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This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order β approaches 1, in addition to the fractional integrals we examined. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ -quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan's lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, some consequences and examples are presented. [ABSTRACT FROM AUTHOR]

This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions by fractional integrals. Some fractional Euler–Maclaurin-type inequalities are established for Lipschitzian functions. Finally, several Euler–Maclaurin-type inequalities are constructed by fractional integrals of bounded variation. [ABSTRACT FROM AUTHOR]

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus. [ABSTRACT FROM AUTHOR]

In this paper, we create a new subclass of convex functions given with tangent functions applying the combination of Babalola operators and Binomial series. Moreover, we obtain several important geometric results, including sharp coefficient bounds, sharp Fekete–Szego inequality, Kruskal inequality, and growth and distortion estimates. Furthermore, for functions with logarithmic coefficients, we compute sharp Fekete–Szego inequality and sharp coefficient bounds. [ABSTRACT FROM AUTHOR]

Subgradient methods are frequently used for optimization problems. However, subgradient techniques are characterized by slow convergence for minimizing ravine convex functions. To accelerate subgradient methods, special linear non-orthogonal transformations of the original space are used. This paper provides an overview of these transformations based on Shor's original idea. Two one-rank linear transformations of Euclidean space are considered. These simple transformations form the basis of variable metric methods for convex minimization that have a natural geometric interpretation in the transformed space. Along with the space transformation, a search direction and a corresponding step size must be defined. Subgradient Fejer-type methods are analyzed to minimize convex functions, and Polyak step size is used for problems with a known optimal objective value. Convergence theorems are provided together with the results of numerical experiments. Directions for future research are discussed. [ABSTRACT FROM AUTHOR]

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s = 1 , λ = α ,   σ (0) = 1 , and w = 0 . In conclusion, the methodology described in this article is expected to stimulate further research in this area. [ABSTRACT FROM AUTHOR]

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new class of convex functions, so-called generalized n-polynomial p-convex functions. We investigate their algebraic properties and provide some relationships between these functions and other types of convex functions. We establish Hermite–Hadamard (H–H) inequality for the newly defined class of functions. Additionally, we derive refinements of H–H inequality for functions whose first derivatives in absolute value at certain power are generalized n-polynomial p-convex. When p = − 1 , our definition evolves into a new definition for the class of convex functions so-called generalized n-polynomial harmonically convex functions. The results obtained in this study generalize regarding those found in the existing literature. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

This paper studies the single-machine group scheduling problem with convex resource allocation and learning effect. The number of jobs in each group is different, and the corresponding common due dates are also different, where the processing time of jobs follows a convex function of resource allocation. Under common due date assignment, the objective is to minimize the weighted sum of earliness, tardiness, common due date, resource consumption, and makespan. To solve the problem, we present the heuristic, simulated annealing, and branch-and-bound algorithms. Computational experiments indicate that the proposed algorithms are effective. [ABSTRACT FROM AUTHOR]

In this paper, we examine a semi-infinite interval-valued optimization problem with vanishing constraints (SIVOPVC) that lacks differentiability and involves constraints that tend to vanish. We give definitions of generalized convex functions along with supportive examples. We investigate duality theorems for the SIVOPVC problem. We establish these theorems by creating duality models, which establish a relationship between SIVOPVC and its corresponding dual models, assuming generalized convexity conditions. Some examples are also given to illustrate the results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the theoretical and numerical analysis of the two-step method, constructed as a modification of Polyak's heavy ball method with the inclusion of an additional momentum parameter. For the quadratic case, the convergence conditions are obtained with the use of the first Lyapunov method. For the non-quadratic case, sufficiently smooth strongly convex functions are obtained, and these conditions guarantee local convergence.An approach to finding optimal parameter values based on the solution of a constrained optimization problem is proposed. The effect of an additional parameter on the convergence rate is analyzed. With the use of an ordinary differential equation, equivalent to the method, the damping effect of this parameter on the oscillations, which is typical for the non-monotonic convergence of the heavy ball method, is demonstrated. In different numerical examples for non-quadratic convex and non-convex test functions and machine learning problems (regularized smoothed elastic net regression, logistic regression, and recurrent neural network training), the positive influence of an additional parameter value on the convergence process is demonstrated. [ABSTRACT FROM AUTHOR]

The current paper proposes and tests algorithms for finding the diameter of a compact convex set and the farthest point in the set to another point. For these two nonconvex problems, I construct Frank–Wolfe and projected gradient ascent algorithms. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, I investigate a homotopy method that gradually deforms a ball into the target set. Motivated by the Frank–Wolfe algorithm, I also find the support function of the intersection of a convex cone and a ball centered at the origin and elaborate a known bisection algorithm for calculating the support function of a convex sublevel set. The Frank–Wolfe and projected gradient algorithms are tested on five compact convex sets: (a) the box whose coordinates range between −1 and 1, (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the Manhattan-norm unit ball, and (e) a sublevel set of the elastic net penalty. Frank–Wolfe and projected gradient ascent are about equally fast on these test problems. Ignoring homotopy, the Frank–Wolfe algorithm is more reliable. However, homotopy allows projected gradient ascent to recover from its failures. [ABSTRACT FROM AUTHOR]

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach by introducing a set of operators known as the left and right generalized tempered fractional integral operators. These operators are utilized to establish new Hermite–Hadamard inequalities for convex functions as well as the multiplication of two convex functions. Additionally, this paper gives two useful identities involving the generalized tempered fractional integral operator for differentiable functions. By leveraging these identities, our results consist of integral inequalities of the Hermite–Hadamard type, which are specifically designed to accommodate convex functions. Furthermore, this study encompasses the identification of several special cases and the recovery of specific known results through comprehensive research. Lastly, this paper offers a range of applications in areas such as matrices, modified Bessel functions and q -digamma functions. [ABSTRACT FROM AUTHOR]

In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of f-divergences can be refined by another f-divergence. [ABSTRACT FROM AUTHOR]

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using G A -convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider different due-date assignment scheduling with group technology and resource allocation on a single machine, where the due date of each job may be different. Under constant processing times, the objective function is to minimize the scheduling cost (i.e., the weighted sum of earliness, tardiness, and due-date assignment cost, where the weights are position dependent). Under some optimal properties, we prove that this problem can be solved in  O (ζ log ζ)  time, where  ζ  is the number of jobs. The problem is also extended to cases which include linear and convex functions of the quantity of resource allocation. The objective function is minimizing the sum of the scheduling cost and the resource-consumption cost. For the special case of linear and convex functions, we show that the problem is polynomially solvable in  O (ζ 3)  time. [ABSTRACT FROM AUTHOR]

In this paper, we find sufficient conditions to be imposed on the parameters of a class of functions related to the Barnes–Mittag-Leffler function that allow us to conclude that it possesses certain geometric properties (such as starlikeness, uniformly starlike (convex), strongly starlike (convex), convexity, and close-to-convexity) in the unit disk. The key tools in some of our proofs are the monotonicity properties of a certain class of functions related to the gamma function. [ABSTRACT FROM AUTHOR]

With the widespread application of spatial data in fields like econometrics and geographic information science, the methods to enhance the robustness of spatial econometric model estimation and variable selection have become a central focus of research. In the context of the spatial error model (SEM), this paper introduces a variable selection method based on exponential square loss and the adaptive lasso penalty. Due to the non-convex and non-differentiable nature of this proposed method, convex programming is not applicable for its solution. We develop a block coordinate descent algorithm, decompose the exponential square component into the difference of two convex functions, and utilize the CCCP algorithm in combination with parabolic interpolation for optimizing problem-solving. Numerical simulations demonstrate that neglecting the spatial effects of error terms can lead to reduced accuracy in selecting zero coefficients in SEM. The proposed method demonstrates robustness even when noise is present in the observed values and when the spatial weights matrix is inaccurate. Finally, we apply the model to the Boston housing dataset. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to present new q-parametrized Hermite–Hadamard-like type integral inequalities for functions whose third quantum derivatives in absolute values are s-convex and  (α , m) -convex, respectively. Two new q-integral identities are presented for three time q-differentiable functions. These lemmas are used like basic elements in our proofs, along with several important tools like q-power mean inequality, and q-Holder's inequality. In a special case, a non-trivial example is considered for a specific parameter and this case illustrates the investigated results. We make links between these findings and several previous discoveries from the literature. [ABSTRACT FROM AUTHOR]

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type. [ABSTRACT FROM AUTHOR]

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. [ABSTRACT FROM AUTHOR]

This paper's major goal is to prove some symmetrical Maclaurin-type integral inequalities inside the framework of multiplicative calculus. In order to accomplish this and after giving some basic tools, we have established a new integral identity. Based on this identity, some symmetrical Maclaurin-type inequalities have been constructed for functions whose multiplicative derivatives are bounded as well as convex. At the end, some applications to special means are provided. [ABSTRACT FROM AUTHOR]

The main results of the study of the convergence rate of quasi-Newton minimization methods were obtained under the assumption that the method operates in the region of the extremum of the function, where there is a stable quadratic representation of the function. Methods based on the quadratic model of the function in the extremum area show significant advantages over classical gradient methods. When solving a specific problem using the quasi-Newton method, a huge number of iterations occur outside the extremum area, unless there is a stable quadratic approximation of the function. In this paper, we study the convergence rate of quasi-Newton-type methods on strongly convex functions with a Lipschitz gradient, without using local quadratic approximations of a function based on the properties of its Hessian. We proved that quasi-Newton methods converge on strongly convex functions with a Lipschitz gradient with the rate of a geometric progression, while the estimate of the convergence rate improves with the increasing number of iterations, which reflects the fact that the learning (adaptation) effect accumulates as the method operates. Another important fact discovered during the theoretical study is the ability of quasi-Newton methods to eliminate the background that slows down the convergence rate. This elimination is achieved through a certain linear transformation that normalizes the elongation of function level surfaces in different directions. All studies were carried out without any assumptions regarding the matrix of second derivatives of the function being minimized. [ABSTRACT FROM AUTHOR]

In this paper, we aim to find unified estimates of fractional integrals involving Mittag–Leffler functions in kernels. The results obtained in terms of fractional integral inequalities are provided for various kinds of convex and related functions. A variant of Hadamard-type inequality is also presented, which shows the upper and lower bounds of fractional integral operators of many kinds. The results of this paper are directly linked with many recently published inequalities. [ABSTRACT FROM AUTHOR]

Hermite–Hadamard inequalities and their refinements have been investigated for a long period of time. In this paper, we obtained refinements of the Hermite–Hadamard inequality of tensorial type for the convex functions of self-adjoint operators in Hilbert spaces. The obtained inequalities generalize the previously obtained inequalities by Dragomir. We also provide useful Lemmas which enabled us to obtain the results. The examples of the obtained inequalities for specific convex functions have been given in the example and consequences section. Symmetry in the upper and lower bounds can be seen in the last Theorem of the paper given, as the upper and lower bounds differ by a constant. [ABSTRACT FROM AUTHOR]

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition a 2 > 0 and without imposing any restrictive growth assumption on the damping term f 1 , using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function ψ , namely ψ ′ (t) ≤ − μ (t) G (ψ (t)) , where G is a convex and increasing function near the origin, and μ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions μ and G, as well as the function F defined by f 0 , which characterizes the growth behavior of f 1 at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a new parameterized identity. Based on this identity we establish some parametrized Simpson-like type symmetric inequalities, for functions whose first derivatives are s- t g s -convex via Reimann–Liouville frational operators. Some special cases are discussed. Applications to numerical quadrature are provided. [ABSTRACT FROM AUTHOR]

Minimax fractional semi-infinite programming is an important research direction for semi-infinite programming, and has a wide range of applications, such as military allocation problems, economic theory, cooperative games, and other fields. Convexity theory plays a key role in many aspects of mathematical programming and is the foundation of mathematical programming research. The relevant theories of semi-infinite programming based on different types of convex functions have their own applicable scope and limitations. It is of great value to study semi-infinite programming on the basis of more generalized convex functions and obtain more general results. In this paper, we defined a new type of generalized convex function, based on the concept of the K−directional derivative, that is, uniform  (B K , ρ) − invex, strictly uniform  (B K , ρ) − invex, uniform  (B K , ρ) − pseudoinvex, strictly uniform  (B K , ρ) − pseudoinvex, uniform  (B K , ρ) − quasiinvex and weakly uniform  (B K , ρ) − quasiinvex function. Then, we studied a class of non-smooth minimax fractional semi-infinite programming problems involving this generalized convexity and obtained sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions. [ABSTRACT FROM AUTHOR]

The objective of the current paper is to find the necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes S T * (γ , β) and K T (γ , β) of analytic functions with negative coefficients. Furthermore, we investigate several inclusion properties of the class Y σ (V , W) associated of the operator I α , c ε defined by this distribution. We also take into consideration an integral operator connected to series of Miller–Ross-type Poisson distributions. Special cases of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we first modify one of the most famous theorems on the principle of differential subordination to hold true for normalized analytic functions with a fixed initial Taylor-Maclaurin coefficient. By using this modified form, we generalize and improve several results, which appeared recently in the literature on the geometric function theory of complex analysis. We also prove some simple conditions for starlikeness, convexity, and the strong starlikeness of several one-parameter families of integral operators, including (for example) a certain μ -convex integral operator and the familiar Bernardi integral operator. [ABSTRACT FROM AUTHOR]

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained. [ABSTRACT FROM AUTHOR]

The purpose of this paper was to define a new class of close-to-convex function, denoted by C V (δ , α) , which is a subclass of all functions that are univalent in D and have positive coefficients normalized by the conditions f (0) = 0 ,     f ′ (0) = 1 , if it satisfies such a condition that is dependent on positive real part. Furthermore, we proved how the power series distribution is essential for determining the sufficient and necessary condition on any function f in class C V (δ , α) . [ABSTRACT FROM AUTHOR]

Sanford S. Miller and Petru T. Mocanu's theory of second-order differential subordinations was extended for the case of third-order differential subordinations by José A. Antonino and Sanford S. Miller in 2011. In this paper, new results are proved regarding third-order differential subordinations that extend the ones involving the classical second-order differential subordination theory. A method for finding a dominant of a third-order differential subordination is provided when the behavior of the function is not known on the boundary of the unit disc. Additionally, a new method for obtaining the best dominant of a third-order differential subordination is presented. This newly proposed method essentially consists of finding the univalent solution for the differential equation that corresponds to the differential subordination considered in the investigation; previous results involving third-order differential subordinations have been obtained mainly by investigating specific classes of admissible functions. The fractional integral of the Gaussian hypergeometric function, previously associated with the theory of fuzzy differential subordination, is used in this paper to obtain an interesting third-order differential subordination by involving a specific convex function. The best dominant is also provided, and the example presented proves the importance of the theoretical results involving the fractional integral of the Gaussian hypergeometric function. [ABSTRACT FROM AUTHOR]

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green's function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area. [ABSTRACT FROM AUTHOR]

The concrete strength criterion is the basis of strength analysis and evaluation under a complex stress state. In this paper, a large number of multiaxial strength tests were carried out, and many mathematical expressions of strength criteria were proposed based on the geometric characteristics and the assumption of a convex function. However, the rationality of the assumption of a convex function limits the use of these strength criteria. In particular, misjudgment will occur near the failure curve surface. Therefore, this paper does not assume the shape function of the criterion in advance. By collecting experimental data and using a machine learning method, it proposes a method of hidden function of failure curve surface. Based on 777 groups of experimental data, the random forest (RF), the back propagation neural network (BP) and the radial basis neural network (RBF) models were used to analyze and verify the feasibility and effectiveness of the method. Subsequently, the results were compared with the Ottosen strength criterion, the Guo Wang strength criterion and the Drucker–Prager (DP) strength criterion. The results show that the consistency between the strength criterion model established by the machine learning algorithm (especially random forest) and the experimental data is higher than the convex function multiaxis strength criterion of the preset failure surface shape. Moreover, the physical significance is clearer, the deficiency of the convex function failure surface hypothesis is avoided and the established multiaxial strength criterion of concrete is more universal. [ABSTRACT FROM AUTHOR]

In this paper, a new Modified Meta-Heuristic algorithm is proposed. This method contains some modifications to improve the performance of the simulated-annealing algorithm (SA). Most authors who deal with improving the SA algorithm presented some improvements and modifications to one or more of the five standard features of the SA algorithm. In this paper, we improve the SA algorithm by presenting some suggestions and modifications to all five standard features of the SA algorithm. Through these suggestions and modifications, we obtained a new algorithm that finds the approximate solution to the global minimum of a non-convex function. The new algorithm contains novel parameters, which are updated at each iteration. Therefore, the variety and alternatives in choosing these parameters demonstrated a noticeable impact on the performance of the proposed algorithm. Furthermore, it has multiple formulas by which the candidate solutions are generated. Diversity in these formulas helped the proposed algorithm to escape a local point while finding the global minimizer of a non-convex function. The efficiency of the proposed algorithm is reported through extensive numerical experiments on some well-known test problems. The performance profiles are used to evaluate and compare the performance of our proposed algorithm against the other five meta-heuristic algorithms. The comparison results between the performance of our suggested algorithm and the other five algorithms indicate that the proposed algorithm is competitive with, and in all cases superior to, the five algorithms in terms of the efficiency, reliability, and effectiveness for finding the global minimizers of non-convex functions. This superiority of the new proposed algorithm is due to those five modified standard features. [ABSTRACT FROM AUTHOR]

This paper mainly studies the distributed optimization problems in a class of undirected networks. The objective function of the problem consists of a smooth convex function and a non-smooth convex function. Each agent in the network needs to optimize the sum of the two objective functions. For this kind of problem, based on the operator splitting method, this paper uses the proximal operator to deal with the non-smooth term and further designs a distributed algorithm that allows the use of uncoordinated step-sizes. At the same time, by introducing the random-block coordinate mechanism, this paper develops an asynchronous iterative version of the synchronous algorithm. Finally, the convergence of the algorithms is proven, and the effectiveness is verified through numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text. [ABSTRACT FROM AUTHOR]

The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann–Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a series of definitions of generalized affine functions for vector-valued functions by use of "linear set". We prove that our generalized affine functions have some similar properties to generalized convex functions. We present examples to show that our generalized affinenesses are different from one another, and also provide an example to show that our definition of presubaffinelikeness is non-trivial; presubaffinelikeness is the weakest generalized affineness introduced in this article. We work with optimization problems that are defined and taking values in linear topological spaces. We devote to the study of constraint qualifications, and derive some optimality conditions as well as a strong duality theorem. Our optimization problems have inequality constraints, equality constraints, and abstract constraints; our inequality constraints are generalized convex functions and equality constraints are generalized affine functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider some relations related to the representations of starlike and convex functions, and obtain some sufficient conditions for starlike and convex functions by using the theory of differential subordination. Actually, we generalize a result by Suffridge for analytic functions with missing coefficients and then we apply that generalization for obtaining the different methods to the implications of starlike or convex functions. Our results generalize and improve the previous results in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to study the bounds of k-integral operators with the Mittag-Leffler kernel in a unified form. To achieve these bounds, the definition of exponentially (α , h − m) − p -convexity is utilized frequently. In addition, a fractional Hadamard type inequality which shows the upper and lower bounds of k-integral operators simultaneously is presented. The results are directly linked with the results of many published articles. [ABSTRACT FROM AUTHOR]

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well. [ABSTRACT FROM AUTHOR]

This paper establishes several new inequalities of Hermite–Hadamard type for | f ′ | q being convex for some fixed q ∈ (0 , 1 ] . As application, some error estimates on special means of real numbers are given. [ABSTRACT FROM AUTHOR]

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings ( U D E T - convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for U D E T - convex   F N V M s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. [ABSTRACT FROM AUTHOR]