Journal articles: 'Convex duality theory'

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Shapiro, Alexander. "On duality theory of convex semi-infinite programming." Optimization 54, no. 6 (December 2005): 535–43. http://dx.doi.org/10.1080/02331930500342823.

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Sun, Wenyu, Chengjin Li, and Raimundo J. B. Sampaio. "On duality theory for non-convex semidefinite programming." Annals of Operations Research 186, no. 1 (March 23, 2011): 331–43. http://dx.doi.org/10.1007/s10479-011-0861-z.

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Richardt, J., F. Karl, and C. Müller. "Connections between fuzzy theory, simulated annealing, and convex duality." Fuzzy Sets and Systems 96, no. 3 (June 1998): 307–34. http://dx.doi.org/10.1016/s0165-0114(96)00301-6.

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Brinkhuis, Ya, and V. M. Tikhomirov. "Duality and calculus of convex objects (theory and applications)." Sbornik: Mathematics 198, no. 2 (February 28, 2007): 171–206. http://dx.doi.org/10.1070/sm2007v198n02abeh003833.

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Lee, Mi Jin, Jong Yeoul Park, and Young Chel Kwon. "Duality in the optimal control for damped hyperbolic systems with positive control." International Journal of Mathematics and Mathematical Sciences 2003, no. 27 (2003): 1703–14. http://dx.doi.org/10.1155/s0161171203209273.

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We study the duality theory for damped hyperbolic equations. These systems have positive controls and convex cost functionals. Our main results lie in the application of duality theorem, that is,inf J=sup K, on various cost functions.

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Ekeland, Ivar. "A duality theory for some non-convex functions of matrices." Ricerche di Matematica 55, no. 1 (July 2006): 1–12. http://dx.doi.org/10.1007/s11587-006-0001-2.

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Park, Jong Yeoul, and Mi Jin Lee. "Duality in the optimal control of hyperbolic equations with positive controls." International Journal of Mathematics and Mathematical Sciences 23, no. 3 (2000): 181–88. http://dx.doi.org/10.1155/s0161171200002015.

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Lan, K. Q., and J. R. L. Webb. "A-properness and fixed point theorems for dissipative type maps." Abstract and Applied Analysis 4, no. 2 (1999): 83–100. http://dx.doi.org/10.1155/s108533759900010x.

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We obtain newA-properness results for demicontinuous, dissipative type mappings defined only on closed convex subsets of a Banach spaceXwith uniformly convex dual and which satisfy a property called weakly inward. The method relies on a new property of the duality mapping in such spaces. New fixed point results are obtained by utilising a theory of fixed point index.

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CHAN, W. L., and S. P. YUNG. "Duality Theory for the Linear-Convex Optimal Control Problem with Delays." IMA Journal of Mathematical Control and Information 4, no. 3 (1987): 251–62. http://dx.doi.org/10.1093/imamci/4.3.251.

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Auslender, A., R. Cominetti, and J. P. Crouziex. "Convex Functions with Unbounded Level Sets and Applications to Duality Theory." SIAM Journal on Optimization 3, no. 4 (November 1993): 669–87. http://dx.doi.org/10.1137/0803034.

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Ben-Tal, A., and M. Teboulle. "Rate distortion theory with generalized information measures via convex programming duality." IEEE Transactions on Information Theory 32, no. 5 (September 1986): 630–41. http://dx.doi.org/10.1109/tit.1986.1057223.

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Tam, Bit-Shun. "On the duality operator of a convex cone." Linear Algebra and its Applications 64 (January 1985): 33–56. http://dx.doi.org/10.1016/0024-3795(85)90265-4.

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Kamenev, G. K. "Duality theory of optimal adaptive methods for polyhedral approximation of convex bodies." Computational Mathematics and Mathematical Physics 48, no. 3 (March 2008): 376–94. http://dx.doi.org/10.1134/s0965542508030056.

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Borwein, J. M., and A. S. Lewis. "Partially finite convex programming, Part I: Quasi relative interiors and duality theory." Mathematical Programming 57, no. 1-3 (May 1992): 15–48. http://dx.doi.org/10.1007/bf01581072.

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Bouchitté, Guy, and Ilaria Fragalà. "A Duality Theory for Non-convex Problems in the Calculus of Variations." Archive for Rational Mechanics and Analysis 229, no. 1 (February 1, 2018): 361–415. http://dx.doi.org/10.1007/s00205-018-1219-3.

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Marti, Johannes, and Riccardo Pinosio. "A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries." Order 37, no. 1 (June 10, 2019): 151–71. http://dx.doi.org/10.1007/s11083-019-09497-0.

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Bartl, Daniel, Patrick Cheridito, Michael Kupper, and Ludovic Tangpi. "Duality for increasing convex functionals with countably many marginal constraints." Banach Journal of Mathematical Analysis 11, no. 1 (January 2017): 72–89. http://dx.doi.org/10.1215/17358787-3750133.

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Yu, Wuyang, Gangsong Leng, and Donghua Wu. "DUAL $L_{p}$ JOHN ELLIPSOIDS." Proceedings of the Edinburgh Mathematical Society 50, no. 3 (October 2007): 737–53. http://dx.doi.org/10.1017/s0013091506000332.

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AbstractIn this paper, the dual $L_p$ John ellipsoids, which include the classical Löwner ellipsoid and the Legendre ellipsoid, are studied. The dual $L_p$ versions of John's inclusion and Ball's volume-ratio inequality are shown. This insight allows for a unified view of some basic results in convex geometry and reveals further the amazing duality between Brunn–Minkowski theory and dual Brunn–Minkowski theory.

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Lewis, Adrian S., and Michael L. Overton. "Eigenvalue optimization." Acta Numerica 5 (January 1996): 149–90. http://dx.doi.org/10.1017/s0962492900002646.

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Optimization problems involving eigenvalues arise in many different mathematical disciplines. This article is divided into two parts. Part I gives a historical account of the development of the field. We discuss various applications that have been especially influential, from structural analysis to combinatorial optimization, and we survey algorithmic developments, including the recent advance of interior-point methods for a specific problem class: semidefinite programming. In Part II we primarily address optimization of convex functions of eigenvalues of symmetric matrices subject to linear constraints. We derive a fairly complete mathematical theory, some of it classical and some of it new. Using the elegant language of conjugate duality theory, we highlight the parallels between the analysis of invariant matrix norms and weakly invariant convex matrix functions. We then restrict our attention further to linear and semidefinite programming, emphasizing the parallel duality theory and comparing primal-dual interior-point methods for the two problem classes. The final section presents some apparently new variational results about eigenvalues of nonsymmetric matrices, unifying known characterizations of the spectral abscissa (related to Lyapunov theory) and the spectral radius (as an infimum of matrix norms).

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Lacker, Daniel. "A non-exponential extension of Sanov’s theorem via convex duality." Advances in Applied Probability 52, no. 1 (March 2020): 61–101. http://dx.doi.org/10.1017/apr.2019.52.

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AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

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Yadav, Tamanna, and S. K. Gupta. "On duality theory for multiobjective semi-infinite fractional optimization model using higher order convexity." RAIRO - Operations Research 55, no. 3 (May 2021): 1343–70. http://dx.doi.org/10.1051/ro/2021064.

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In the article, a semi-infinite fractional optimization model having multiple objectives is first formulated. Due to the presence of support functions in each numerator and denominator with constraints, the model so constructed is also non-smooth. Further, three different types of dual models viz Mond-Weir, Wolfe and Schaible are presented and then usual duality results are proved using higher-order (K × Q) − (ℱ, α, ρ, d)-type I convexity assumptions. To show the existence of such generalized convex functions, a nontrivial example has also been exemplified. Moreover, numerical examples have been illustrated at suitable places to justify various results presented in the paper. The formulation and duality results discussed also generalize the well known results appeared in the literature.

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Kong, Xiangyu, Yinfeng Zhang, and GuoLin Yu. "Optimality and duality in set-valued optimization utilizing limit sets." Open Mathematics 16, no. 1 (October 19, 2018): 1128–39. http://dx.doi.org/10.1515/math-2018-0095.

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AbstractThis paper deals with optimality conditions and duality theory for vector optimization involving non-convex set-valued maps. Firstly, under the assumption of nearly cone-subconvexlike property for set-valued maps, the necessary and sufficient optimality conditions in terms of limit sets are derived for local weak minimizers of a set-valued constraint optimization problem. Then, applications to Mond-Weir type and Wolfe type dual problems are presented.

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Sun, Xiangkai, Xian-Jun Long, and Liping Tang. "Regularity conditions and Farkas-type results for systems with fractional functions." RAIRO - Operations Research 54, no. 5 (July 28, 2020): 1369–84. http://dx.doi.org/10.1051/ro/2019070.

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This paper deals with some new versions of Farkas-type results for a system involving cone convex constraint, a geometrical constraint as well as a fractional function. We first introduce some new notions of regularity conditions in terms of the epigraphs of the conjugate functions. By using these regularity conditions, we obtain some new Farkas-type results for this system using an approach based on the theory of conjugate duality for convex or DC optimization problems. Moreover, we also show that some recently obtained results in the literature can be rediscovered as special cases of our main results.

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Rao, Murali, and Zoran Vondraćek. "Nonlinear potentials in function spaces." Nagoya Mathematical Journal 165 (March 2002): 91–116. http://dx.doi.org/10.1017/s0027763000008163.

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We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

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Dumir, V. C., R. J. Hans-Gill, and J. B. Wilker. "Contributions to a General Theory of View-Obstruction Problems." Canadian Journal of Mathematics 45, no. 3 (June 1, 1993): 517–36. http://dx.doi.org/10.4153/cjm-1993-027-9.

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AbstractIn the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

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Dette, Holger. "A note on some peculiar nonlinear extremal phenomena of the Chebyshev polynomials." Proceedings of the Edinburgh Mathematical Society 38, no. 2 (June 1995): 343–55. http://dx.doi.org/10.1017/s001309150001912x.

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We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials (where Pj(x) is a polynomial of degree j) under the restriction that the sup-norm of is bounded on the interval [ −b, b] (b>0). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval [ −b, b].

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Kurkina, M. V., S. P. Semenov, V. V. Slavsky, O. V. Samarina, O. A. Petuhova, A. A. Petrov, A. A. Finogenov, and V. A. Samarin. "Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals." Izvestiya of Altai State University, no. 4(114) (September 9, 2020): 96–101. http://dx.doi.org/10.14258/izvasu(2020)4-15.

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In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

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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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Johnston, Nathaniel, and David W. Kribs. "A family of norms with applications in quantum information theory." Quantum Information and Computation 11, no. 1&2 (January 2011): 104–23. http://dx.doi.org/10.26421/qic11.1-2-8.

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We consider the problem of computing the family of operator norms recently introduced. We develop a family of semidefinite programs that can be used to exactly compute them in small dimensions and bound them in general. Some theoretical consequences follow from the duality theory of semidefinite programming, including a new constructive proof that for all r there are non-positive partial transpose Werner states that are r-undistillable. Several examples are considered via a MATLAB implementation of the semidefinite program, including the case of Werner states and randomly generated states via the Bures measure, and approximate distributions of the norms are provided. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states.

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Sriperumbudur, Bharath K., and Gert R. G. Lanckriet. "A Proof of Convergence of the Concave-Convex Procedure Using Zangwill's Theory." Neural Computation 24, no. 6 (June 2012): 1391–407. http://dx.doi.org/10.1162/neco_a_00283.

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The concave-convex procedure (CCCP) is an iterative algorithm that solves d.c. (difference of convex functions) programs as a sequence of convex programs. In machine learning, CCCP is extensively used in many learning algorithms, including sparse support vector machines (SVMs), transductive SVMs, and sparse principal component analysis. Though CCCP is widely used in many applications, its convergence behavior has not gotten a lot of specific attention. Yuille and Rangarajan analyzed its convergence in their original paper; however, we believe the analysis is not complete. The convergence of CCCP can be derived from the convergence of the d.c. algorithm (DCA), proposed in the global optimization literature to solve general d.c. programs, whose proof relies on d.c. duality. In this note, we follow a different reasoning and show how Zangwill's global convergence theory of iterative algorithms provides a natural framework to prove the convergence of CCCP. This underlines Zangwill's theory as a powerful and general framework to deal with the convergence issues of iterative algorithms, after also being used to prove the convergence of algorithms like expectation-maximization and generalized alternating minimization. In this note, we provide a rigorous analysis of the convergence of CCCP by addressing two questions: When does CCCP find a local minimum or a stationary point of the d.c. program under consideration? and when does the sequence generated by CCCP converge? We also present an open problem on the issue of local convergence of CCCP.

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ZĂlinescu, C. "Duality results involving functions associated to nonempty subsets of locally convex spaces." Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 103, no. 2 (September 2009): 219–34. http://dx.doi.org/10.1007/bf03191905.

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Jameson, G. J. O. "2-convexity and 2-concavity in Schatten ideals." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 4 (November 1996): 697–702. http://dx.doi.org/10.1017/s0305004100001651.

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The properties p-convexity and q-concavity are fundamental in the study of Banach sequence spaces (see [L-TzII]), and in recent years have been shown to be of great significance in the theory of the corresponding Schatten ideals ([G-TJ], [LP-P] and many other papers). In particular, the notions 2-convex and 2-concave are meaningful in Schatten ideals. It seems to have been noted only recently [LP-P] that a Schatten ideal has either of these properties if the underlying sequence space has. One way of establishing this is to use the fact that if (E, ‖ ‖E) is 2-convex, then there is another Banach sequence space (F, ‖ ‖F) such that ‖x;‖ = ‖x2‖F for all x ε E. The 2-concave case can then be deduced using duality, though this raises some difficulties, for example when E is inseparable.

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GUEYE, M. M., M. SENE, M. NDIAYE, and N. DJITTE. "Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces." Creative Mathematics and Informatics 29, no. 1 (January 30, 2020): 27–36. http://dx.doi.org/10.37193/cmi.2020.01.04.

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Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

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BAUSCHKE, HEINZ H., JONATHAN M. BORWEIN, and PATRICK L. COMBETTES. "ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES." Communications in Contemporary Mathematics 03, no. 04 (November 2001): 615–47. http://dx.doi.org/10.1142/s0219199701000524.

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The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed and several useful characterizations are given. The proofs rely on new results on the more subtle behavior of subdifferentials and directional derivatives at boundary points of the domain. In weak Asplund spaces, a new formula allows the recovery of the subdifferential from nearby gradients. Finally, it is shown that every Legendre function on a reflexive Banach space is zone consistent, a fundamental property in the analysis of optimization algorithms based on Bregman distances. Numerous illustrating examples are provided.

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Lewis, A. S. "Consistency of Moment Systems." Canadian Journal of Mathematics 47, no. 5 (October 1, 1995): 995–1006. http://dx.doi.org/10.4153/cjm-1995-052-2.

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AbstractAn important question in the study of moment problems is to determine when a fixed point in ℝn lies in the moment cone of vectors , with μ a nonnegative measure. In associated optimization problems it is also important to be able to distinguish between the interior and boundary of the moment cone. Recent work of Dachuna-Castelle, Gamboa and Gassiat derived elegant computational characterizations for these problems, and for related questions with an upper bound on μ. Their technique involves a probabilistic interpretation and large deviations theory. In this paper a purely convex analytic approach is used, giving a more direct understanding of the underlying duality, and allowing the relaxation of their assumptions.

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Bartels, Sören. "Error estimates for a class of discontinuous Galerkin methods for nonsmooth problems via convex duality relations." Mathematics of Computation 90, no. 332 (June 3, 2021): 2579–602. http://dx.doi.org/10.1090/mcom/3656.

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Gersten, S. M., and T. R. Riley. "SOME DUALITY CONJECTURES FOR FINITE GRAPHS AND THEIR GROUP THEORETIC CONSEQUENCES." Proceedings of the Edinburgh Mathematical Society 48, no. 2 (May 23, 2005): 389–421. http://dx.doi.org/10.1017/s0013091503000890.

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AbstractWe pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class $c$ nilpotent groups admit degree $c+1$ polynomial isoperimetric functions.Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions—the analogues of non-deterministic space and time.

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Sain, Debmalya. "On best approximations to compact operators." Proceedings of the American Mathematical Society 149, no. 10 (July 21, 2021): 4273–86. http://dx.doi.org/10.1090/proc/15494.

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We study best approximations to compact operators between Banach spaces and Hilbert spaces, from the point of view of Birkhoff-James orthogonality and semi-inner-products. As an application of the present study, some distance formulae are presented in the space of compact operators. The special case of bounded linear functionals as compact operators is treated separately and some applications to best approximations in reflexive, strictly convex and smooth Banach spaces are discussed. An explicit example is presented in ℓ p n \ell _p^{n} spaces, where 1 > p > ∞ , 1 > p > \infty , to illustrate the applicability of the methods developed in this article. A comparative analysis of the results presented in this article with the well-known classical duality principle in approximation theory is conducted to demonstrate the advantage in the former case, from a computational point of view.

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Bai, Zhiquan, Tongtong Wang, Piming Ma, Yanbo Ma, and Kyungsup Kwak. "Fair Resource Allocation with QoS Guarantee in Secure Multiuser TDMA Networks." Wireless Communications and Mobile Computing 2018 (July 11, 2018): 1–10. http://dx.doi.org/10.1155/2018/1489659.

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We investigate a secure multiuser time division multiple access (TDMA) system with statistical delay quality of service (QoS) guarantee in terms of secure effective capacity. An optimal resource allocation policy is proposed to minimize the β-fair cost function of the average user power under the individual QoS constraint, which also balances the energy efficiency and fairness among the users. First, convex optimization problems associated with the resource allocation policy are formulated. Then, a subgradient iteration algorithm based on the Lagrangian duality theory and the dual decomposition theory is employed to approach the global optimal solutions. Furthermore, considering the practical channel conditions, we develop a stochastic subgradient iteration algorithm which is capable of dynamically learning the intended wireless channels and acquiring the global optimal solution. It is shown that the proposed optimal resource allocation policy depends on the delay QoS requirement and the channel conditions. The optimal policy can save more power and achieve the balance of the energy efficiency and the fairness compared with the other resource allocation policies.

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Wang, Heng, Aijun Liu, and Xiaofei Pan. "Optimization of Joint Power and Bandwidth Allocation in Multi-Spot-Beam Satellite Communication Systems." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/683604.

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Multi-spot-beam technique has been widely applied in modern satellite communication systems. However, the satellite power and bandwidth resources in a multi-spot-beam satellite communication system are scarce and expensive; it is urgent to utilize the resources efficiently. To this end, dynamically allocating the power and bandwidth is an available way. This paper initially formulates the problem of resource joint allocation as a convex optimization problem, taking into account a compromise between the maximum total system capacity and the fairness among the spot beams. A joint bandwidth and power allocation iterative algorithm based on duality theory is then proposed to obtain the optimal solution of this optimization problem. Compared with the existing separate bandwidth or power optimal allocation algorithms, it is shown that the joint allocation algorithm improves both the total system capacity and the fairness among spot beams. Moreover, it is easy to be implemented in practice, as the computational complexity of the proposed algorithm is linear with the number of spot beams.

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Tsao, H. S. Jacob, and Shu-Cherng Fang. "Linear programming with inequality constraints via entropic perturbation." International Journal of Mathematics and Mathematical Sciences 19, no. 1 (1996): 177–84. http://dx.doi.org/10.1155/s0161171296000257.

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A dual convex programming approach to solving linear programs with inequality constraints through entropic perturbation is derived. The amount of perturbation required depends on the desired accuracy of the optimum. The dual program contains only non-positivity constraints. Anϵ-optimal solution to the linear program can be obtained effortlessly from the optimal solution of the dual program. Since cross-entropy minimization subject to linear inequality constraints is a special case of the perturbed linear program, the duality result becomes readily applicable. Many standard constrained optimization techniques can be specialized to solve the dual program. Such specializations, made possible by the simplicity of the constraints, significantly reduce the computational effort usually incurred by these methods. Immediate applications of the theory developed include an entropic path-following approach to solving linear semi-infinite programs with an infinite number of inequality constraints and the widely used entropy optimization models with linear inequality and/or equality constraints.

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Hansen, G., I. Herburt, H. Martini, and M. Moszyńska. "Starshaped sets." Aequationes mathematicae 94, no. 6 (May 20, 2020): 1001–92. http://dx.doi.org/10.1007/s00010-020-00720-7.

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Abstract This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines.

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Zalmai, G. J. "Parameter-free sufficient optimality conditions and duality models for minmax fractional subset programming problems with generalized(F ϱ, θ)-convex functions." Computers & Mathematics with Applications 45, no. 10-11 (May 2003): 1507–35. http://dx.doi.org/10.1016/s0898-1221(03)00134-2.

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Zhao, Yanchun, Shiqiang Hu, and Yongsheng Yang. "Inverse kinematics for the variable geometry truss manipulator via a Lagrangian dual method." International Journal of Advanced Robotic Systems 13, no. 6 (November 28, 2016): 172988141666677. http://dx.doi.org/10.1177/1729881416666779.

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This article studies the inverse kinematics problem of the variable geometry truss manipulator. The problem is cast as an optimization process which can be divided into two steps. Firstly, according to the information about the location of the end effector and fixed base, an optimal center curve and the corresponding distribution of the intermediate platforms along this center line are generated. This procedure is implemented by solving a non-convex optimization problem that has a quadratic objective function subject to quadratic constraints. Then, in accordance with the distribution of the intermediate platforms along the optimal center curve, all lengths of the actuators are calculated via the inverse kinematics of each variable geometry truss module. Hence, the approach that we present is an optimization procedure that attempts to generate the optimal intermediate platform distribution along the optimal central curve, while the performance index and kinematic constraints are satisfied. By using the Lagrangian duality theory, a closed-form optimal solution of the original optimization is given. The numerical simulation substantiates the effectiveness of the introduced approach.

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Lee, Hyunok. "Recent Applications of Nonparametric Programming Methods." Northeastern Journal of Agricultural and Resource Economics 21, no. 2 (October 1992): 113–20. http://dx.doi.org/10.1017/s0899367x00002622.

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Nonparametric techniques have recently come into vogue in agricultural economics: Applications abound in both consumer and producer models of the agricultural economy. Moreover, several distinct approaches to nonparametric analysis exist. There are nonparametric statistical techniques, semiparametric estimation techniques, nonparametric revealed-preference analysis of consumption data, and nonparametric analysis of production data. Both revealed-preference analysis and nonparametric analysis of production data rely on the basic fact, which provides the foundation for much of modern duality theory, that convex sets can be completely characterized by their supporting hyperplanes. This observation allows one to apply simple mathematical programming (in particular, linear programming) methods to analyze production and consumption data. My task today is to provide an overview of nonparametric programming approaches to production data. Thus, I will not address any of the other topics cited above. However, I would be remiss if I did not mention the close connection between these subject areas and what I intend to survey today. Moreover, one should also recognize that very closely related to the literature on nonparametric programming analysis of production data are the fields of estimation of efficiency frontier via statistical methods. (A useful survey here is Lovell and Schmidt).

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Wang, Heng, Aijun Liu, Xiaofei Pan, and Jianfei Yang. "Optimization of Power Allocation for Multiusers in Multi-Spot-Beam Satellite Communication Systems." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/780823.

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In recent years, multi-spot-beam satellite communication systems have played a key role in global seamless communication. However, satellite power resources are scarce and expensive, due to the limitations of satellite platform. Therefore, this paper proposes optimizing the power allocation of each user in order to improve the power utilization efficiency. Initially the capacity allocated to each user is calculated according to the satellite link budget equations, which can be achieved in the practical satellite communication systems. The problem of power allocation is then formulated as a convex optimization, taking account of a trade-off between the maximization of the total system capacity and the fairness of power allocation amongst the users. Finally, an iterative algorithm based on the duality theory is proposed to obtain the optimal solution to the optimization. Compared with the traditional uniform resource allocation or proportional resource allocation algorithms, the proposed optimal power allocation algorithm improves the fairness of power allocation amongst the users. Moreover, the computational complexity of the proposed algorithm is linear with both the numbers of the spot beams and users. As a result, the proposed power allocation algorithm is easy to be implemented in practice.

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Ali, Azhar, Amna Baig, Ghulam Mujtaba Awan, Wali Ullah Khan, Zain Ali, and Guftaar Ahmad Sardar Sidhu. "Efficient Resource Management for Sum Capacity Maximization in 5G NOMA Systems." Applied System Innovation 2, no. 3 (August 7, 2019): 27. http://dx.doi.org/10.3390/asi2030027.

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The modern cellular technologies are expected to provide high data rates and massive connectivity in fifth generation (5G) systems; however, this may be impossible through traditional radio access techniques. Recently, non-orthogonal multiple access (NOMA) has emerged as one of the promising cellular techniques for modern cellular communications with its ability to provide access for multiple users to the network over the same system resources. This paper studies resource management problem for downlink transmission of multiuser NOMA system. Our objective is to optimize both frequency and power resources for sum capacity maximization while taking into account each user minimum capacity requirement. Firstly, the problem of resource management decouples into two subproblems, that is, efficient sub-channel assignment and optimal power allocation, respectively. Secondly, for given power at base station, we design two sub-optimal algorithms for sub-channel assignment based on user channel condition and user minimum capacity requirement, respectively. Lastly, for any given sub-channel assignment, the problem first transforms into standard convex optimization problem and then we employ duality theory. To evaluate our proposed NOMA scheme, the enhanced version of existing NOMA optimization scheme is also presented as a benchmark. Results demonstrate that the proposed NOMA resource management scheme outperforms the benchmark NOMA optimization scheme in terms of sum capacity.

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Ha, Manh Linh, Dinh Huy Nguyen, and Thi Thanh Truc Nguyen. "Optimality conditions for set-valued optimization problems via scalarization function." Science & Technology Development Journal - Engineering and Technology 3, SI3 (December 27, 2020): first. http://dx.doi.org/10.32508/stdjet.v3isi3.642.

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One of the most important and popular topics in optimization problems is to find its optimal solutions, especially Pareto optimal points, a well-known solution introduced in multi-objective optimization. This topic is one of the oldest challenges in many issues related to science, engineering and other fields. Many important practical-problems in science and engineering can be expressed in terms of multi-objective/ set-valued optimization problems in order to achieve the proper results/ properties. To find the Pareto solutions, a corresponding scalarization problem has been established and studied. The relationships between the primal problem and its scalarization one should be investigated for finding optimal solutions. It can be shown that, under some suitable conditions, the solutions of the corresponding scalarization problem have uniform spread and have a close relationship to Pareto optimal solutions for the primal one. Scalarization has played an essential role in studying not only numerical methods but also duality theory. It can be usefully applied to get relationships/ important results between other fields, for example optimization, convex analysis and functional analysis. In scalarization, we ussually use a kind of scalarized-functions. One of the first and the most popular scalarized-functions used in scalarization method is the Gerstewitz function. In the paper, we mention some problems in set-valued optimization. Then, we propose an application of the Gerstewitz function to these problems. In detail, we establish some optimality conditions for Pareto/ weak solutions of unconstrained/ constrained set-valued optimization problems by using the Gerstewitz function. The study includes the consideration of problems in theoretical approach. Some examples are given to illustrate the obtained results.

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Totaro, Paolo, and Domenico Ninno. "Biological Recursion and Digital Systems: Conceptual Tools for Analysing Man-Machine Interaction." Theory, Culture & Society 37, no. 5 (May 3, 2020): 27–49. http://dx.doi.org/10.1177/0263276420915264.

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The theory of numbers, the theory of computation and well-known biological and neurological studies on cognition and consciousness all indicate the concept of recursion as their common denominator. Mathematical recursion owes its meaning and properties to a dual relationship between its results, which always constitute a sequence, and the operator that generated them, which is instead invariant. This article proposes that this duality in recursion originates from the duality between the biological homeostatic equilibrium in living systems and the adaptive physico-chemical changes required to sustain such equilibria. Such duality gives order and meaning to the experiences of a living system. One of the many implications of this innovative perspective is that this duality can decouple computational results from our intuitive order relations, and that this can cause a rarefaction of the capacity of digital systems to convey communication and favour adaptation to the environment.

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Chorny, Victor. "“Presence” in the Broad Present. Gumbrecht, H. U. (2020). Production of Presence. What Meaning Cannot Convey. Kharkiv: IST Publishing." Sententiae 40, no. 1 (April 1, 2021): 67–78. http://dx.doi.org/10.31649/sent40.01.067.

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This review of the Ukrainian translation of H. U. Gumbrecht’s best-known work brings out the strengths and weaknesses of the translation and the peculiar reception of Gumbrecht’s key ideas (“presence” and “the broad present”) in Ukraine. It also critically assesses Gumbrecht’s own original and often contradictory points. I question the relevance of Gumrecht’s meaning / presence distinction for reconstructing the history of the philosophical tradition, as well as for analysing our complex relation to the world. I also demonstrate the weakness of his biased attempts to paint his opponents as relativists. Besides, I contrast Gumbrecht’s meaning / presence dualism with John Dewey’s theory of experience. The latter conceives experience as a dialectical relation between “doing” and “undergoing”. This juxtaposition shows that Gumbrecht’s theory cannot give a satisfactory account of the mechanisms of everyday or aesthetic experience due to its lack of consistent “everyday” epistemology. Moreover, his vague concept of “presence” and its unequivocal appraisal conflict with his own concept of the chronotope of “broad” or “complex” present, as presented in the selected essays of The Time Is Out of Joint. Eventually, I conclude that Gumbrecht’s eclectic terminological apparatus, as well as uncritical and biased reconstruction of the tradition preclude any serious philosophical engagement. However, it does not undermine the significance of his particular insights and theoretical instruments (such as “the broad present”) for cultural analysis.

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Journal articles on the topic 'Convex duality theory'

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