Foundations of Mathematical Optimization

Pallaschke, Diethard; Rolewicz, S.

doi:10.1007/978-94-017-1588-1

Cited by 155 publications

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“…For more see [24]. It is also straightforward to show the basic properties such as linearity and convexity of the soft approximator.…”

Section: Soft Analysismentioning

confidence: 99%

Soft Vibrational Force on Stock Market Networks

Balcı¹,

Akgüller²

2016

OALib

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Stock market networks commonly involve uncertainty, and the theory of soft sets emerges as a powerful tool to handle it. In this study, we present a soft analogue of the differential of a vibrational potential function acting on a stock market network as vibrational force. For this purpose, we first study the vibrational potential function operating on each vertex by using the Laplacian of the neighborhood graph, then applied the soft approximator for the soft sets where the data points are embedded to Euclidean n space. We used the data of the globally operating leading stock markets of 17 countries and presented the results respect to them.

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“…For more see [24]. It is also straightforward to show the basic properties such as linearity and convexity of the soft approximator.…”

Section: Soft Analysismentioning

confidence: 99%

Soft Vibrational Force on Stock Market Networks

Balcı¹,

Akgüller²

2016

OALib

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“…His approach is connected with the following problem: given a monotone operator M : X =t X*, it is possible to get a closed convex function q on X x X* such that q*(x*,x) = q(x,x*) for any (x,x*) € X x X* and f M ^ q ^ p M , where f M is the Fitzpatrick representation of M and p M = f M . A positive answer is provided here in the broader framework of generalised convexity and generalised monotonicity (see [1,14,15,27,29]). For the study of maximal monotone operators and their representations by convex functions (on spaces which are larger than X x X*), we refer to the recent monograph by Simons [28].…”

Section: Introductionmentioning

confidence: 99%

“…Among dualities, a familiar class is formed by conjugacies (or conjugations), that is, dualities for which [14] and have been studied by a number of authors (see [1,6,11,16,19,18,15,22,23,26] and the references therein). It has also be shown by When W is a reflexive Banach space X, Y = X* and c is the evaluation (x,x*) >->• x*(x), this duality is close to the classical Fenchel conjugacy since it is composed of this conjugacy with the interchange of variables (x*,x) >-* (x,x*).…”

Section: Introductionmentioning

confidence: 99%

Autoconjugate functions and representations of monotone operators

Penot

2003

Bull. Austral. Math. Soc.

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“…With such a family one can associate a notion of convexity, a polarity, and a conjugacy, as we now recall. We refer to [15], [26], [27], [54], [77], [85], [97], [99] for extensive treatments and axiomatic approaches; for the notion of polarity see also [29], [57], [58], [60], [97]. We say that a subset A of X is E-convex if it is an intersection of a family of elementary subsets of X.…”

Section: A General Framework: Dualities and Polaritiesmentioning

confidence: 99%

“…Here we adopt a more general framework since in microeconomics one usually set the problems in orthants or cones rather than in the whole spaces. The just defined conjugacy has the advantage of entering into the general framework of the Fenchel-Moreau conjugacy (see [53], [6] and many others references such as [27], [39], [38], [58], [66], [54], [77], [85]) for which the conjugate of f is given by relation (3) where c : X × Y → R := R ∪ {−∞, ∞} is an appropriate coupling function. In order to see that, it suffices to take for c the opposite of the indicator function ι F of (the graph of) F := (X × Y ) \E −1 , given by ι F (x, y) = 0 for (x, y) ∈ F, +∞ otherwise.…”

Section: A General Framework: Dualities and Polaritiesmentioning

confidence: 99%

The bearing of duality on microeconomics

Penot

Advances in Mathematical Economics

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We present some observations about links between some classical theories of microeconomics and dualities which have been used in optimization theory and in the study of firstorder Hamilton-Jacobi equations. We introduce a variant of the classical indirect utility function called the wary indirect utility function and a variant of the expenditure function. We focus the attention on the links between these functions, observing that they have better relationships with the direct functions than their classical forms and we give economic interpretations of them.Mathematics Classification Scheme 2000 91B16, 91B38, 91B02, 91B08. Keywords: conjugacy, cost function, duality, expenditure function, indirect utility function, polarity, price, subdifferential, utility function. Dedicated to the memory of Philippe MichelThe bearing of duality on a number of questions in micro-economic theory has been known for several years (see [14] .).Here we start with a conjugacy deduced from a polarity. This simple instance of duality encompasses the passage from an utility function to an indirect utility function. The observation that this passage stems from a polarity allows to deduce characterizations and some properties from general results. A similar fact occurs for production functions. The passage from an utility function to an expenditure function requires a more sophisticated duality. It appears that this duality has been used recently for the study of Hamilton-Jacobi equations ([1 . We devote some attention to a comparison between the approach used in that field and the classical scheme of mathematical economics.The main results point out that a natural variant of the inverse utility function gives rise to a simpler correspondence. Similarly, a natural variant of the expenditure function yields a function (called the wary expenditure function) which has better properties and whose characterization

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Foundations of Mathematical Optimization

Cited by 155 publications

References 0 publications

Soft Vibrational Force on Stock Market Networks

Soft Vibrational Force on Stock Market Networks

Autoconjugate functions and representations of monotone operators

The bearing of duality on microeconomics

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