• Title/Summary/Keyword: Convex and Concave Function

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SVN-Ostrowski Type Inequalities for (α, β, γ, δ) -Convex Functions

  • Maria Khan;Asif Raza Khan;Ali Hassan
    • International Journal of Computer Science & Network Security
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    • v.24 no.1
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    • pp.85-94
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    • 2024
  • In this paper, we present the very first time the generalized notion of (α, β, γ, δ) - convex (concave) function in mixed kind, which is the generalization of (α, β) - convex (concave) functions in 1st and 2nd kind, (s, r) - convex (concave) functions in mixed kind, s - convex (concave) functions in 1st and 2nd kind, p - convex (concave) functions, quasi convex(concave) functions and the class of convex (concave) functions. We would like to state the well-known Ostrowski inequality via SVN-Riemann Integrals for (α, β, γ, δ) - convex (concave) function in mixed kind. Moreover we establish some SVN-Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are (α, β, γ, δ)-convex (concave) functions in mixed kind by using different techniques including Hölder's inequality and power mean inequality. Also, various established results would be captured as special cases with respect to convexity of function.

A NOTE ON PRECONVEXITY SPACES

  • Min, Won-Keun
    • Honam Mathematical Journal
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    • v.29 no.4
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    • pp.589-595
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    • 2007
  • In this paper, we introduce the concepts of the convexity hull and co-convex sets on preconvexity spaces. We study some properties for the co-convexity hull and characterize c-convex functions and c-concave functions by using the co-convexity hull and the convexity hull.

ON SPHERICALLY CONCAVE FUNCTIONS

  • KIM SEONG-A
    • The Pure and Applied Mathematics
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    • v.12 no.3 s.29
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    • pp.229-235
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    • 2005
  • The notions of spherically concave functions defined on a subregion of the Riemann sphere P are introduced in different ways in Kim & Minda [The hyperbolic metric and spherically convex regions. J. Math. Kyoto Univ. 41 (2001), 297-314] and Kim & Sugawa [Charaterizations of hyperbolically convex regions. J. Math. Anal. Appl. 309 (2005), 37-51]. We show continuity of the concave function defined in the latter and show that the two notions of the concavity are equivalent for a function of class $C^2$. Moreover, we find more characterizations for spherically concave functions.

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Analysis of D2D Utility Function with the Interference Majorization

  • Oh, Changyoon
    • Journal of the Korea Society of Computer and Information
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    • v.25 no.7
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    • pp.75-83
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    • 2020
  • We consider the D2D utility optimization problem in the cellular system. More specifically, we develop a concave function decision rule which reduces the complexity of non-convex optimization problem. Typically, utility function, which is a function of the signal and the interference, is non-convex. In this paper, we analyze the utility function from the interference perspective. We introduce the 'relative interference' and the 'interference majorization'. The relative interference captures the level of interference at D2D receiver's perspective. The interference majorization approximates the interference by applying the major interference. Accordingly, we propose a concave function decision rule, and the corresponding convex optimization solution. Simulation results show that the utility function is concave when the relative interference is less than 0.1, which is a typical D2D usage scenario. We also show that the proposed convex optimization solution can be applied for such relative interference cases.

A GENERALIZED SINGULAR FUNCTION

  • Baek, In-Soo
    • Journal of the Chungcheong Mathematical Society
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    • v.23 no.4
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    • pp.657-661
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    • 2010
  • We study a singular function which we call a generalized cylinder convex(concave) function induced from different generalized dyadic expansion systems on the unit interval. We show that the generalized cylinder convex(concave)function is a singular function and the length of its graph is 2. Using a local dimension set in the unit interval, we give some characterization of the distribution set using its derivative, which leads to that this singular function is nowhere differentiable in the sense of topological magnitude.

FENCHEL DUALITY THEOREM IN MULTIOBJECTIVE PROGRAMMING PROBLEMS WITH SET FUNCTIONS

  • Liu, Sanming;Feng, Enmin
    • Journal of applied mathematics & informatics
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    • v.13 no.1_2
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    • pp.139-152
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    • 2003
  • In this paper, we characterize a vector-valued convex set function by its epigraph. The concepts of a vector-valued set function and a vector-valued concave set function we given respectively. The definitions of the conjugate functions for a vector-valued convex set function and a vector-valued concave set function are introduced. Then a Fenchel duality theorem in multiobjective programming problem with set functions is derived.

An Additive Sparse Penalty for Variable Selection in High-Dimensional Linear Regression Model

  • Lee, Sangin
    • Communications for Statistical Applications and Methods
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    • v.22 no.2
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    • pp.147-157
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    • 2015
  • We consider a sparse high-dimensional linear regression model. Penalized methods using LASSO or non-convex penalties have been widely used for variable selection and estimation in high-dimensional regression models. In penalized regression, the selection and prediction performances depend on which penalty function is used. For example, it is known that LASSO has a good prediction performance but tends to select more variables than necessary. In this paper, we propose an additive sparse penalty for variable selection using a combination of LASSO and minimax concave penalties (MCP). The proposed penalty is designed for good properties of both LASSO and MCP.We develop an efficient algorithm to compute the proposed estimator by combining a concave convex procedure and coordinate descent algorithm. Numerical studies show that the proposed method has better selection and prediction performances compared to other penalized methods.

A Concave Function Minimization Algorithm Under 0-1 Knapsack Constraint using Strong Valid Inequalities (유효 절단 부등식을 이용한 오목함수 0-1 배낭제약식 문제의 해법)

  • 오세호
    • Journal of the Korean Operations Research and Management Science Society
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    • v.22 no.3
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    • pp.11-22
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    • 1997
  • The aim of this paper is to develop the B & B type algorithms for globally minimizing concave function under 0-1 knapsack constraint. The linear convex envelope underestimating the concave object function is introduced for the bounding operations which locate the vertices of the solution set. And the simplex containing the solution set is sequentially partitioned into the subsimplices over which the convex envelopes are calculated in the candidate problems. The adoption of cutting plane method enhances the efficiency of the algorithm. These mean valid inequalities with respect to the integer solution which eliminate the nonintegral points before the bounding operation. The implementations are effectively concretized in connection with the branching stategys.

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