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Journal articles on the topic 'Convex Function and Duality'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

Egudo, Richard R. "Multiobjective fractional duality." Bulletin of the Australian Mathematical Society 37, no. 3 (1988): 367–78. http://dx.doi.org/10.1017/s0004972700026988.

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The concept of efficiency (Pareto optimum) is used to formulate duality for multiobjective fractional programming problems. We consider programs where the components of the objective function have non-negative and convex numerators while the denominators are concave and positive. For this case the Mond-Weir extension of Bector dual analogy is given. We also give the Schaible type vector dual. The case where functions are ρ-convex (weakly or strongly convex) is also considered.

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2

Wibowo, Ratno Bagus Edy, Marjono, and Eko Dedi Pramana. "Legendre-Fenchel duality in m-convexity." Hilbert Journal of Mathematical Analysis 2, no. 2 (2024): 099–105. http://dx.doi.org/10.62918/hjma.v2i2.23.

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3

Zhang, Jun. "Divergence Function, Duality, and Convex Analysis." Neural Computation 16, no. 1 (2004): 159–95. http://dx.doi.org/10.1162/08997660460734047.

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From a smooth, strictly convex function Φ: Rn → R, a parametric family of divergence function DΦ(α) may be introduced: [Formula: see text] for x, y, ε int dom(Φ) and for α ε R, with DΦ(±1 defined through taking the limit of α. Each member is shown to induce an α-independent Riemannian metric, as well as a pair of dual α-connections, which are generally nonflat, except for α = ±1. In the latter case, D(±1)Φ reduces to the (nonparametric) Bregman divergence, which is representable using and its convex conjugate Φ * and becomes the canonical divergence for dually flat spaces (Amari, 1982, 1985; A

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4

Hassan, Mansur, and Adam Baharum. "Modified Courant-Beltrami penalty function and a duality gap for invex optimization problem." International Journal for Simulation and Multidisciplinary Design Optimization 10 (2019): A10. http://dx.doi.org/10.1051/smdo/2019010.

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In this paper, we modified a Courant-Beltrami penalty function method for constrained optimization problem to study a duality for convex nonlinear mathematical programming problems. Karush-Kuhn-Tucker (KKT) optimality conditions for the penalized problem has been used to derived KKT multiplier based on the imposed additional hypotheses on the constraint function g. A zero-duality gap between an optimization problem constituted by invex functions with respect to the same function η and their Lagrangian dual problems has also been established. The examples have been provided to illustrate and pr

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5

Scott, C. H., T. R. Jefferson, and E. Sirri. "On duality for convex minimization with nested maxima." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 26, no. 4 (1985): 517–22. http://dx.doi.org/10.1017/s0334270000004690.

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AbstractIn this paper, we consider convex programs with linear constraints where the objective function involves nested maxima of linear functions as well as a convex function. A dual program is constructed which has interpretational significance and may be easier to solve than the primal formulation. A numerical example is given to illustrate the method.

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6

Kailey, N., and S. Sonali. "Higher-order symmetric duality in nondifferentiable multiobjective optimization over cones." Filomat 33, no. 3 (2019): 711–24. http://dx.doi.org/10.2298/fil1903711k.

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In this paper, a new pair of higher-order nondifferentiable multiobjective symmetric dual programs over arbitrary cones is formulated, where each of the objective functions contains a support function of a compact convex set. We identify a function lying exclusively in the class of higher-order K-?-convex and not in the class of K-?-bonvex function already existing in literature. Weak, strong and converse duality theorems are then established under higher-order K-?-convexity assumptions. Self duality is obtained by assuming the functions involved to be skew-symmetric. Several known results are

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7

Mishra, M. S., S. Nanda, and D. Acharya. "Strong pseudo-convexity and symmetric duality in nonlinear programming." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 27, no. 2 (1985): 238–44. http://dx.doi.org/10.1017/s0334270000004884.

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AbstractIn this note, the weak duality theorem of symmetric duality in nonlinear programming and some related results are established under weaker (strongly Pseudo-convex/strongly Pseudo-concave) assumptions. These results were obtained by Bazaraa and Goode [1] under (stronger) convex/concave assumptions on the function.

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8

Fang, D. H. "Stable Zero Lagrange Duality for DC Conic Programming." Journal of Applied Mathematics 2012 (2012): 1–17. http://dx.doi.org/10.1155/2012/606457.

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We consider the problems of minimizing a DC function under a cone-convex constraint and a set constraint. By using the infimal convolution of the conjugate functions, we present a new constraint qualification which completely characterizes the Farkas-type lemma and the stable zero Lagrange duality gap property for DC conical programming problems in locally convex spaces.

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9

Dubey, Ramu, and S. K. Gupta. "On duality for a second-order multiobjective fractional programming problem involving type-I functions." Georgian Mathematical Journal 26, no. 3 (2019): 393–404. http://dx.doi.org/10.1515/gmj-2017-0038.

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Abstract The purpose of this paper is to study a nondifferentiable multiobjective fractional programming problem (MFP) in which each component of objective functions contains the support function of a compact convex set. For a differentiable function, we introduce the class of second-order {(C,\alpha,\rho,d)-V} -type-I convex functions. Further, Mond–Weir- and Wolfe-type duals are formulated for this problem and appropriate duality results are proved under the aforesaid assumptions.

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10

Volle, M., J. E. Martínez-Legaz, and J. Vicente-Pérez. "Duality for Closed Convex Functions and Evenly Convex Functions." Journal of Optimization Theory and Applications 167, no. 3 (2013): 985–97. http://dx.doi.org/10.1007/s10957-013-0395-4.

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11

Jeyakumar, V., and B. Mond. "On generalised convex mathematical programming." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 34, no. 1 (1992): 43–53. http://dx.doi.org/10.1017/s0334270000007372.

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AbstractThe sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:where the funtion f and g satisfyfor some η: X0 × X0 → ℝnIt is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractio

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12

Dworczak, Piotr, and Anton Kolotilin. "The persuasion duality." Theoretical Economics 19, no. 4 (2024): 1701–55. http://dx.doi.org/10.3982/te5900.

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We present a unified duality approach to Bayesian persuasion. The optimal dual variable, interpreted as a price function on the state space, is shown to be a supergradient of the concave closure of the objective function at the prior belief. Strong duality holds when the objective function is Lipschitz continuous. When the objective depends on the posterior belief through a set of moments, the price function induces prices for posterior moments that solve the corresponding dual problem. Thus, our general approach unifies known results for one‐dimensional moment persuasion, while yielding new r

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13

Li, Juwen, Zezhong Wu, Rong Zhou та Shengyu He. "The KKT Optimality Conditions and Duality for Constrained Programming Problem with Generalized α- Convex Fuzzy Functions". Scholars Journal of Physics, Mathematics and Statistics 10, № 02 (2023): 63–86. http://dx.doi.org/10.36347/sjpms.2023.v10i02.003.

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This paper mainly studies the mixed constraint interval programming problem under the generalized convex fuzzy mapping. Firstly, this paper give the concepts of fuzzy mappings, such as quasiconvex, strictly quasiconvex, pseudoconvex and strictly pseudoconvex. Then, the relation of generalized convex fuzzy mapping is studied and some properties are obtained. Finally, the necessary and sufficient KKT conditions are given, and the duality problem is established. The weak duality, strong duality and inverse duality theorem of fuzzy interval programming are proved.

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14

KOSHI, Shozo. "Convergence of convex functions and duality." Hokkaido Mathematical Journal 14, no. 3 (1985): 399–414. http://dx.doi.org/10.14492/hokmj/1381757647.

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15

Cui, Zhenyu, and Jun Deng. "Shortfall risk through Fenchel duality." International Journal of Financial Engineering 05, no. 02 (2018): 1850019. http://dx.doi.org/10.1142/s2424786318500196.

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In this paper, we propose a Fenchel duality approach to study the minimization problem of the shortfall risk. We consider a general increasing and strictly convex loss function, which may be more general than the situation of convex risk measures usually assumed in the literature. We first translate the associated stochastic optimization problem to an equivalent static optimization problem, and then obtain the explicit structure of the optimal randomized test for both complete and incomplete markets. For the incomplete market case, to the best of our knowledge, we obtain for the first time the

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16

Patel, Raman. "Mixed-type duality for multiobjective fractional variational control problems." International Journal of Mathematics and Mathematical Sciences 2005, no. 1 (2005): 109–24. http://dx.doi.org/10.1155/ijmms.2005.109.

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The concept of mixed-type duality has been extended to the class of multiobjective fractional variational control problems. A number of duality relations are proved to relate the efficient solutions of the primal and its mixed-type dual problems. The results are obtained forρ-convex (generalizedρ-convex) functions. The results generalize a number of duality results previously obtained for finite-dimensional nonlinear programming problems under various convexity assumptions.

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17

Craven, B. D., and B. M. Glover. "Invex functions and duality." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 39, no. 1 (1985): 1–20. http://dx.doi.org/10.1017/s1446788700022126.

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AbstractFor both differentiable and nondifferentiable functions defined in abstract spaces we characterize the generalized convex property, here called cone-invexity, in terms of Lagrange multipliers. Several classes of such functions are given. In addition an extended Kuhn-Tucker type optimality condition and a duality result are obtained for quasidifferentiable programming problems.

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18

Zhang, Xiaomin, and Zezhong Wu. "Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems." Advances in Operations Research 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/708979.

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Some assumptions for the objective functions and constraint functions are given under the conditions of convex and generalized convex, which are based on theF-convex,ρ-convex, and(F,ρ)-convex. The sufficiency of Kuhn-Tucker optimality conditions and appropriate duality results are proved involving(F,ρ)-convex,(F,α,ρ,d)-convex, and generalized(F,α,ρ,d)-convex functions.

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19

Krishnan, Arjun, Firas Rassoul-Agha, and Timo Seppäläinen. "Geodesic length and shifted weights in first-passage percolation." Communications of the American Mathematical Society 3, no. 5 (2023): 209–89. http://dx.doi.org/10.1090/cams/18.

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We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the ℓ 1 \ell ^1 distance to the

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20

Molchanov, Ilya. "Continued fractions built from convex sets and convex functions." Communications in Contemporary Mathematics 17, no. 05 (2015): 1550003. http://dx.doi.org/10.1142/s0219199715500030.

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In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

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21

Hiriart-Urruty, J. B. "A General Formula on the Conjugate of the Difference of Functions." Canadian Mathematical Bulletin 29, no. 4 (1986): 482–85. http://dx.doi.org/10.4153/cmb-1986-076-7.

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AbstractGiven an arbitrary function g :X → (-∞, +∞] and a lowersemicontinuous convex function h:X → (-∞, +∞], we give the general expression of the conjugate (g — h)* of g - h in terms of g* and h*. As a consequence, we get Toland's duality theorem:

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22

Jeyakumar, V. "On subgradient duality with strong and weak convex functions." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 2 (1986): 143–52. http://dx.doi.org/10.1017/s1446788700027130.

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AbstractA duality theorem of Wolfe for nonlinear differentiable programs is extended to nondifferentiable programs with strong and weak convex functions, by replacing gradients by local subgradient. A converse duality theorem is also proved.

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23

SEGAL, ALEXANDER, and BOAZ A. SLOMKA. "PROJECTIONS OF LOG-CONCAVE FUNCTIONS." Communications in Contemporary Mathematics 14, no. 05 (2012): 1250036. http://dx.doi.org/10.1142/s0219199712500368.

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Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝn containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were re

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24

Zặlinescu, C. "A comparison of constraint qualifications in infinite-dimensional convex programming revisited." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 40, no. 3 (1999): 353–78. http://dx.doi.org/10.1017/s033427000001095x.

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In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formulainf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good f

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25

Goebel, Rafal. "Lyapunov Functions and Duality for Convex Processes." SIAM Journal on Control and Optimization 51, no. 4 (2013): 3332–50. http://dx.doi.org/10.1137/120900174.

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26

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 (2007): 509–15. http://dx.doi.org/10.1007/s10957-007-9234-9.

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27

Kaur, Arshpreet, and MaheshKumar Sharma. "Higher order symmetric duality for multiobjective fractional programming problems over cones." Yugoslav Journal of Operations Research, no. 00 (2021): 12. http://dx.doi.org/10.2298/yjor200615012k.

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This article studies a pair of higher order nondifferentiable symmetric fractional programming problem over cones. First, higher order cone convex function is introduced. Then using the properties of this function, duality results are set up, which give the legitimacy of the pair of primal dual symmetric model.

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28

Weir, T., and B. Mond. "Proper efficiency and duality for vector valued optimization problems." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 1 (1987): 21–34. http://dx.doi.org/10.1017/s1446788700028937.

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AbstractThe duality results of Wolfe for scalar convex programming problems and some of the more recent duality results for scalar nonconvex programming problems are extended to vector valued programs. Weak duality is established using a ‘Pareto’ type relation between the primal and dual objective functions.

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29

Bila, Samet, and Refail Kasımbeyli. "ON THE WEAK SUBDIFFERENTIAL, AUGMENTED NORMAL CONES AND DUALITY IN NONCONVEX OPTIMIZATION." Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 13, no. 1 (2025): 67–76. https://doi.org/10.20290/estubtdb.1632350.

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This article studies the properties of the weak subdifferential for nonsmooth and nonconvex analysis studied. This study presents a formulation that is directly involved in convex analysis carried out in the nonconvex case. In this work, we present a theory that applies epigraphs to obtain augmented normal cones. The perturbation function plays a crucial role in establishing optimality conditions. This study demonstrates that positively homogeneous and lower semicontinuous functions are weakly subdifferentiable. Moreover, under specific conditions related to the objective function, the constra

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30

Antczak, Tadeusz, Vinay Singh та Mohan Subba. "Optimality and duality results for (h,φ)-nondifferentiable multiobjective programming problems with (h,φ) – (b,f,ρ) -convex functions". Filomat 36, № 12 (2022): 4139–56. http://dx.doi.org/10.2298/fil2212139a.

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Generalized algebraic operations introduced by Ben-Tal [5] are used to define new classes of generalized convex functions, namely (h,?)?(b,F,?) -convex functions and generalized (h,?)?(b,F,?)-convex functions in the vectorial case. Further, optimality and duality results are proved for the considered (h,?)- nondifferentiable multiobjective programming problem under assumptions that the functions involved are (generalized) (h,?)- (b,F,?)-convex.

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31

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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32

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 (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 mai

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33

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 (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 sev

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34

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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35

Gupta, Anjana, Aparna Mehra, and Davinder Bhatia. "Approximate convexity in vector optimisation." Bulletin of the Australian Mathematical Society 74, no. 2 (2006): 207–18. http://dx.doi.org/10.1017/s0004972700035656.

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Approximate convex functions are characterised in terms of Clarke generalised gradient. We apply this characterisation to derive optimality conditions for quasi efficient solutions of nonsmooth vector optimisation problems. Two new classes of generalised approximate convex functions are defined and mixed duality results are obtained.

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36

Choi, Hyungjin, Umesh Vaidya, and Yongxin Chen. "A Convex Data-Driven Approach for Nonlinear Control Synthesis." Mathematics 9, no. 19 (2021): 2445. http://dx.doi.org/10.3390/math9192445.

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We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the m

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37

Chai, Yanfei. "Robust strong duality for nonconvex optimization problem under data uncertainty in constraint." AIMS Mathematics 6, no. 11 (2021): 12321–38. http://dx.doi.org/10.3934/math.2021713.

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<abstract><p>This paper deals with the robust strong duality for nonconvex optimization problem with the data uncertainty in constraint. A new weak conjugate function which is abstract convex, is introduced and three kinds of robust dual problems are constructed to the primal optimization problem by employing this weak conjugate function: the robust augmented Lagrange dual, the robust weak Fenchel dual and the robust weak Fenchel-Lagrange dual problem. Characterizations of inequality (1.1) according to robust abstract perturbation weak conjugate duality are established by using the

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38

Jayswal, Anurag, Ashish Kumar Prasad, and Krishna Kummari. "Nondifferentiable Minimax Programming Problems in Complex Spaces Involving Generalized Convex Functions." Journal of Optimization 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/297015.

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We start our discussion with a class of nondifferentiable minimax programming problems in complex space and establish sufficient optimality conditions under generalized convexity assumptions. Furthermore, we derive weak, strong, and strict converse duality theorems for the two types of dual models in order to prove that the primal and dual problems will have no duality gap under the framework of generalized convexity for complex functions.

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39

SUNEJA, S. K., and MEETU BHATIA. "CONE CONVEX AND RELATED FUNCTIONS IN OPTIMIZATION OVER TOPOLOGICAL VECTOR SPACES." Asia-Pacific Journal of Operational Research 24, no. 06 (2007): 741–54. http://dx.doi.org/10.1142/s0217595907001504.

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In this paper cone convex and related functions have been studied. The concept of cone semistrictly convex functions on topological vector spaces is introduced as a generalization of semistrictly convex functions. Certain properties of these functions have been established and their interrelations with cone convex and cone subconvex functions have been explored. Assuming the functions to be cone subconvex, sufficient optimality conditions are proved for a vector valued minimization problem over topological vector spaces, involving Gâteaux derivatives. A Mond-Weir type dual is associated and we

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40

Kapoor, Muskan, Surjeet Kaur Suneja, and Meetu Bhatia Grover. "Higher order optimality and duality in fractional vector optimization over cones." Tamkang Journal of Mathematics 48, no. 3 (2017): 273–87. http://dx.doi.org/10.5556/j.tkjm.48.2017.2311.

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In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

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41

Krivosheev, Aleksandr Sergeevich, and Olesya Aleksandrovna Krivosheeva. "Interpolation and fundamental principle." Ufa Mathematical Journal 16, no. 3 (2024): 54–64. https://doi.org/10.13108/2024-16-3-54.

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In this work we study the spaces of functions analytic in convex domains in the complex plane. We consider subspaces of such spaces, which are invariant with respect to the differentiation operator. We study the fundamental principle problem for an invariant subspace, that is, the problem on representing all its elements by a series of eigenfunctions and generalized eigenfunctions of the differentiation operator in this subspace, which are the exponentials and exponential monomials. We provide a complete description of the space of sequences of the coefficients of the series, by which we repre

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42

Jayswal, Anurag, I. M. Stancu-Minasian, and Dilip Kumar. "Minmax fractional programming problem involving generalized convex functions." Journal of Numerical Analysis and Approximation Theory 41, no. 1 (2012): 47–61. http://dx.doi.org/10.33993/jnaat411-968.

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In the present study we focus our attention on a minmax fractional programming problem and its second order dual problem. Duality results are obtained for the considered dual problem under the assumptions of second order \(\left( {F,\alpha ,\rho ,d}\right) \) -type I functions.

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43

Cholamjiak, Prasit, Yeol Je Cho, and Suthep Suantai. "Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions." Analele Universitatii "Ovidius" Constanta - Seria Matematica 21, no. 1 (2013): 183–200. http://dx.doi.org/10.2478/auom-2013-0011.

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Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

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44

Craven, B. D. "A note on nondifferentiable symmetric duality." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 28, no. 1 (1986): 30–35. http://dx.doi.org/10.1017/s0334270000005178.

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Under suitable hypotheses on the function f, the two constrained minimization problems:are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to:in which T and S are convex cones, S* is the dual cone of S, and ∂y denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This

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Ronglu, Li, and Wang Junming. "Invariants in abstract mapping pairs." Journal of the Australian Mathematical Society 76, no. 3 (2004): 369–82. http://dx.doi.org/10.1017/s1446788700009927.

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AbstractIn a topological vector space, duality invariant is a very important property, some famous theorems, such as the Mackey-Arens theorem, the Mackey theorem, the Mazur theorem and the Orlicz-Pettis theorem, all show some duality invariants.In this paper we would like to show an important improvement of the invariant results, which are related to sequential evaluation convergence of function series. Especially, a very general invariant result is established for an abstract mapping pair (Φ, B(Φ, X)) consisting of a nonempty set Φ and B(Φ, X) = {f ∈ XΦ: f (Φ) is bounded}, where X is a locall

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Li, Lifeng, Sanyang Liu, and Jianke Zhang. "Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/383692.

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Interval-valued univex functions are introduced for differentiable programming problems. Optimality and duality results are derived for a class of generalized convex optimization problems with interval-valued univex functions.

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Bhardwaj, Vinod Kumar. "Optimization of convex functions with fenchel biconjugation and duality." International Journal of Advanced Technology and Engineering Exploration 5, no. 42 (2018): 83–88. http://dx.doi.org/10.19101/ijatee.2018.542013.

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Martínez-Legaz, J. E., and B. F. Svaiter. "Minimal convex functions bounded below by the duality product." Proceedings of the American Mathematical Society 136, no. 03 (2007): 873–79. http://dx.doi.org/10.1090/s0002-9939-07-09176-9.

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Drapeau, Samuel, Andreas H. Hamel, and Michael Kupper. "Complete Duality for Quasiconvex and Convex Set-Valued Functions." Set-Valued and Variational Analysis 24, no. 2 (2015): 253–75. http://dx.doi.org/10.1007/s11228-015-0332-9.

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Jung, Jong. "Convergence of iterative algorithms for continuous pseudocontractive mappings." Filomat 30, no. 7 (2016): 1767–77. http://dx.doi.org/10.2298/fil1607767j.

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In this paper, we prove strong convergence of a path for a convex combination of a pseudocontractive type of operators in a real reflexive Banach space having a weakly continuous duality mapping J? with gauge function ?. Using path convergency, we establish strong convergence of an implicit iterative algorithm for a pseudocontractive mapping combined with a strongly pseudocontractive mapping in the same Banach space.

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