Convex Functions and Duality in Optimization Problems and Dynamics

1969 ◽  
pp. 117-141 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Functions ◽  
Optimization Problems ◽  
Duality In Optimization

scholarly journals Some Inequalities Related to Interval-Valued η h -Convex Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Lei Chen ◽  
Muhammad Shoaib Saleem ◽  
Muhammad Sajid Zahoor ◽  
Rahat Bano
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Optimization Problems ◽  
Extended Version ◽  
Interval Valued

Convexity plays an important role in many areas of mathematics, especially in the study of optimization problems where they are distinguished by a number of convenient properties. Our aim is to introduce a more extended version of convexity. In this paper, we introduced interval-valued generalized η h convex function and proved Hermite–Hadamard-, Jensen-, and Ostrowski-type inequalities in this generalization. The presented results are generalizations of many existing results of literature.

scholarly journals Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

2021 ◽  
Vol 7 (3) ◽  
pp. 4338-4358
Author(s):  
Muhammad Bilal Khan ◽  
◽  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Kamsing Nonlaopon ◽  
...  
Keyword(s):  
Convex Functions ◽  
Optimization Problems ◽  
Fuzzy Optimization ◽  
Order Relation ◽  
Inclusion Relation ◽  
Nonconvex Functions ◽  
New Directions ◽  
Fuzzy Order ◽  
Interval Valued ◽  
Fuzzy Interval

<abstract> <p>The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

scholarly journals On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Qi Li ◽  
Muhammad Shoaib Saleem ◽  
Peiyu Yan ◽  
Muhammad Sajid Zahoor ◽  
Muhammad Imran
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Convex Functions ◽  
Fractional Integral ◽  
Optimization Problems ◽  
Fractional Integral Operator ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Special Means ◽  
New Inequalities

The theory of convex functions plays an important role in the study of optimization problems. The fractional calculus has been found the best to model physical and engineering processes. The aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. In this paper, we present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are also presented. Moreover, we present some applications of the proposed inequalities to special means.

scholarly journals On Extended Convex Functions via Incomplete Gamma Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Yan Zhao ◽  
M. Shoaib Saleem ◽  
Shahid Mehmood ◽  
Zabidin Salleh
Keyword(s):  
Convex Functions ◽  
Optimization Problems ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. In this paper, firstly we introduce the notion of h -exponential convex functions. This notion can be considered as generalizations of many existing definitions of convex functions. Then, we establish some well-known inequalities for the proposed notion via incomplete gamma functions. Precisely speaking, we established trapezoidal, midpoint, and He’s inequalities for h -exponential and harmonically exponential convex functions via incomplete gamma functions. Moreover, we gave several remarks to prove that our results are more generalized than the existing results in the literature.

scholarly journals Efficient Proximal Mapping Computation for Low-Rank Inducing Norms

2021 ◽  
Author(s):  
Christian Grussler ◽  
Pontus Giselsson
Keyword(s):  
Convex Functions ◽  
Optimization Problems ◽  
Binary Search ◽  
Splitting Methods ◽  
Nuclear Norm ◽  
Low Rank ◽  
Proximal Mapping ◽  
Unitarily Invariant Norms ◽  
Sparsity Constraint

AbstractLow-rank inducing unitarily invariant norms have been introduced to convexify problems with a low-rank/sparsity constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. The simpler problem can often be solved analytically as demonstrated for the so-called low-rank inducing Frobenius and spectral norms. The framework also allows to compute the proximal mapping of increasing convex functions composed with these norms as well as projections onto their epigraphs.

scholarly journals Strongly and Semi Strongly E_h-b-Vex Functions: Applications to Optimization Problems

2019 ◽  
pp. 2022-2029
Author(s):  
Saba Nasser Majeed
Keyword(s):  
Objective Function ◽  
Nonlinear Optimization ◽  
Convex Functions ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Nonlinear Optimization Problem ◽  
Optimality Properties

In this paper, we propose new types of non-convex functions called strongly --vex functions and semi strongly --vex functions. We study some properties of these proposed functions. As an application of these functions in optimization problems, we discuss some optimality properties of the generalized nonlinear optimization problem for which we use, as an objective function, strongly --vex function and semi strongly --vex function.

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