Twice-differentiable Functions Research Articles

New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions

This paper delves into an inquiry that centers on the exploration of fractional adaptations of Milne-type inequalities by employing the framework of twice-differentiable convex mappings. Leveraging the fundamental tenets of convexity, H\"{o}lder's inequality, and the power-mean inequality, a series of novel inequalities are deduced. These newly acquired inequalities are fortified through insightful illustrative examples, bolstered by rigorous proofs. Furthermore, to lend visual validation, graphical representations are meticulously crafted for the showcased examples.

Hermite–Hadamard–Mercer Inequalities Associated with Twice-Differentiable Functions with Applications

In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain pre-existing inequalities found in the literature. Finally, we provide applications to the newly established inequalities to verify their usefulness.

On conformable fractional versions of trapezoid-type inequalities according to twice-differentiable functions

This paper proves an equality for the case of twice-differentiable convex mappings with respect to the conformable fractional integrals. With the help of this equality, several trapezoid-type inequalities are established by convex functions involving conformable fractional integrals. Sundry significant inequalities are acquired with taking advantage of the convexity, the Holder inequality, and the ¨ power mean inequality. Furthermore, we present several new results connected with trapezoid-type inequalities by using the special choices of obtained results.

Midpoint-type inequalities via twice-differentiable functions on tempered fractional integrals

In this paper, we obtain an equality involving tempered fractional integrals for twice-differentiable functions. By using this equality, we establish several left Hermite–Hadamard-type inequalities for the case of tempered fractional integrals. Moreover, we derive our results by using special cases of obtained theorems.

Training recurrent neural networks by sequential least squares and the alternating direction method of multipliers

This paper proposes a novel algorithm for training recurrent neural network models of nonlinear dynamical systems from an input/output training dataset. Arbitrary convex and twice-differentiable loss functions and regularization terms are handled by sequential least squares and either a line-search (LS) or a trust-region method of Levenberg–Marquardt (LM) type for ensuring convergence. In addition, to handle non-smooth regularization terms such as ℓ1, ℓ0, and group-Lasso regularizers, as well as to impose possibly non-convex constraints such as integer and mixed-integer constraints, we combine sequential least squares with the alternating direction method of multipliers (ADMM). We call the resulting algorithm NAILS (nonconvex ADMM iterations and least squares) in the case line search (LS) is used, or NAILM if a trust-region method (LM) is employed instead. The training method, which is also applicable to feedforward neural networks as a special case, is tested in three nonlinear system identification problems.

Conformable fractional versions of Hermite–Hadamard-type inequalities for twice-differentiable functions

In this paper, new inequalities for the left and right sides of the Hermite–Hadamard inequality are acquired for twice-differentiable mappings. Conformable fractional integrals are used to derive these inequalities. Furthermore, we provide our results by using special cases of obtained theorems.

Fractional inequalities of corrected Euler–Maclaurin-type for twice-differentiable functions

Fractional inequalities of corrected Euler–Maclaurin-type for twice-differentiable functions

Fractional midpoint-type inequalities for twice-differentiable functions

In this research article, we obtain an identity for twice differentiable functions whose second derivatives in absolute value are convex. By using this identity, we prove several left Hermite-Hadamardtype inequalities for the case of Riemann-Liouville fractional integrals. Furthermore, we provide our results by using special cases of obtained theorems.

Ostrowski-type inequalities pertaining to Atangana–Baleanu fractional operators and applications containing special functions

The objective of this article is to incorporate the concept of the Ostrowski inequality with the Atangana–Baleanu fractional integral operator. A novel integral identity for twice-differentiable functions is established after a rigorous investigation of several basic definitions and existing ideas related to inequalities and fractional calculus. Following that, numerous Ostrowski-type inequalities are provided based on this identity, which uses Mittag–Leffler as its kernel structure. Some specific applications, such as q-digamma functions and modified Bessel functions, are also investigated. Choosing s=1, we also analyze new results for convex functions as special cases. Our findings corroborate some well-documented inequalities.

Consecutive Quadratic Residues and Primitive Roots in the Sequences Formed by Twice-differentiable Functions

Consecutive Quadratic Residues and Primitive Roots in the Sequences Formed by Twice-differentiable Functions

Post-Quantum Midpoint-Type Inequalities Associated with Twice-Differentiable Functions

In this study, first we establish a p,q-integral identity involving the second p,q-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-p,q-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature.

Some New Simpson’s-Formula-Type Inequalities for Twice-Differentiable Convex Functions via Generalized Fractional Operators

From the past to the present, various works have been dedicated to Simpson’s inequality for differentiable convex functions. Simpson-type inequalities for twice-differentiable functions have been the subject of some research. In this paper, we establish a new generalized fractional integral identity involving twice-differentiable functions, then we use this result to prove some new Simpson’s-formula-type inequalities for twice-differentiable convex functions. Furthermore, we examine a few special cases of newly established inequalities and obtain several new and old Simpson’s-formula-type inequalities. These types of analytic inequalities, as well as the methodologies for solving them, have applications in a wide range of fields where symmetry is crucial.

Some Hermite–Hadamard-Type Fractional Integral Inequalities Involving Twice-Differentiable Mappings

The theory of fractional analysis has been a focal point of fascination for scientists in mathematical science, given its essential definitions, properties, and applications in handling real-life problems. In the last few decades, many mathematicians have shown their considerable interest in the theory of fractional calculus and convexity due to their wide range of applications in almost all branches of applied sciences, especially in numerical analysis, physics, and engineering. The objective of this article is to establish Hermite-Hadamard type integral inequalities by employing the k-Riemann-Liouville fractional operator and its refinements, whose absolute values are twice-differentiable h-convex functions. Moreover, we also present some special cases of our presented results for different types of convexities. Moreover, we also study how q-digamma functions can be applied to address the newly investigated results. Mathematical integral inequalities of this class and the arrangements associated have applications in diverse domains in which symmetry presents a salient role.

Kloosterman sums with twice-differentiable functions

We bound Kloosterman-like sums of the shape \[ \sum_{n=1}^N \exp(2\pi i (x \floor{f(n)}+ y \floor{f(n)}^{-1})/p), \] with integers parts of a real-valued, twice-differentiable function $f$ is satisfying a certain limit condition on $f''$, and $\floor{f(n)}^{-1}$ is meaning inversion modulo~$p$. As an immediate application, we obtain results concerning the distribution of modular inverses inverses $\floor{f(n)}^{-1} \pmod{p}$. The results apply, in particular, to Piatetski-Shapiro sequences $ \floor{t^c}$ with $c\in(1,\frac{4}{3})$. The proof is an adaptation of an argument used by Banks and the first named author in a series of papers from 2006 to 2009.

Solving global optimization problems using reformulations and signomial transformations

In this paper, a framework for reformulating nonconvex mixed-integer nonlinear programming (MINLP) problems containing twice-differentiable (C2) functions to convex relaxed form is discussed. To provide flexibility and for utilizing more effective transformation strategies, the twice-differentiable functions can be partitioned into convex, signomial and general nonconvex functions. The latter two can then be convexified using lifting transformations in combination with approximations using piecewise linear functions (PLFs). However, since there are many degrees of freedom in how to select the set of transformations, an optimization-based method is proposed for finding an optimal set. The lifting transformations are based on single-variable power and exponential transformations for signomials. For nonconvex C2-functions the α reformulation (αR) technique as well as more generally the method of difference of convex functions can be applied. In the αR, the αBB convex underestimator can be used. The framework is utilized in the α signomial global optimization (αSGO) algorithm to find the ϵ-global solution to a nonconvex problem by iteratively updating the approximations provided by the PLFs. The framework can also be used to directly obtain a convex relaxation of any nonconvex MINLP problem of the specified type to a determined accuracy.

Generalized Sparse Learning of Linear Models Over the Complete Subgraph Feature Set.

Supervised learning over graphs is an intrinsically difficult problem: simultaneous learning of relevant features from the complete subgraph feature set, in which enumerating all subgraph features occurring in given graphs is practically intractable due to combinatorial explosion. We show that 1) existing graph supervised learning studies, such as Adaboost, LPBoost, and LARS/LASSO, can be viewed as variations of a branch-and-bound algorithm with simple bounds, which we call Morishita-Kudo bounds; 2) We present a direct sparse optimization algorithm for generalized problems with arbitrary twice-differentiable loss functions, to which Morishita-Kudo bounds cannot be directly applied; 3) We experimentally showed that i) our direct optimization method improves the convergence rate and stability, and ii) L1-penalized logistic regression (L1-LogReg) by our method identifies a smaller subgraph set, keeping the competitive performance, iii) the learned subgraphs by L1-LogReg are more size-balanced than competing methods, which are biased to small-sized subgraphs.

Optimal recovery of isotropic classes of twice-differentiable functions defined on [formula omitted]-dimensional Euclidean space

In the paper, inspired by the works of V. F. Babenko, S. V. Borodachov and D. S. Skorokhodov, we consider the problem of optimal recovery of isotropic classes of twice-differentiable multivariate functions defined on Euclidean space Rd, and get some exact results. This problem is connected with the optimal covering of Rd in discrete geometry. What Babenko et al. considered is the same kind of the optimal recovery problem of an isotropic class of twice-differentiable multivariate functions defined on a compact set of Rd and an isotropic class of twice-differentiable periodic functions.

Hermite–Hadamard-type inequalities for r-convex functions based on the use of Riemann–Liouville fractional integrals

By using two fundamental fractional integral identities, we deduce some new Hermite–Hadamard-type inequalities for differentiable r-convex functions and twice-differentiable r-convex functions involving Riemann–Liouville fractional integrals.

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called ?BB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.

Functional ANOVA, ultramodularity and monotonicity: Applications in multiattribute utility theory

Utility function properties as monotonicity and concavity play a fundamental role in reflecting a decision-maker’s preference structure. These properties are usually characterized via partial derivatives. However, elicitation methods do not necessarily lead to twice-differentiable utility functions. Furthermore, while in a single-attribute context concavity fully reflects risk aversion, in multiattribute problems such correspondence is not one-to-one. We show that Tsetlin and Winkler’s multivariate risk attitudes imply ultramodularity of the utility function. We demonstrate that geometric properties of a multivariate utility function can be successfully studied by utilizing an integral function expansion (functional ANOVA). The necessary and sufficient conditions under which monotonicity and/or ultramodularity of single-attribute functions imply the monotonicity and/or ultramodularity of the corresponding multiattribute function under additive, preferential and mutual utility independence are then established without reliance on the utility function differentiability. We also investigate the relationship between the presence of interactions among the attributes of a multiattribute utility function and the decision-maker’s multivariate risk attitudes.