Academic literature on the topic 'Fenchel duality theorem'
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Journal articles on the topic "Fenchel duality theorem"
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Rodrigues, B. C. "The Fenchel duality theorem in Fréchet spaees." Optimization 21, no. 1 (January 1990): 13–22. http://dx.doi.org/10.1080/02331939008843516.
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Hamel, Andreas H. "A Fenchel–Rockafellar duality theorem for set-valued optimization." Optimization 60, no. 8-9 (August 2011): 1023–43. http://dx.doi.org/10.1080/02331934.2010.534794.
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Reich, Simeon, and Stephen Simons. "Fenchel duality, Fitzpatrick functions and the Kirszbraun–Valentine extension theorem." Proceedings of the American Mathematical Society 133, no. 9 (March 22, 2005): 2657–60. http://dx.doi.org/10.1090/s0002-9939-05-07983-9.
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Liu, Sanming, and Enmin Feng. "Fenchel duality theorem in multiobjective programming problems with set functions." Journal of Applied Mathematics and Computing 13, no. 1-2 (September 2003): 139–52. http://dx.doi.org/10.1007/bf02936081.
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Boţ, R. I., S. M. Grad, and G. Wanka. "Fenchel’s Duality Theorem for Nearly Convex Functions." Journal of Optimization Theory and Applications 132, no. 3 (June 15, 2007): 509–15. http://dx.doi.org/10.1007/s10957-007-9234-9.
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Boccuto, Antonio. "Hahn-Banach and Sandwich Theorems for Equivariant Vector Lattice-Valued Operators and Applications." Tatra Mountains Mathematical Publications 76, no. 1 (December 1, 2020): 11–34. http://dx.doi.org/10.2478/tmmp-2020-0015.
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AbstractWe prove Hahn-Banach, sandwich and extension theorems for vector lattice-valued operators, equivariant with respect to a given group G of homomorphisms. As applications and consequences, we present some Fenchel duality and separation theorems, a version of the Moreau-Rockafellar formula and some Farkas and Kuhn-Tucker-type optimization results. Finally, we prove that the obtained results are equivalent to the amenability of G.
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Boccuto, Antonio. "Subdifferential calculus for invariant linear ordered vector space-valued operators and applications." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 4 (May 30, 2016): 6160–70. http://dx.doi.org/10.24297/jam.v12i4.386.
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We give a direct proof of sandwich-type theorems for linear invariant partially ordered vector space operators in the setting of convexity. As consequences, we deduce equivalence results between sandwich, Hahn-Banach, separation and Krein-type extension theorems, Fenchel duality, Farkas and Kuhn-Tucker-type minimization results and subdifferential formulas in the context of invariance. As applications, we give Tarski-type extension theorems and related examples for vector lattice-valued invariant probabilities, defined on suitable kinds of events.
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Zhou, Yuying, and Gang Li. "The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions." Numerical Algebra, Control & Optimization 4, no. 1 (2014): 9–23. http://dx.doi.org/10.3934/naco.2014.4.9.
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Roos, Kees, Marleen Balvert, Bram L. Gorissen, and Dick den Hertog. "A Universal and Structured Way to Derive Dual Optimization Problem Formulations." INFORMS Journal on Optimization 2, no. 4 (October 2020): 229–55. http://dx.doi.org/10.1287/ijoo.2019.0034.
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The dual problem of a convex optimization problem can be obtained in a relatively simple and structural way by using a well-known result in convex analysis, namely Fenchel’s duality theorem. This alternative way of forming a strong dual problem is the subject of this paper. We recall some standard results from convex analysis and then discuss how the dual problem can be written in terms of the conjugates of the objective function and the constraint functions. This is a didactically valuable method to explicitly write the dual problem. We demonstrate the method by deriving dual problems for several classical problems and also for a practical model for radiotherapy treatment planning, for which deriving the dual problem using other methods is a more tedious task. Additional material is presented in the appendices, including useful tables for finding conjugate functions of many functions.
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Carpio, Ronaldo, and Takashi Kamihigashi. "Fast value iteration: an application of Legendre-Fenchel duality to a class of deterministic dynamic programming problems in discrete time." Journal of Difference Equations and Applications 26, no. 2 (January 31, 2020): 209–22. http://dx.doi.org/10.1080/10236198.2020.1713770.
Full textDissertations / Theses on the topic "Fenchel duality theorem"
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Lorenz, Nicole. "Application of the Duality Theory." Doctoral thesis, Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-94108.
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The aim of this thesis is to present new results concerning duality in scalar optimization. We show how the theory can be applied to optimization problems arising in the theory of risk measures, portfolio optimization and machine learning.
First we give some notations and preliminaries we need within the thesis. After that we recall how the well-known Lagrange dual problem can be derived by using the general perturbation theory and give some generalized interior point regularity conditions used in the literature. Using these facts we consider some special scalar optimization problems having a composed objective function and geometric (and cone) constraints. We derive their duals, give strong duality results and optimality condition using some regularity conditions. Thus we complete and/or extend some results in the literature especially by using the mentioned regularity conditions, which are weaker than the classical ones. We further consider a scalar optimization problem having single chance constraints and a convex objective function. We also derive its dual, give a strong duality result and further consider a special case of this problem. Thus we show how the conjugate duality theory can be used for stochastic programming problems and extend some results given in the literature.
In the third chapter of this thesis we consider convex risk and deviation measures. We present some more general measures than the ones given in the literature and derive formulas for their conjugate functions. Using these we calculate some dual representation formulas for the risk and deviation measures and correct some formulas in the literature. Finally we proof some subdifferential formulas for measures and risk functions by using the facts above.
The generalized deviation measures we introduced in the previous chapter can be used to formulate some portfolio optimization problems we consider in the fourth chapter. Their duals, strong duality results and optimality conditions are derived by using the general theory and the conjugate functions, respectively, given in the second and third chapter. Analogous calculations are done for a portfolio optimization problem having single chance constraints using the general theory given in the second chapter. Thus we give an application of the duality theory in the well-developed field of portfolio optimization.
We close this thesis by considering a general Support Vector Machines problem and derive its dual using the conjugate duality theory. We give a strong duality result and necessary as well as sufficient optimality conditions. By considering different cost functions we get problems for Support Vector Regression and Support Vector Classification. We extend the results given in the literature by dropping the assumption of invertibility of the kernel matrix. We use a cost function that generalizes the well-known Vapnik's ε-insensitive loss and consider the optimization problems that arise by using this. We show how the general theory can be applied for a real data set, especially we predict the concrete compressive strength by using a special Support Vector Regression problem.
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Csetnek, Ernö Robert. "Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operators." Doctoral thesis, Universitätsbibliothek Chemnitz, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200902025.
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The aim of this work is to present several new results concerning
duality in scalar convex optimization, the formulation of sequential
optimality conditions and some applications of the duality to the theory
of maximal monotone operators.
After recalling some properties of the classical generalized
interiority notions which exist in the literature, we give some
properties of the quasi interior and quasi-relative interior,
respectively. By means of these notions we introduce several
generalized interior-point regularity conditions which guarantee
Fenchel duality. By using an approach due to Magnanti, we derive
corresponding regularity conditions expressed via the quasi
interior and quasi-relative interior which ensure Lagrange
duality. These conditions have the advantage to be applicable in
situations when other classical regularity conditions fail.
Moreover, we notice that several duality results given in the
literature on this topic have either superfluous or contradictory
assumptions, the investigations we make offering in this sense an
alternative.
Necessary and sufficient sequential optimality conditions for a
general convex optimization problem are established via
perturbation theory. These results are applicable even in the
absence of regularity conditions. In particular, we show that
several results from the literature dealing with sequential
optimality conditions are rediscovered and even improved.
The second part of the thesis is devoted to applications of the
duality theory to enlargements of maximal monotone operators in
Banach spaces. After establishing a necessary and sufficient
condition for a bivariate infimal convolution formula, by
employing it we equivalently characterize the
$\varepsilon$-enlargement of the sum of two maximal monotone
operators. We generalize in this way a classical result
concerning the formula for the $\varepsilon$-subdifferential of
the sum of two proper, convex and lower semicontinuous functions.
A characterization of fully enlargeable monotone operators is also
provided, offering an answer to an open problem stated in the
literature. Further, we give a regularity condition for the
weak$^*$-closedness of the sum of the images of enlargements of
two maximal monotone operators.
The last part of this work deals with enlargements of positive sets in SSD spaces. It is shown that many results from the literature concerning enlargements of maximal monotone operators can be generalized to the setting of Banach SSD spaces.
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Hodrea, Ioan Bogdan. "Farkas - type results for convex and non - convex inequality systems." Doctoral thesis, [S.l. : s.n.], 2008. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200800075.
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Lorenz, Nicole. "Application of the Duality Theory: New Possibilities within the Theory of Risk Measures, Portfolio Optimization and Machine Learning." Doctoral thesis, 2011. https://monarch.qucosa.de/id/qucosa%3A19760.
Full textAbstract:
The aim of this thesis is to present new results concerning duality in scalar optimization. We show how the theory can be applied to optimization problems arising in the theory of risk measures, portfolio optimization and machine learning.
First we give some notations and preliminaries we need within the thesis. After that we recall how the well-known Lagrange dual problem can be derived by using the general perturbation theory and give some generalized interior point regularity conditions used in the literature. Using these facts we consider some special scalar optimization problems having a composed objective function and geometric (and cone) constraints. We derive their duals, give strong duality results and optimality condition using some regularity conditions. Thus we complete and/or extend some results in the literature especially by using the mentioned regularity conditions, which are weaker than the classical ones. We further consider a scalar optimization problem having single chance constraints and a convex objective function. We also derive its dual, give a strong duality result and further consider a special case of this problem. Thus we show how the conjugate duality theory can be used for stochastic programming problems and extend some results given in the literature.
In the third chapter of this thesis we consider convex risk and deviation measures. We present some more general measures than the ones given in the literature and derive formulas for their conjugate functions. Using these we calculate some dual representation formulas for the risk and deviation measures and correct some formulas in the literature. Finally we proof some subdifferential formulas for measures and risk functions by using the facts above.
The generalized deviation measures we introduced in the previous chapter can be used to formulate some portfolio optimization problems we consider in the fourth chapter. Their duals, strong duality results and optimality conditions are derived by using the general theory and the conjugate functions, respectively, given in the second and third chapter. Analogous calculations are done for a portfolio optimization problem having single chance constraints using the general theory given in the second chapter. Thus we give an application of the duality theory in the well-developed field of portfolio optimization.
We close this thesis by considering a general Support Vector Machines problem and derive its dual using the conjugate duality theory. We give a strong duality result and necessary as well as sufficient optimality conditions. By considering different cost functions we get problems for Support Vector Regression and Support Vector Classification. We extend the results given in the literature by dropping the assumption of invertibility of the kernel matrix. We use a cost function that generalizes the well-known Vapnik's ε-insensitive loss and consider the optimization problems that arise by using this. We show how the general theory can be applied for a real data set, especially we predict the concrete compressive strength by using a special Support Vector Regression problem.
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Zhong, Yangfan. "Joint Source-Channel Coding Reliability Function for Single and Multi-Terminal Communication Systems." Thesis, 2008. http://hdl.handle.net/1974/1207.
Full textAbstract:
Traditionally, source coding (data compression) and channel coding (error protection) are performed separately and sequentially, resulting in what we call a tandem (separate) coding system. In
practical implementations, however, tandem coding might involve a large delay and a high coding/decoding complexity, since one needs to remove the redundancy in the source coding part and then insert certain redundancy in the channel coding part. On the other hand, joint source-channel coding (JSCC), which coordinates source and channel coding or combines them into a single step, may offer substantial improvements over the tandem coding approach.
This thesis deals with the fundamental Shannon-theoretic limits for a variety of communication systems via JSCC. More specifically, we investigate the reliability function (which is the largest rate at which the coding probability of error vanishes exponentially with
increasing blocklength) for JSCC for the following discrete-time communication systems: (i) discrete memoryless systems; (ii) discrete memoryless systems with perfect channel feedback; (iii) discrete memoryless systems with source side information; (iv) discrete systems with Markovian memory; (v) continuous-valued
(particularly Gaussian) memoryless systems; (vi) discrete asymmetric 2-user source-channel systems.
For the above systems, we establish upper and lower bounds for the JSCC reliability function and we analytically compute these bounds. The conditions for which the upper and lower bounds coincide are also provided. We show that the conditions are satisfied for a large class of source-channel systems, and hence exactly determine the reliability function. We next provide a systematic comparison between the JSCC reliability function and the tandem coding reliability function (the reliability function resulting from separate source and channel coding). We show that the JSCC reliability function is substantially larger than the tandem coding
reliability function for most cases. In particular, the JSCC reliability function is close to twice as large as the tandem coding reliability function for many source-channel pairs. This exponent gain provides a theoretical underpinning and justification for JSCC design as opposed to the widely used tandem coding method, since
JSCC will yield a faster exponential rate of decay for the system error probability and thus provides substantial reductions in
complexity and coding/decoding delay for real-world communication systems.Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-05-13 22:31:56.425
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Hodrea, Ioan Bogdan. "Farkas - type results for convex and non - convex inequality systems." Doctoral thesis, 2007. https://monarch.qucosa.de/id/qucosa%3A18859.
Full textAbstract:
As the title already suggests the aim of the present work is to present Farkas -
type results for inequality systems involving convex and/or non - convex functions.
To be able to give the desired results, we treat optimization problems which involve
convex and composed convex functions or non - convex functions like DC functions
or fractions.
To be able to use the fruitful Fenchel - Lagrange duality approach, to the primal
problem we attach an equivalent problem which is a convex optimization problem.
After giving a dual problem to the problem we initially treat, we provide weak
necessary conditions which secure strong duality, i.e., the case when the optimal
objective value of the primal problem coincides with the optimal objective value of
the dual problem and, moreover, the dual problem has an optimal solution.
Further, two ideas are followed. Firstly, using the weak and strong duality
between the primal problem and the dual problem, we are able to give necessary
and sufficient optimality conditions for the optimal solutions of the primal problem.
Secondly, provided that no duality gap lies between the primal problem and its
Fenchel - Lagrange - type dual we are able to demonstrate some Farkas - type
results and thus to underline once more the connections between the theorems of
the alternative and the theory of duality. One statement of the above mentioned
Farkas - type results is characterized using only epigraphs of functions.
We conclude our investigations by providing necessary and sufficient optimality
conditions for a multiobjective programming problem involving composed convex
functions. Using the well-known linear scalarization to the primal multiobjective
program a family of scalar optimization problems is attached. Further to each of
these scalar problems the Fenchel - Lagrange dual problem is determined. Making
use of the weak and strong duality between the scalarized problem and its dual the
desired optimality conditions are proved. Moreover, the way the dual problem of
the scalarized problem looks like gives us an idea about how to construct a vector
dual problem to the initial one. Further weak and strong vector duality assertions
are provided.
Books on the topic "Fenchel duality theorem"
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Chambers, Robert G. Competitive Agents in Certain and Uncertain Markets. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190063016.001.0001.
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This book uses concepts from optimization theory to develop an integrated analytic framework for treating consumer, producer, and market equilibrium analyses as special cases of a generic optimization problem. The same framework applies to both stochastic and non-stochastic decision settings, so that the latter is recognized as an (important) special case of the former. The analytic techniques are borrowed from convex analysis and variational analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. The book shows how virtually identical conjugate analyses form the basis for modeling economic behavior in each of the areas studied. The basic analytic concepts are borrowed from convex analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. It is demonstrated how virtually identical conjugate analyses form the basis for modelling economic behaviour in each of the areas studied.
Book chapters on the topic "Fenchel duality theorem"
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Chambers, Robert G. "Differentials and Convex Analysis." In Competitive Agents in Certain and Uncertain Markets, 7–64. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190063016.003.0002.
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Mathematical tools necessary to the argument are presented and discussed. The focus is on concepts borrowed from the convex analysis and variational analysis literatures. The chapter starts by introducing the notions of a correspondence, upper hemi-continuity, and lower hemi-continuity. Superdifferential and subdifferential correspondences for real-valued functions are then introduced, and their essential properties and their role in characterizing global optima are surveyed. Convex sets are introduced and related to functional concavity (convexity). The relationship between functional concavity (convexity), superdifferentiability (subdifferentiability), and the existence of (one-sided) directional derivatives is examined. The theory of convex conjugates and essential conjugate duality results are discussed. Topics treated include Berge's Maximum Theorem, cyclical monotonicity of superdifferential (subdifferential) correspondences, concave (convex) conjugates and biconjugates, Fenchel's Inequality, the Fenchel-Rockafellar Conjugate Duality Theorem, support functions, superlinear functions, sublinear functions, the theory of infimal convolutions and supremal convolutions, and Fenchel's Duality Theorem.
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Chambers, Robert G. "Equilibrium, Efficiency, and Welfare." In Competitive Agents in Certain and Uncertain Markets, 211–34. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190063016.003.0007.
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Competitive equilibria are studied in both partial-equilibrium and general-equilibrium settings for economies characterized by consumers with incomplete preference structures. Market equilibrium determination is developed as solving a zero-maximum problem for a supremal convolution whose dual, by Fenchel's Duality Theorem, coincides with a zero-minimum for an infimal convolution that characterizes Pareto optima. The First and Second Welfare Theorems are natural consequences. The maximization of the sum of consumer surplus and producer surplus is studied in this analytic setting, and the implications of nonsmooth preference structures or technologies for equilibrium determination are discussed.