Weak Duality

1983 ◽  
pp. 20-36
Author(s):  
Klaus Glashoff ◽  
Sven-Åke Gustafson
Keyword(s):  
Weak Duality

Weak duality and the asymptotic determination of mass spectra

1970 ◽  
Vol 65 (4) ◽  
pp. 683-688
Author(s):  
D. Atkinson ◽  
K. Dietz
Keyword(s):  
Mass Spectra ◽  
Weak Duality ◽  
Asymptotic Determination

scholarly journals A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 372
Author(s):  
Liu He ◽  
Qi-Lin Wang ◽  
Ching-Feng Wen ◽  
Xiao-Yan Zhang ◽  
Xiao-Bing Li
Keyword(s):  
Existence Theorem ◽  
Lower Order ◽  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Strong Duality ◽  
Higher Order ◽  
Weak Duality ◽  
Duality Theorems ◽  
Benson Proper Efficiency

In this paper, we introduce the notion of higher-order weak adjacent epiderivative for a set-valued map without lower-order approximating directions and obtain existence theorem and some properties of the epiderivative. Then by virtue of the epiderivative and Benson proper efficiency, we establish the higher-order Mond-Weir type dual problem for a set-valued optimization problem and obtain the corresponding weak duality, strong duality and converse duality theorems, respectively.

Lagrangean conditions and quasiduality

1977 ◽  
Vol 16 (3) ◽  
pp. 325-339 ◽  
Author(s):  
B.D. Craven
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Local Minimum ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Weak Duality ◽  
Partially Ordered Vector Space ◽  
Counter Example ◽  
Cone Constraints ◽  
Constrained Minimization Problem

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

Strong and weak duality principles for fractal dimension in Euclidean space

1995 ◽  
Vol 118 (3) ◽  
pp. 393-410 ◽  
Author(s):  
Colleen D. Cutler
Keyword(s):  
Strong Duality ◽  
The Other ◽  
Duality Principle ◽  
Weak Duality ◽  
Pointwise Dimension ◽  
Analytic Subset ◽  
Analytic Set ◽  
Borel Measures ◽  
Dual Representations ◽  
Packing Dimensions

AbstractTricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

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