Convex Functions Research Articles

Characteristics of concurrent flame spread over convex charring materials

In real forest fire scenarios, there are a large number of convex and concave terrains at the junction of flat land and slopes. In this paper, a typical charring material, kraft paper, is selected as the experimental sample, and the variation law of key characteristic parameters of flame spread over convex solid surface are investigated. The experimental results show that all the flame spread processes have obvious deceleration behaviors, and the influence mechanism of flame tangent angle on the flame spread deceleration of convex surfaces is revealed. The characteristic length of the pyrolysis zone shows a tendency of increasing and then decreasing with time. The variation curves of radiant heat flux present two peaks. Combining with the characteristic length, the reason for the occurrence of two peaks is explained. The exponential relationship between the instantaneous mass loss rate and the characteristic length of the pyrolysis zone is obtained. Based on the range of power exponents, the heat transfer controlling mechanism in the pyrolysis zone is classified into three categories for all the working conditions. Moreover, a piecewise power-law relationship between the dimensionless maximum flame height and the dimensionless heat release rate is established. The results of this study will fill the research gap of flame spread of solid materials with curved surfaces, and will be of great academic value for the improvement and development of theories on building and forest fire safety.

An Approach to Generalized (η1,η2)-Convex Functions via Local Fractional Calculus and some Applications

An Approach to Generalized (η1,η2)-Convex Functions via Local Fractional Calculus and some Applications

Local quadratic convergence of the SQP method for an optimal control problem governed by a regularized fracture propagation model

We prove local quadratic convergence of the sequential quadratic programming (SQP) method for an optimal control problem of tracking type governed by one time step of the Euler-Lagrange equation of a time discrete regularized fracture or damage energy minimization problem. This lower-level energy minimization problem % describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process and a viscous regularization term. Conditions on the latter, corresponding to a time step restriction, guarantee strict convexity and thus unique solvability of the Euler Lagrange equations. Nonetheless, these are quasilinear and the control problem is nonconvex. For the convergence proof with $L^\infty$ localization of the SQP-method, we follow the approach from \cite{Troeltzsch:1999}, utilizing strong regularity of generalized equations and arguments from \cite{HoppeNeitzel:2020} for $L^2$-localization. 2020 Mathematics Subject Classification 90C55, 49M41, 49M15.

Узагальнені обернені нерівності Єнсена-Штеффенсена та пов’язані нерівності

We compare two linear functionals that are negative on convex functions. Further, using Green's functions we give some new conditions for reversed Jensen-Steffensen and related inequalities to hold. Using Green's function we also give refinement of Levinson type generalization of reversed Jensen-Steffensen and related inequalities. The acquired results are then used for constructing mean-value theorems.

Plane SV-waves scattering by seawater convex surface topography and underlying lined tunnel in a saturated half-space

Analytical solutions of boundary-valued problems for aboveground topography and underground buried structures, especially for sites with overlying seawater, have always been challenging. Based on the region decomposition technique (RDT) and Hankel integral transform method (HITM), SV-waves scattering by the convex circular topography with seawater and underlying tunnel in a saturated half-space is solved. The primary problem is tackling additional scattered waves introduced by the water–soil interface and convex topography, while ensuring boundary conditions are all met. To this end, scattering wave potential functions in cylindrical coordinates are first transformed into the Hankel integral form in Cartesian coordinates. This greatly facilitates the straightforward application of free boundary conditions at the soil–seawater flat interface, rather than making a large circular arc approximate assumption, which, in turn, avoids the resultant accumulating errors. Secondly, scattering problem to be solved is mathematically and analytically formulated as solving a 5 × 5 linear equation system comprised of wave functions inside the convex topography and lining tunnel while satisfying all physical continuous boundary conditions. This is the main innovation and resulting consequence from the theoretical derivation process. Finally, validation of numerical examples between this paper, existing studies and theoretical analysis further demonstrates the reliability and general adaptability of solutions. Parametric studies of the incident SV waves are conducted to investigate the spatially varying seismic response. The effects of SV-waves incident angles and frequencies, tunnel burial depth, tunnel lining thickness, saturated soil porosity, and seawater depth are investigated and analyzed in detail. This study provides a feasible scheme of investigating the dynamic response of complex submarine sites, which is of great significance for both theoretical value and engineering purposes.

MB38(M=Be and Zn): A Quasi-Planar Structure Rather Than A Core-Shell Octahedral Borospherene Structure.

In a recent paper (ChemPhysChem, 2023, 24, e202200947), based on the results computed using DFT method, the perfect core-shell octahedral configuration Be@B38 and Zn@B38 was reported to be the global minima of the MB38(M=Be and Zn) clusters. However, this paper presents the lower energy structures of MB38(M=Be and Zn) clusters as a quasi-planar configuration, the Be atom is found to reside on the convex surface of the quasi-planar B38 isomer, while the Zn atom tends to be attached to the top three B atoms of the quasi-planar B38 isomer. Our results show that quasi-planar MB38(M=Be and Zn) at DFT method have lower energy than core-shell octahedral configuration M@B38(M=Be and Zn). Natural atomic charges, valence electron density, electron localization function (ELF) analyses identify the MB38(M=Be and Zn) to be charge transfer complexes (Be2+B382-and Be1+B381-) and suggest primarily the electrostatic interactions between doped atom and B38 fragment. The photoelectron spectra of the corresponding anionic structures were simulated, providing theoretical basis for future structural identification.

Quantum symmetric integral inequalities for convex functions

In the context of quantum symmetric calculus, this study proposed more refined version of Ostrowski and Hermite–Hadamard type inequalities. The function involved in these inequalities are convex functions. In order to reach the target, left and right quantum symmetric derivative and corresponding integral are used. Furthermore, the Hölder inequality is established in the frame work of left and right quantum symmetric integral. The new results refined the results about integral inequalities that exist in the literature.

Inexact Fixed-Point Proximity Algorithm for the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm. Specifically, the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm regularization term. Such an ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} models, in terms of approximation accuracy and sparsity of the solutions.

Inexact Fixed-Point Proximity Algorithm for the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm. Specifically, the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm regularization term. Such an ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} models, in terms of approximation accuracy and sparsity of the solutions.

A Nesterov Type Algorithm with Double Tikhonov Regularization: Fast Convergence of the Function Values and Strong Convergence to the Minimal Norm Solution

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from argminf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {argmin}f$$\\end{document}. We also show fast convergence for the potential energies f(xn)-minf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(x_n)-\ ext {min}f$$\\end{document} and f(yn)-minf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(y_n)-\ ext {min}f$$\\end{document}, where (xn),(yn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(x_n),\\,(y_n)$$\\end{document} are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function.

On the complexity of a quadratic regularization algorithm for minimizing nonsmooth and nonconvex functions

In this paper, we consider the problem of minimizing the function f ( x ) = g 1 ( x ) + g 2 ( x ) − h ( x ) over R n , where g 1 is a proper and lower semicontinuous function, g 2 is continuously differentiable with a Hölder continuous gradient and h is a convex function that may be nondifferentiable. This problem has important practical applications but is challenging to solve due to the presence of nonconvexities and nonsmoothness. To address this issue, we propose an algorithm based on a proximal gradient method that uses a quadratic approximation of the function g 2 and a nonconvex regularization term. We show that the number of iterations required to reach our stopping criterion is O ( max { ϵ − β + 1 β , η 2 β ϵ − 2 ( β + 1 ) β } ) . Our approach offers a promising strategy for solving this challenging optimization problem and has potential applications in various fields. Numerical examples are provided to illustrate the theoretical results.

Accelerated optimization through time-scale analysis of inertial dynamics with asymptotic vanishing and Hessian-driven dampings

ABSTRACT Gradient optimization algorithms can be effectively studied from the perspective of ordinary differential equations (ODEs), where these algorithms are obtained through the temporal discretization of the continuous dynamics. This approach provides a powerful tool to comprehend acceleration phenomena in optimization, thanks to time-scale techniques and Lyapunov analysis. In the context of a Hilbert setting, our study focuses on the rapid optimization properties of inertial dynamics, which combine asymptotically vanishing viscous damping with Hessian-driven damping. These dynamics are derived as high-resolution ODEs from both the Nesterov accelerated gradient method and the Ravine method. For a general differentiable convex function f, choosing the viscous damping coefficient in the form of α / t with 3 $ ]]> α > 3 guarantees an inverse quadratic convergence rate of the values, i.e. o ( 1 / t 2 ) , the weak convergence of trajectories towards optimal solutions, and the fast convergence of gradients towards zero. By carefully scaling the dynamics temporally and based on the tuning of the damping coefficient in front of the Hessian term, we identify the limit dynamic as α becomes large. In particular, we examine the case where the limit dynamic corresponds to the Levenberg-Marquardt regularization of Newton's continuous method. This explanation accounts for the aforementioned fast convergence properties and provides fresh insights into the complexity of these methods, which play a central role in optimization for high-dimensional problems. The interplay between numerical algorithms and continuous-time ODEs plays a fundamental role in our analysis for the design and understanding of accelerated optimization algorithms. Numerical experiments are presented to illustrate and support the theoretical results.

Fine Representation of Hessian of Convex Functions and Ricci Tensor on RCD Spaces

AbstractIt is known that on RCD spaces one can define a distributional Ricci tensor $$\textbf{Ric}$$ Ric . Here we give a fine description of this object by showing that it admits the polar decomposition $$\begin{aligned} \textbf{Ric}=\omega \,|\textbf{Ric}| \end{aligned}$$ Ric = ω | Ric | for a suitable non-negative measure $$|\textbf{Ric}|$$ | Ric | and unitary tensor field $$\omega $$ ω . The regularity of both the mass measure and of the polar vector are also described. The representation provided here allows to answer some open problems about the structure of the Ricci tensor in such singular setting. Our discussion also covers the case of Hessians of convex functions and, under suitable assumptions on the base space, of the Sectional curvature operator.

Starlike and Convex functions associated with Hypergeometric Matrix functions

Starlike and Convex functions associated with Hypergeometric Matrix functions

New cyclic Jensen type bounds via weighted HH inequalities and Montgomery identity with application to information inequalities

In this article a new method is introduced to give weighted estimations of the differences of cyclic refinements of Jensen functionals. We utilize weighted form of Hermite–Hadamard (HH) inequality along with the approximations of Montgomery two- and one-point formulae with Peano type kernel along with the consequences of the n-times differentiable convex functions. As a result, we present new upper and lower bounds that are also verified with concrete examples to show the correctness of the bounds obtained for special cases. As a result, we provide estimations in terms of cyclic power means. We also improvise our results to give new uniform estimations for useful distance functions in information theory. Finally, some estimations of Zipf and Hybrid Zipf law are provided as well.

Hermite-Hadamard inequalities for generalized (m-F)-convex function in the framework of local fractional integrals

This work presents new versions of the Hermite-Hadamard Inequality, for (m−F)-convex functions, defined on fractal sets Rς ( 0 ς ≤ 1). So, we show some new results for twice differentiable functions using local fractional calculus, as well as some new definitions. We will construct these new integral inequality using the generalized H¨older-integral inequality and the power mean integral inequality. Furthermore, we present some new inequalities for the midpoint and trapezoid formulas in a novel type of fractal calculus. The conclusions in this paper are substantial advancements and generalizations of prior research reported in the literature.

The Swedish Production of Apartments as a Function of GNP, Building Costs and Population Changes: Generation of Intelligent Media Content via Big Data Analytics

The Swedish Production of Apartments as a Function of GNP, Building Costs and Population Changes: Generation of Intelligent Media Content via Big Data Analytics

Inequalities for strongly s-convex functions via Atangana-Baleanu fractional integral operators

It is more convenient to use fractional derivatives and integrals to express and represent rapid changes than to use integer derivatives and integrals. For this reason, fractional analysis has been found worthy of study in many fields. In recent years, fractional derivatives and integrals have been discussed together with inequality theory and the studies have attracted attention. In this article, we discuss new Hermite-Hadamard type approximations for strongly convex functions with the help of Atangana-Baleanu fractional integral operators. Additionally, new upper bounds have been obtained using various auxiliary inequalities with the help of twice differentiable strongly convex functions.

Refinement of Jensen Mercer and Hermite–Hadamard-Mercer type inequalities for generalized convex functions on co-ordinates with their computational analysis

In the current study, the Jensen-Mercer inequality is extended to co-ordinated h-convex functions. Additionally, a novel inequality is employed to derive Hermite–Hadamard-Mercer type inequalities for h-convex functions defined on the co-ordinates of a rectangle in the plane. These developments not only reinforce the core tenets of convex analysis but also expand the applicability of Hermite-Hadamard-Mercer type inequalities to generalized convex functions on co-ordinates. This provides valuable tools for data analysis and optimization problem-solving. The practical utility and efficacy of this generalized inequality in real-world scenarios involving co-ordinates are demonstrated through a computational study. Mathematics Subject Classification: 26D10, 26D15, 26A51, 34A08.

Hermite–Hadamard-type inequalities for strongly $$(\alpha ,m)$$-convex functions via quantum calculus

Hermite–Hadamard-type inequalities for strongly $$(\alpha ,m)$$-convex functions via quantum calculus