Journal articles: 'Analyse convexe'

1

Granas, Andrzej. "Sur un principe géométrique en analyse convexe." Studia Mathematica 101, no. 1 (1991): 1–18. http://dx.doi.org/10.4064/sm-101-1-1-18.

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2

Deguire, Paul, and Andrzej Granas. "Sur une certaine alternative non-linéaire en analyse convexe." Studia Mathematica 83, no. 2 (1986): 127–38. http://dx.doi.org/10.4064/sm-83-2-127-138.

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3

Barré, C., J. M. André, P. Jonnard, and C. Bonnelle. "Analyse spatiale d'une source X à l'aide d'un spectromètre à cristal convexe." X-Ray Spectrometry 24, no. 5 (September 1995): 260–66. http://dx.doi.org/10.1002/xrs.1300240510.

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4

Clément, Benoît. "Analyse par intervalles et optimisation convexe pour résoudre un problème général de faisabilité d’une contrainte robuste." Journal Européen des Systèmes Automatisés 46, no. 4-5 (July 30, 2012): 381–95. http://dx.doi.org/10.3166/jesa.46.381-395.

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5

Popovici, Nicolae. "Convexité au sens direct ou inverse et applications dans l'optimisation vectorielle." Journal of Numerical Analysis and Approximation Theory 29, no. 1 (February 1, 2000): 75–82. http://dx.doi.org/10.33993/jnaat291-656.

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(in English) The aim of this paper is to study vector optimization poblems involving objective functions which are convex in some direct or inverse sense (i.e. a special class of cone-quasiconvex functions). In particular, it is shown that the image of the objective function is a cone-convex set, property which is important from the scalarization point of view in vector optimization. (in French) Le but de cet article est d'etudier les problemes d'optimisation vectorielles ayant des fonctions objectifs convexes au sens direct ou inverse. Il s'agit notamment de donner des conditions suffisantes pour que l'image d'un ensemble convexe par des fonction objectifs cone-quasiconvexes soit un ensemble cone-convexe, propriete qui joue un role important dans la plupart des methodes de scalarisation utilisees dans l'optimisation vectorielle.

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Jin, Chi, Praneeth Netrapalli, Rong Ge, Sham M. Kakade, and Michael I. Jordan. "On Nonconvex Optimization for Machine Learning." Journal of the ACM 68, no. 2 (March 2021): 1–29. http://dx.doi.org/10.1145/3418526.

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Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial. For modern machine learning, where the dimension can be in the millions, such dependence would be catastrophic. We analyze perturbed versions of GD and SGD and show that they are truly efficient—their dimension dependence is only polylogarithmic. Indeed, these algorithms converge to second-order stationary points in essentially the same time as they take to converge to classical first-order stationary points.

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7

Klaričić Bakula, Milica, and Kazimierz Nikodem. "Converse Jensen inequality for strongly convex set-valued maps." Journal of Mathematical Inequalities, no. 2 (2018): 545–50. http://dx.doi.org/10.7153/jmi-2018-12-40.

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Acu, Ana-Maria, Margareta Heilmann, and Ioan Rasa. "Strong Converse Results for Linking Operators and Convex Functions." Journal of Function Spaces 2020 (December 5, 2020): 1–5. http://dx.doi.org/10.1155/2020/4049167.

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We consider a family B n , ρ c of operators which is a link between classical Baskakov operators (for ρ = ∞ ) and their genuine Durrmeyer type modification (for ρ = 1 ). First, we prove that for fixed n , c and a fixed convex function f , B n , ρ c f is decreasing with respect to ρ . We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators B n , ρ c applied to convex functions.

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9

Ceng, Lu-Chuan, and Ching-Feng Wen. "Hybrid Gradient-Projection Algorithm for Solving Constrained Convex Minimization Problems with Generalized Mixed Equilibrium Problems." Journal of Function Spaces and Applications 2012 (2012): 1–26. http://dx.doi.org/10.1155/2012/678353.

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It is well known that the gradient-projection algorithm (GPA) for solving constrained convex minimization problems has been proven to have only weak convergence unless the underlying Hilbert space is finite dimensional. In this paper, we introduce a new hybrid gradient-projection algorithm for solving constrained convex minimization problems with generalized mixed equilibrium problems in a real Hilbert space. It is proven that three sequences generated by this algorithm converge strongly to the unique solution of some variational inequality, which is also a common element of the set of solutions of a constrained convex minimization problem, the set of solutions of a generalized mixed equilibrium problem, and the set of fixed points of a strict pseudocontraction in a real Hilbert space.

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Klaričić Bakula, Milica, and Kazimierz Nikodem. "On the converse Jensen inequality for strongly convex functions." Journal of Mathematical Analysis and Applications 434, no. 1 (February 2016): 516–22. http://dx.doi.org/10.1016/j.jmaa.2015.09.032.

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11

Kamada, Yuichiro, and Fuhito Kojima. "Voter Preferences, Polarization, and Electoral Policies." American Economic Journal: Microeconomics 6, no. 4 (November 1, 2014): 203–36. http://dx.doi.org/10.1257/mic.6.4.203.

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In most variants of the Hotelling-Downs model of election, it is assumed that voters have concave utility functions. This assumption is arguably justified in issues such as economic policies, but convex utilities are perhaps more appropriate in others, such as moral or religious issues. In this paper, we analyze the implications of convex utility functions in a two-candidate probabilistic voting model with a polarized voter distribution. We show that the equilibrium policies diverge if and only if voters' utility function is sufficiently convex. If two or more issues are involved, policies converge in “concave issues” and diverge in “convex issues.” (JEL D72)

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12

Shehu, Yekini, and Jerry N. Ezeora. "Path Convergence and Approximation of Common Zeroes of a Finite Family ofm-Accretive Mappings in Banach Spaces." Abstract and Applied Analysis 2010 (2010): 1–14. http://dx.doi.org/10.1155/2010/285376.

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LetEbe a real Banach space which is uniformly smooth and uniformly convex. LetKbe a nonempty, closed, and convex sunny nonexpansive retract ofE, whereQis the sunny nonexpansive retraction. IfEadmits weakly sequentially continuous duality mappingj, path convergence is proved for a nonexpansive mappingT:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite family ofm-accretive mappings ofKtoE. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings fromKtoEunder certain mild conditions.

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13

Neilan, Michael, Abner J. Salgado, and Wujun Zhang. "Numerical analysis of strongly nonlinear PDEs." Acta Numerica 26 (May 1, 2017): 137–303. http://dx.doi.org/10.1017/s0962492917000071.

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We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

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14

Zhu, Yajie, and Mingchuan Zhang. "An Adaptive Variance Reduction Zeroth-Order Algorithm for Finite-Sum Optimization." Frontiers in Computing and Intelligent Systems 3, no. 3 (May 17, 2023): 66–70. http://dx.doi.org/10.54097/fcis.v3i3.8568.

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The unconstrained finite-sum optimization problem is a common type of problem in the field of optimization, and there is currently limited research on zeroth-order optimization algorithms. To solve unconstrained finite-sum optimization problems for non-convex function, we propose a zeroth-order optimization algorithm with adaptive variance reduction, called ZO-AdaSPIDER for short. Then, we analyze the convergence performance of the algorithm. The theoretical results show that ZO-AdaSPIDER algorithm can converge to -stationary point when facing non-convex function, and its convergence rate is .

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15

Cong, Xin-rong, and Long-suo Li. "Analysis of Robust Stability for a Class of Stochastic Systems via Output Feedback: The LMI Approach." Journal of Function Spaces and Applications 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/873578.

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This paper investigates the robust stability for a class of stochastic systems with both state and control inputs. The problem of the robust stability is solved via static output feedback, and we convert the problem to a constrained convex optimization problem involving linear matrix inequality (LMI). We show how the proposed linear matrix inequality framework can be used to select a quadratic Lyapunov function. The control laws can be produced by assuming the stability of the systems. We verify that all controllers can robustly stabilize the corresponding system. Further, the numerical simulation results verify the theoretical analysis results.

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16

Wei, Li, and Ruilin Tan. "Iterative Schemes for Finite Families of Maximal Monotone Operators Based on Resolvents." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/451279.

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The purpose of this paper is to present two iterative schemes based on the relative resolvent and the generalized resolvent, respectively. And, it is shown that the iterative schemes converge weakly to common solutions for two finite families of maximal monotone operators in a real smooth and uniformly convex Banach space and one example is demonstrated to explain that some assumptions in the main results are meaningful, which extend the corresponding works by some authors.

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17

Volle, M. "Sur quelques formules de dualit� convexe et non convexe." Set-Valued Analysis 2, no. 1-2 (1994): 369–79. http://dx.doi.org/10.1007/bf01027112.

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18

Sobel, D. "Do the desires of rational agents converge?" Analysis 59, no. 3 (July 1, 1999): 137–47. http://dx.doi.org/10.1093/analys/59.3.137.

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19

Reich, Simeon, and Hong-Kun Xu. "An iterative approach to a constrained least squares problem." Abstract and Applied Analysis 2003, no. 8 (2003): 503–12. http://dx.doi.org/10.1155/s1085337503212082.

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A constrained least squares problem in a Hilbert spaceHis considered. The standard Tikhonov regularization method is used. In the case where the set of the constraints is the nonempty intersection of a finite collection of closed convex subsets ofH, an iterative algorithm is designed. The resulting sequence is shown to converge strongly to the unique solution of the regularized problem. The net of the solutions to the regularized problems strongly converges to the minimum norm solution of the least squares problem if its solution set is nonempty.

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20

EHRT, JULIA, and JÖRG HÄRTERICH. "ASYMPTOTIC BEHAVIOR OF SPATIALLY INHOMOGENEOUS BALANCE LAWS." Journal of Hyperbolic Differential Equations 02, no. 03 (September 2005): 645–72. http://dx.doi.org/10.1142/s0219891605000579.

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We study the longtime behavior of spatially inhomogeneous scalar balance laws with periodic initial data and a convex flux. Our main result states that for a large class of initial data the entropy solution will either converge uniformly to some steady state or to a discontinuous time-periodic solution. This extends results of Lyberopoulos, Sinestrari and Fan and Hale obtained in the spatially homogeneous case. The proof is based on the method of generalized characteristics together with ideas from dynamical systems theory. A major difficulty consists of finding the periodic solutions which determine the asymptotic behavior. To this end we introduce a new tool, the Rankine–Hugoniot vector field, which describes the motion of a (hypothetical) shock with certain prescribed left and right states. We then show the existence of periodic solutions of the Rankine–Hugoniot vector field and prove that the actual shock curves converge to these periodic solutions.

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21

Moretti, L. "Why the Converse Consequence Condition cannot be accepted." Analysis 63, no. 4 (October 1, 2003): 297–300. http://dx.doi.org/10.1093/analys/63.4.297.

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22

Yuan, Deming. "Asynchronous Gossip-Based Gradient-Free Method for Multiagent Optimization." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/618641.

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This paper considers the constrained multiagent optimization problem. The objective function of the problem is a sum of convex functions, each of which is known by a specific agent only. For solving this problem, we propose an asynchronous distributed method that is based on gradient-free oracles and gossip algorithm. In contrast to the existing work, we do not require that agents be capable of computing the subgradients of their objective functions and coordinating their step size values as well. We prove that with probability 1 the iterates of all agents converge to the same optimal point of the problem, for a diminishing step size.

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23

Yan, Bing, Hsin Yuan Chen, Peter B. Luh, Simon Wang, and Joey Chang. "Litho Machine Scheduling With Convex Hull Analyses." IEEE Transactions on Automation Science and Engineering 10, no. 4 (October 2013): 928–37. http://dx.doi.org/10.1109/tase.2013.2277812.

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24

Ciliberto, Carlo, Massimiliano Pontil, and Dimitrios Stamos. "Reexamining low rank matrix factorization for trace norm regularization." Mathematics in Engineering 5, no. 3 (2022): 1–22. http://dx.doi.org/10.3934/mine.2023053.

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<abstract><p>Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem. In practice this approach works well, and it is often computationally faster than standard convex solvers such as proximal gradient methods. Nevertheless, it is not guaranteed to converge to a global optimum, and the optimization can be trapped at poor stationary points. In this paper we show that it is possible to characterize all critical points of the non-convex problem. This allows us to provide an efficient criterion to determine whether a critical point is also a global minimizer. Our analysis suggests an iterative meta-algorithm that dynamically expands the parameter space and allows the optimization to escape any non-global critical point, thereby converging to a global minimizer. The algorithm can be applied to problems such as matrix completion or multitask learning, and our analysis holds for any random initialization of the factor matrices. Finally, we confirm the good performance of the algorithm on synthetic and real datasets.</p></abstract>

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25

Cortez, Miguel Vivas, Ali Althobaiti, Abdulrahman F. Aljohani, and Saad Althobaiti. "Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications." Axioms 13, no. 7 (July 12, 2024): 471. http://dx.doi.org/10.3390/axioms13070471.

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Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down (U·D) relations and over newly defined class U·D-ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of U·D-relations, recent findings have been developed using fuzzy Aumman’s, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields.

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26

Wang, Wei, Ming Jin, Shanghua Li, and Xinyu Cao. "AUV-Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions." Journal of Function Spaces 2017 (2017): 1–6. http://dx.doi.org/10.1155/2017/5309698.

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In this paper, we apply theUV-algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affine matrix-valued mapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to applyUV-algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and theU-Lagrangian of the object function at a given point will be addressed particularly. Finally, theUV-algorithm will be presented to get the approximate solution of the primal problem by solving the approximate problem.

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27

Merhav, Neri. "Reversing Jensen’s Inequality for Information-Theoretic Analyses." Information 13, no. 1 (January 13, 2022): 39. http://dx.doi.org/10.3390/info13010039.

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In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several application examples that are relevant to information theory. Moreover, the main ideas behind the derivations turn out to be applicable to generate bounds to expectations of multivariate convex/concave functions, as well as functions that are not necessarily convex or concave.

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28

Drewniak, Józef. "Convex and strongly convex fuzzy sets." Journal of Mathematical Analysis and Applications 126, no. 1 (August 1987): 292–300. http://dx.doi.org/10.1016/0022-247x(87)90093-x.

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29

Shang, Yanling, and Jing Xie. "Global Stabilization of Nonholonomic Chained Form Systems with Input Delay." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/156457.

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This paper investigates the global stabilization problem for a class of nonholonomic systems in chained form with input delay. A particular transformation is introduced to convert the original time-delay system into a delay-free form. Then, by using input-state-scaling technique and the method of sliding mode control, a constructive design procedure for state feedback control is given, which can guarantee that all the system states globally asymptotically converge to the origin. An illustrative example is also provided to demonstrate the effectiveness of the proposed scheme.

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30

Guo, Shuya, Yu-Ming Chu, Ghulam Farid, Sajid Mehmood, and Waqas Nazeer. "Fractional Hadamard and Fejér-Hadamard Inequalities Associated with Exponentially s,m-Convex Functions." Journal of Function Spaces 2020 (August 7, 2020): 1–10. http://dx.doi.org/10.1155/2020/2410385.

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The aim of this paper is to present the fractional Hadamard and Fejér-Hadamard inequalities for exponentially s,m-convex functions. To establish these inequalities, we will utilize generalized fractional integral operators containing the Mittag-Leffler function in their kernels via a monotone function. The presented results in particular contain a number of fractional Hadamard and Fejér-Hadamard inequalities for s-convex, m-convex, s,m-convex, exponentially convex, exponentially s-convex, and convex functions.

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31

Sababheh, Mohammad, Shigeru Furuichi, and Hamid Reza Moradi. "Composite convex functions." Journal of Mathematical Inequalities, no. 3 (2021): 1267–85. http://dx.doi.org/10.7153/jmi-2021-15-85.

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32

Feldman, Nathan S., and Paul J. McGuire. "Convex-cyclic matrices, convex-polynomial interpolation and invariant convex sets." Operators and Matrices, no. 2 (2017): 465–92. http://dx.doi.org/10.7153/oam-11-31.

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33

Ubhaya, Vasant A. "Uniform approximation by quasi-convex and convex functions." Journal of Approximation Theory 55, no. 3 (December 1988): 326–36. http://dx.doi.org/10.1016/0021-9045(88)90099-8.

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34

Yaghoubi, Hassan, Assef Zare, Mohammad Rasouli, and Roohallah Alizadehsani. "Novel Frequency-Based Approach to Analyze the Stability of Polynomial Fractional Differential Equations." Axioms 12, no. 2 (January 31, 2023): 147. http://dx.doi.org/10.3390/axioms12020147.

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This paper proposes a novel approach for analyzing the stability of polynomial fractional-order systems using the frequency-distributed fractional integrator model. There are two types of frequency and temporal stabilization methods for fractional-order systems that global and semi-global stability conditions attain using the sum-of-squares (SOS) method. Substantiation conditions of global and asymptotical stability are complicated for fractional polynomial systems. According to recent studies on nonlinear fractional-order systems, this paper concentrates on polynomial fractional-order systems with any degree of nonlinearity. Global stability conditions are obtained for polynomial fractional-order systems (PFD) via the sum-of-squares approach and the frequency technique employed. This method can be effective in nonlinear systems where the linear matrix inequality (LMI) approach is incapable of response. This paper proposes to solve non-convex SOS-designed equations and design framework key ideas to avoid conservative problems. A Lyapunov polynomial function is determined to address the problem of PFD stabilization conditions and stability established using sufficiently expressed conditions. The main goal of this article is to present an analytical method based on the optimization method for fractional order models in the form of frequency response. This method can convert it into an optimization problem, and by changing the solution of the optimization problem, the stability of the fractional-order system can be improved.

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35

Huang, Mingyang, Chenglin Liu, and Liang Shan. "Containment Control of First-Order Multi-Agent Systems under PI Coordination Protocol." Algorithms 14, no. 7 (July 14, 2021): 209. http://dx.doi.org/10.3390/a14070209.

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This paper investigates the containment control problem of discrete-time first-order multi-agent system composed of multiple leaders and followers, and we propose a proportional-integral (PI) coordination control protocol. Assume that each follower has a directed path to one leader, and we consider several cases according to different topologies composed of the followers. Under the general directed topology that has a spanning tree, the frequency-domain analysis method is used to obtain the sufficient convergence condition for the followers achieving the containment-rendezvous that all the followers reach an agreement value in the convex hull formed by the leaders. Specially, a less conservative sufficient condition is obtained for the followers under symmetric and connected topology. Furthermore, it is proved that our proposed protocol drives the followers with unconnected topology to converge to the convex hull of the leaders. Numerical examples show the correctness of the theoretical results.

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36

Mauldin, R. Daniel. "Coalition convex preference orders are almost surely convex." Journal of Mathematical Analysis and Applications 114, no. 2 (March 1986): 548–51. http://dx.doi.org/10.1016/0022-247x(86)90106-x.

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Ubhaya, Vasant A. "Lp approximation by quasi-convex and convex functions." Journal of Mathematical Analysis and Applications 139, no. 2 (May 1989): 574–85. http://dx.doi.org/10.1016/0022-247x(89)90130-3.

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38

Guseinov, Kh G., S. A. Duzce, and O. Ozer. "Convex extensions of the convex set valued maps." Journal of Mathematical Analysis and Applications 314, no. 2 (February 2006): 672–88. http://dx.doi.org/10.1016/j.jmaa.2005.04.019.

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39

Gabriyelyan, S. "Locally convex properties of free locally convex spaces." Journal of Mathematical Analysis and Applications 480, no. 2 (December 2019): 123453. http://dx.doi.org/10.1016/j.jmaa.2019.123453.

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40

Özcan, Serap. "Some Integral Inequalities for Harmonically (α,s)-Convex Functions." Journal of Function Spaces 2019 (July 30, 2019): 1–8. http://dx.doi.org/10.1155/2019/2394021.

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In the paper, the author introduces a new class of harmonically convex functions, which is called harmonically α,s-convex functions and establishes some new integral inequalities of the Hermite-Hadamard type for harmonically α,s-convex functions. The properties of the newly introduced class of harmonically convex functions are also investigated. Finally, some applications to special means are given.

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41

Adil Khan, Muhammad, Yu-Ming Chu, Artion Kashuri, Rozana Liko, and Gohar Ali. "Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations." Journal of Function Spaces 2018 (May 29, 2018): 1–9. http://dx.doi.org/10.1155/2018/6928130.

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We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

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Kareem, Z. H., and L. N. M. Tawfiq. "Recent modification of decomposition method for solving wave-like equation." Journal of Interdisciplinary Mathematics 26, no. 5 (2023): 809–20. http://dx.doi.org/10.47974/jim-1235.

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In this article exact solution for nonlinear wave-like equations with variable coefficients will be obtain by using reliable manner depend on combined Laplace transform with decomposition technique and the results has shown a high-precision, smooth and the series solution is converge rapidly to exact analytic solution compared with other classic approaches. Suggested approach not needs any discretization by data of domain or presents assumption or neglect for a perturbation parameter in problems and not need to use any assumption to convert the non-linear terms into linear. Two examples of strongly nonlinear 2-dimensional space high order have been presented to show the convergence of solution obtained by suggested method to the exact.

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43

Beg, Ismat. "Ordered Convex Metric Spaces." Journal of Function Spaces 2021 (October 25, 2021): 1–4. http://dx.doi.org/10.1155/2021/7552451.

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The aim of this article is to introduce a new notion of ordered convex metric spaces and study some basic properties of these spaces. Several characterizations of these spaces are proven that allow making geometric interpretations of the new concepts.

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Gordji, Madjid Eshaghi, Mohsen Rostamian Delavar, and M. De La Sen. "On φ-convex functions." Journal of Mathematical Inequalities, no. 1 (2016): 173–83. http://dx.doi.org/10.7153/jmi-10-15.

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45

Arnold, Anton, and Jean Dolbeault. "Refined convex Sobolev inequalities." Journal of Functional Analysis 225, no. 2 (August 2005): 337–51. http://dx.doi.org/10.1016/j.jfa.2005.05.003.

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46

Xu, Shaoyuan, and Yan Han. "Fixed Point Theorems of Superlinear Operators with Applications." Journal of Function Spaces 2022 (May 18, 2022): 1–8. http://dx.doi.org/10.1155/2022/2965300.

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In this paper, by using the partial order method and monotone iterative techniques, the existence and uniqueness of fixed points for a class of superlinear operators are studied, without requiring any compactness or continuity. As corollaries, the new fixed point theorems for α -convex operators α > 1 , e -convex operators, positive α homogeneous operator α > 1 , generalized e -convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.

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47

Song, Ying-Qing, Muhammad Adil Khan, Syed Zaheer Ullah, and Yu-Ming Chu. "Integral Inequalities Involving Strongly Convex Functions." Journal of Function Spaces 2018 (June 11, 2018): 1–8. http://dx.doi.org/10.1155/2018/6595921.

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We study the notions of strongly convex function as well as F-strongly convex function. We present here some new integral inequalities of Jensen’s type for these classes of functions. A refinement of companion inequality to Jensen’s inequality established by Matić and Pečarić is shown to be recaptured as a particular instance. Counterpart of the integral Jensen inequality for strongly convex functions is also presented. Furthermore, we present integral Jensen-Steffensen and Slater’s inequality for strongly convex functions.

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Jung, Chahn Yong, Muhammad Shoaib Saleem, Waqas Nazeer, Muhammad Sajid Zahoor, Attia Latif, and Shin Min Kang. "Unification of Generalized and p-Convexity." Journal of Function Spaces 2020 (January 9, 2020): 1–6. http://dx.doi.org/10.1155/2020/4016386.

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In the present note, we will introduce the definition of generalized p convex function. We will investigate some properties of generalized p convex function. Moreover, we will develop Jensen’s type, Schur type, and Hermite Hadamard type inequalities for generalized p convex function.

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Zeng, Fancheng, Guanqiu Qi, Zhiqin Zhu, Jian Sun, Gang Hu, and Matthew Haner. "Convex Neural Networks Based Reinforcement Learning for Load Frequency Control under Denial of Service Attacks." Algorithms 15, no. 2 (January 23, 2022): 34. http://dx.doi.org/10.3390/a15020034.

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With the increase in the complexity and informatization of power grids, new challenges, such as access to a large number of distributed energy sources and cyber attacks on power grid control systems, are brought to load-frequency control. As load-frequency control methods, both aggregated distributed energy sources (ADES) and artificial intelligence techniques provide flexible solution strategies to mitigate the frequency deviation of power grids. This paper proposes a load-frequency control strategy of ADES-based reinforcement learning under the consideration of reducing the impact of denial of service (DoS) attacks. Reinforcement learning is used to evaluate the pros and cons of the proposed frequency control strategy. The entire evaluation process is realized by the approximation of convex neural networks. Convex neural networks are used to convert the nonlinear optimization problems of reinforcement learning for long-term performance into the corresponding convex optimization problems. Thus, the local optimum is avoided, the optimization process of the strategy utility function is accelerated, and the response ability of controllers is improved. The stability of power grids and the convergence of convex neural networks under the proposed frequency control strategy are studied by constructing Lyapunov functions to obtain the sufficient conditions for the steady states of ADES and the weight convergence of actor–critic networks. The article uses the IEEE14, IEEE57, and IEEE118 bus testing systems to verify the proposed strategy. Our experimental results confirm that the proposed frequency control strategy can effectively reduce the frequency deviation of power grids under DoS attacks.

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Youness, E. A., and Tarek Emam. "Strongly E-convex sets and strongly E-convex functions." Journal of Interdisciplinary Mathematics 8, no. 1 (January 2005): 107–17. http://dx.doi.org/10.1080/09720502.2005.10700394.

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Journal articles on the topic 'Analyse convexe'

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