generalized convexity
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2022 ◽  
Vol 6 (1) ◽  
pp. 28
Author(s):  
Tao Yan ◽  
Ghulam Farid ◽  
Hafsa Yasmeen ◽  
Chahn Yong Jung

In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.


2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ana-Maria Acu ◽  
Ioan Cristian Buscu ◽  
Ioan Rasa

<p style='text-indent:20px;'>Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended.</p>


Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3182
Author(s):  
Gabriela Cristescu ◽  
Vlad-Florin Drăgoi ◽  
Sorin Horaţiu Hoară

Some properties of generalized convexity for sets and functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based on both interpolation conditions and shape knowledge. It is proved that the approximant objects preserve both the high-order convexity and some extremum properties of the exact reliability polynomials. It leads to pointing out the area of the network where the maximum number of paths is achieved. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Haijun Wang ◽  
Huihui Wang

AbstractIn this paper, we study the duality theorems of a nondifferentiable semi-infinite interval-valued optimization problem with vanishing constraints (IOPVC). By constructing the Wolfe and Mond–Weir type dual models, we give the weak duality, strong duality, converse duality, restricted converse duality, and strict converse duality theorems between IOPVC and its corresponding dual models under the assumptions of generalized convexity.


Author(s):  
Guolin Yu ◽  
Siqi Li ◽  
Xiao Pan ◽  
Wenyan Han

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Zareen A. Khan ◽  
Saima Rashid ◽  
Rehana Ashraf ◽  
Dumitru Baleanu ◽  
Yu-Ming Chu

AbstractIn the paper, we extend some previous results dealing with the Hermite–Hadamard inequalities with fractal sets and several auxiliary results that vary with local fractional derivatives introduced in the recent literature. We provide new generalizations for the third-order differentiability by employing the local fractional technique for functions whose local fractional derivatives in the absolute values are generalized convex and obtain several bounds and new results applicable to convex functions by using the generalized Hölder and power-mean inequalities.As an application, numerous novel cases can be obtained from our outcomes. To ensure the feasibility of the proposed method, we present two examples to verify the method. It should be pointed out that the investigation of our findings in fractal analysis and inequality theory is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.


2020 ◽  
Vol 36 (3) ◽  
pp. 443-452
Author(s):  
JUTAMAS KERDKAEW ◽  
RABIAN WANGKEEREE ◽  
GUE MYUNG LEE

In this paper, we investigate the robust optimization problem involving nonsmooth and nonconvex real-valued functions. We firstly establish a necessary condition for the local robust weak sharp solution of considered problem under a constraint qualification. These optimality conditions are presented in terms of multipliers and Mordukhovich subdifferentials of the related functions. Then, by employing the robust version of the (KKT) condition, and some appropriate generalized convexity conditions, we also obtain some sufficient conditions for the global robust weak sharp solutions of the problem. In addition, some examples are presented for illustrating or supporting the results.


2020 ◽  
Vol 13 (3) ◽  
pp. 63-83
Author(s):  
Artion Kashuri ◽  
Muhammad Aamir Ali ◽  
Mujahid Abbas ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Wenyan Han ◽  
Guolin Yu ◽  
Tiantian Gong

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.


2020 ◽  
Vol 13 (3) ◽  
pp. 219-255 ◽  
Author(s):  
Adolfo Arroyo-Rabasa ◽  
Guido De Philippis ◽  
Filip Rindler

AbstractWe show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.


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