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1.
In this paper we work in separated locally convex spaces where we give equivalent statements for the formulae of the conjugate function of the sum of a convex lower‐semicontinuous function and the precomposition of another convex lower‐semicontinuous function which is also K ‐increasing with a K ‐convex K ‐epi‐closed function, where K is a nonempty closed convex cone. These statements prove to be the weakest constraint qualifications given so far under which the formulae for the subdifferential of the mentioned sum of functions are valid. Then we deliver constraint qualifications inspired from them that guarantee some conjugate duality assertions. Two interesting special cases taken from the literature conclude the paper. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
2.
In this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping. This condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior-point regularity conditions given so far in the literature. Moreover, it provides a weak sufficient condition for Fenchel duality regarding convex optimization problems in infinite dimensional spaces. As an application, we discuss the strong conical hull intersection property (CHIP) for a finite family of closed convex sets. 相似文献
3.
The subject of this paper is the systematic study of second order notions concerning differentiable functions with Lipschitz
derivative. The results and notions are motivated by recent papers of Cominetti, Correa and Hiriart-Urruty. The first goal
of this paper is the comparison of several known second order directional derivatives. The second goal is the introduction
of a generalized Hessian which is a set of certain symmetric bilinear forms. The relation of this generalized Hessian to other
existing second order derivatives is also described.
The research was supported by a grant from the National Science Foundation NSF-66-2270, which is gratefully acknowledged.
Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T-016846 and by the Humboldt
Foundation. 相似文献
4.
Throughout this paper, the authors introduce a new condition, defined by Assumption S′, which establishes a necessary and sufficient condition for the validity of the strong duality between a convex optimization problem and its Lagrange dual. This work will be focused on the context of emptiness of the interior of the ordering cone and convexity of the equality constraints. Moreover, this new condition will be necessary and sufficient for the infinite dimensional Lagrange multiplier rule. This new principle will find application to the elastic–plastic torsion problem, to the continuum model of transportation and to a problem with quadratic equality constraint with connected to evolutionary illumination and visibility problems. 相似文献
5.
V. Jeyakumar 《Optimization Letters》2008,2(1):15-25
A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal
(i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it
plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality
requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality
have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical
hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas
of convex optimization such as constrained approximation and error bounds.
Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful
comments 相似文献
6.
Seán Dineen 《Journal of Functional Analysis》2010,259(2):545-560
Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c0, the original Tsirelson space T∗ and the Tsirelson∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains. 相似文献
7.
Lagrangian relaxation is often an efficient tool to solve (large-scale) optimization problems, even nonconvex. However it
introduces a duality gap, which should be small for the method to be really efficient. Here we make a geometric study of the
duality gap. Given a nonconvex problem, we formulate in a first part a convex problem having the same dual. This formulation
involves a convexification in the product of the three spaces containing respectively the variables, the objective and the
constraints. We apply our results to several relaxation schemes, especially one called “Lagrangean decomposition” in the combinatorial-optimization
community, or “operator splitting” elsewhere. We also study a specific application, highly nonlinear: the unit-commitment
problem.
Received: June 1997 / Accepted: December 2000?Published online April 12, 2001 相似文献
8.
In the present paper, two types of second order dual models are formulated for a minmax fractional programming problem. The
concept of η-bonvexity/generalized η-bonvexity is adopted in order to discuss weak, strong and strict converse duality theorems.
The research of Z. Husain is supported by the Department of Atomic Energy, Government of India, under the NBHM Post-Doctoral
Fellowship Program No. 40/9/2005-R&D II/1739. 相似文献
9.
D. B. Rokhlin 《Siberian Mathematical Journal》2009,50(1):162-166
Consider a closed convex cone C in a Banach ideal space X on some measure space with σ-finite measure. We prove that the fulfilment of the conditions C∩X + = {0} and C??X + guarantees the existence of a strictly positive continuous functional on X whose restriction to C is nonpositive. 相似文献
10.
In this paper, following the method in the proof of the composition duality principle due to Robinson and using some basic
properties of the ε-subdifferential and the conjugate function of a convex function, we establish duality results for an ε-variational inequality problem. Then, we give Fenchel duality results for the ε-optimal solution of an unconstrained convex optimization problem. Moreover, we present an example to illustrate our Fenchel
duality results for the ε-optimal solutions.
The authors thank the referees for valuable suggestions and comments. This work was supported by Grant No. R01-2003-000-10825-0
from the Basic Research Program of KOSEF. 相似文献
11.
In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent
and we study the notion ofD-compactness and D-boundedness in probabilistic normed spaces. 相似文献
12.
T. Q. Son J. J. Strodiot V. H. Nguyen 《Journal of Optimization Theory and Applications》2009,141(2):389-409
In this paper, ε-optimality conditions are given for a nonconvex programming problem which has an infinite number of constraints. The objective
function and the constraint functions are supposed to be locally Lipschitz on a Banach space. In a first part, we introduce
the concept of regular ε-solution and propose a generalization of the Karush-Kuhn-Tucker conditions. These conditions are up to ε and are obtained by weakening the classical complementarity conditions. Furthermore, they are satisfied without assuming
any constraint qualification. Then, we prove that these conditions are also sufficient for ε-optimality when the constraints are convex and the objective function is ε-semiconvex. In a second part, we define quasisaddlepoints associated with an ε-Lagrangian functional and we investigate their relationships with the generalized KKT conditions. In particular, we formulate
a Wolfe-type dual problem which allows us to present ε-duality theorems and relationships between the KKT conditions and regular ε-solutions for the dual. Finally, we apply these results to two important infinite programming problems: the cone-constrained
convex problem and the semidefinite programming problem. 相似文献
13.
E. L. Peterson 《Journal of Optimization Theory and Applications》1978,26(1):15-41
Extensions of the ordinary Lagrangian are used both in saddle-point characterizations of optimality and in a development of duality theory.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-73-2516. 相似文献
14.
A new contraction principle in menger spaces 总被引:2,自引:0,他引:2
In the present work we introduce a new type of contraction mapping by using a specific function and obtain certain fixed point results in Menger spaces. The work is in line with the research for generalizing the Banach's contraction principle. We extend the notion of altering distance function to Menger Spaces and obtain fixed point results. 相似文献
15.
W. K. Min 《Acta Mathematica Hungarica》2008,121(3):283-292
We introduce and study the concepts of weak neighborhood systems, weak neighborhood spaces, ω(ψ, ψ′)-continuity, ω-continuity and ω*-continuity on WNS’s.
This work was supported by a grant from Research Institute for Basic Science at Kangwon National University. 相似文献
16.
Altannar Chinchuluun Dehui Yuan Panos M. Pardalos 《Annals of Operations Research》2007,154(1):133-147
In this paper, we consider nondifferentiable multiobjective fractional programming problems. A concept of generalized convexity,
which is called (C,α,ρ,d)-convexity, is first discussed. Based on this generalized convexity, we obtain efficiency conditions for multiobjective fractional
programming (MFP). Furthermore, we establish duality results for three types of dual problems of (MFP) and present the corresponding
duality theorems. 相似文献
17.
D. P. Bertsekas 《Journal of Optimization Theory and Applications》2008,139(2):209-225
We consider the problem
where x
i
are multidimensional subvectors of x, f
i
are convex functions, and S is a subspace. Monotropic programming, extensively studied by Rockafellar, is the special case where the subvectors x
i
are the scalar components of x. We show a strong duality result that parallels Rockafellar’s result for monotropic programming, and contains other known
and new results as special cases. The proof is based on the use of ε-subdifferentials and the ε-descent method, which is used here as an analytical vehicle.
Work partially supported by the National Science Foundation Grant No. CCR-9731273. 相似文献
18.
Using a generalization of the classical ballot theorem, Niu and Cooper [7] established a duality relation between the joint
distribution of several variables associated with the busy cycle inM/G/1 (with a modified first service) and the corresponding joint distribution of several related variables in its dualGI/M/1. In this note, we generalize this duality relation toGI/G/1 queues with modified first services; this clarifies the original result, and shows that the generalized ballot theorem
is superfluous for the duality relation. 相似文献
19.
This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K
0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X
2 is not a sufficient condition of K
0(B(X)) = 0. 相似文献
20.
A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces
The purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem. Under suitable conditions, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces. The results presented in the paper improve and extend some recent results. 相似文献