. 369-379 ◽  
Author(s):  
P. Kanniappan ◽  
Sundaram M. A. Sastry
Keyword(s):  
Convex Function ◽  
Objective Function ◽  
Programming Problem ◽  
Convex Programming ◽  
Duality Theorem ◽  
Sufficient Optimality Conditions ◽  
Convex Programming Problem ◽  
Lower Semi ◽  
Continuous Convex Function ◽  
Necessary And Sufficient

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.

scholarly journals A comparison of constraint qualifications in infinite-dimensional convex programming revisited

1999 ◽  
Vol 40 (3) ◽  
pp. 353-378 ◽  
Author(s):  
C. Zặlinescu
Keyword(s):  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Strong Duality ◽  
Sufficient Condition ◽  
Subdifferential Calculus ◽  
Infinite Dimensional ◽  
Qualification Conditions ◽  
The One

In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formulainf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good formulae for conjugates are equivalent and they can be used to obtain formulae for ε-subdifferentials, using a technique developed in [17] and extensively used in [46].

scholarly journals Optimality conditions for a cone-convex programming problem

1979 ◽  
Vol 27 (2) ◽  
pp. 141-162 ◽  
Author(s):  
Hélène M. Massam
Keyword(s):  
Vector Space ◽  
Optimality Conditions ◽  
Programming Problem ◽  
Convex Programming ◽  
Convex Cone ◽  
Constraint Qualifications ◽  
Closed Convex Cone ◽  
Convex Programming Problem ◽  
Nonnegative Orthant ◽  
Locally Convex

AbstractOptimality conditions without constraint qualifications are given for the convex programming problem: Maximize f(x) such that g(x) ∈ B, where f maps X into R and is concave, g maps X into Rm and is B-concave, X is a locally convex topological vector space and B is a closed convex cone containing no line. In the case when B is the nonnegative orthant, the results reduce to some of those obtained recently by Ben-Israel, Ben-Tal and Zlobec.

scholarly journals A duality theorem for nondifferentiable convex programming with operatorial constraints

1980 ◽  
Vol 22 (1) ◽  
pp. 145-152 ◽  
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.

scholarly journals Facial reduction for a cone-convex programming problem

1981 ◽  
Vol 30 (3) ◽  
pp. 369-380 ◽  
Author(s):  
Jon M. Borwein ◽  
Henry Wolkowicz
Keyword(s):  
Dimensional Space ◽  
Convex Programming Problem ◽  
Finite Dimensional Space ◽  
Facial Reduction ◽  
Finite Dimensional ◽  
Arbitrary Convex ◽  
Lagrange Multiplier Theorem ◽  
Image Position ◽  
Characterization Of Optimality

AbstractIn this paper we study the abstract convex programwhere S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S (on Ω). We use the concept of a minimal cone for (P) to correct and strengthen a previous characterization of optimality for (P), see Theorem 3.2. The results presented here are used in a sequel to provide a Lagrange multiplier theorem for (P) which holds without any constraint qualification.

scholarly journals On an optimal control problem for a parabolic inclusion

1996 ◽  
Vol 143 ◽  
pp. 195-217
Author(s):  
Bui an Ton
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Convex Function ◽  
Control Problem ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Lower Semi ◽  
Continuous Convex Function ◽  
Effective Domain

Let H, U be two real Hilbert spaces and let g be a proper lower semi-continuous convex function from L2 (0, T;H) into R+. For each t in [0, T], let φ(t,.) be a proper l.s.c. convex function from H into R with effective domain Dφ(t,.)) and let h be a l.s.c. convex function from a closed convex subset u of U into L2(0, T;H) withfor all u in U. The constants γ and C are positive.

PROJECTIONS OF LOG-CONCAVE FUNCTIONS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250036
Author(s):  
ALEXANDER SEGAL ◽  
BOAZ A. SLOMKA
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Duality Mapping ◽  
Concave Functions ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Natural Notion ◽  
Linear Sections ◽  
Funct Anal

Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝn containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

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