3,797 results
Search Results
2. Constrained minimum variance and covariance steering based on affine disturbance feedback control parameterization.
- Author
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Balci, Isin M. and Bakolas, Efstathios
- Subjects
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STOCHASTIC control theory , *MINIMUM variance estimation , *COVARIANCE matrices , *UNCERTAIN systems , *CONVEX functions , *PARAMETERIZATION , *LINEAR matrix inequalities - Abstract
This paper deals with finite‐horizon minimum‐variance and covariance steering problems subject to constraints. The goal of the minimum variance problem is to steer the state mean of an uncertain system to a prescribed vector while minimizing the trace of its terminal state covariance whereas the goal in the covariance steering problem is to steer the covariance matrix of the terminal state to a prescribed positive definite matrix. The paper proposes a solution approach that relies on a stochastic version of the affine disturbance feedback control parametrization. In this control policy parametrization, the control input at each stage is expressed as an affine function of the history of disturbances that have acted upon the system. It is shown that this particular parametrization reduces the stochastic optimal control problems considered in this paper into tractable convex programs or difference of convex functions programs with essentially the same decision variables. In addition, the paper proposes a variation of this control parametrization that relies on truncated histories of past disturbances, which allows for sub‐optimal controllers to be designed that strike a balance between performance and computational cost. The suboptimality of the truncated policies is formally analyzed and closed form expressions are provided for the performance loss due to the use of the truncation scheme. Finally, the paper concludes with a comparative analysis of the truncated versions of the proposed policy parametrization and other standard policy parametrizations through numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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3. Insight into the gas–liquid transition from the Berthelot model.
- Author
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Mi, Li-Qin, Li, Dandan, Li, Shanshan, and Li, Zhong-Heng
- Subjects
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THERMODYNAMICS , *FIRST-order phase transitions , *EQUATIONS of state , *PHASE transitions , *CONVEX functions , *LATENT heat - Abstract
We extend the parametric method developed for the van der Waals model by Lekner [Am. J. Phys. 50(2), 161–163 (1982)] to other equations of state, particularly the Berthelot model, thereby making the testing of these equations of state much faster and simpler. We systematically investigate important properties of first-order phase transitions in the Berthelot model. Thermodynamic properties near the critical point are discussed and the predictions of the Berthelot and van der Waals models are compared with experimental data. The Berthelot equation affords an improved fit to the density–temperature coexistence curve for many substances when compared to the van der Waals equation. A failure of the Berthelot model is its prediction of latent heat and heat capacities that are convex functions at lower temperatures. We also examine two modifications of the Berthelot equation of state that, like the van der Waals model, are also solvable by the parameter method. These, which we call the cPF and dPF models, reduce to the van der Waals and Berthelot models in different limits of their parameters. They give improved fits to the experimental data away from the critical point but involve an additional fitting parameter. Editor's note: While the van der Waals equation of state provides a simple model for phase transitions, it fails to achieve a good quantitative fit for properties near phase transitions in most substances. A closely related model, the Berthelot model, still has only two free parameters, but it allows the attraction between molecules to depend not only on volume but also on temperature. This paper builds on the parametric expressions for the van der Waals gas derived in a 1982 paper in this journal by John Lekner. It shows that similar expressions derived from the Berthelot model provide a much better fit to the data. This derivation could be shared with students in intermediate or advanced thermodynamics courses. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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4. Panoptic Segmentation with Convex Object Representation.
- Author
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Yao, Zhicheng, Wang, Sa, Zhu, Jinbin, and Bao, Yungang
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DEEP learning , *COMPUTER vision , *OBJECT-oriented methods (Computer science) , *VECTORS (Calculus) , *CONVEX functions - Abstract
The accurate representation of objects holds pivotal significance in the realm of panoptic segmentation. Presently, prevalent object representation methodologies, including box-based, keypoint-based and query-based techniques, encounter a challenge known as the 'representation confusion' issue in specific scenarios, often resulting in the mislabeling of instances. In response, this paper introduces Convex Object Representation (COR), a straightforward yet highly effective approach to address this problem. COR leverages a CNN-based Euclidean Distance Transform to convert the target instance into a convex heatmap. Simultaneously, it offers a parallel embedding method for encoding the object. Subsequently, COR characterizes objects based on the distinctive embedding vectors of their convex vertices. This paper seamlessly integrates COR into a state-of-the-art query-based panoptic segmentation framework. Experimental findings validate that COR successfully mitigates the representation confusion predicament, enhancing segmentation accuracy. The COR-augmented methods exhibit notable improvements of +1.3 and +0.7 points in PQ on the Cityscapes validation and MS COCO panoptic 2017 validation datasets, respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Further Hermite–Hadamard-Type Inequalities for Fractional Integrals with Exponential Kernels.
- Author
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Li, Hong, Meftah, Badreddine, Saleh, Wedad, Xu, Hongyan, Kiliçman, Adem, and Lakhdari, Abdelghani
- Subjects
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CONVEX functions , *DIFFERENTIABLE functions , *INTEGRAL inequalities , *INTEGRAL operators , *INTEGRALS - Abstract
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]
- Published
- 2024
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6. On Approximate Variational Inequalities and Bilevel Programming Problems.
- Author
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Upadhyay, Balendu Bhooshan, Stancu-Minasian, Ioan, Poddar, Subham, and Mishra, Priyanka
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BILEVEL programming , *SUBDIFFERENTIALS , *CONVEX functions - Abstract
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]
- Published
- 2024
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7. A subclass of analytic functions with negative coefficient defined by generalizing Srivastava-Attiya operator.
- Author
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Hamaad, Suha J., Juma, Abdul Rahman S., and Ebrahim, Hassan H.
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ANALYTIC functions , *CONVEX functions , *GENERALIZATION - Abstract
The primary goal of this paper is to introduce and investigate a novel subclass of analytic functions in the open unit disk by generalizing the Srivastava-Attiya operator. So by using the generalization we have introduced a subclass of analytic function with negative coefficients in the unit disk. We have referred to the previous studies that used the Sirvastava-Attiya operator and generalized it, explained the functions of the class 퓐 and the basic definitions that included this paper. We used some important lemmas from previous studies to prove our results, and we obtained some important geometric properties of the analytical functions. We proved the theorem of growth and destortion, and we showed the cofficient bound, extreme points of the functions in this class, in addition to the radii of the starlike, convex and close-to-convex functions of order 휑. Finally, we defined the 훼 −neighborhood and showed the relationship between the functions that belong to the class S ⌣ λ p , μ q , t s , a , λ (γ , ρ , l , σ) and the functions that belong to the class S ⌣ λ p , μ q , t s , a , λ , ω (γ , ρ , l , σ). [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. On strongly generalized convex stochastic processes.
- Author
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Sharma, Nidhi, Mishra, Rohan, and Hamdi, Abdelouahed
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STOCHASTIC processes , *CONVEX functions , *INTEGRAL inequalities - Abstract
In this paper, we introduce the notion of strongly generalized convex functions which is called as strongly η-convex stochastic processes. We prove the Hermite-Hadamard, Ostrowski type inequality, and obtain some important inequalities for above processes. Some previous results are special cases of the results obtained in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Analytical and geometrical approach to the generalized Bessel function.
- Author
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Bulboacă, Teodor and Zayed, Hanaa M.
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INTEGRAL transforms , *BESSEL functions , *CONVEX functions , *MATHEMATICAL notation - Abstract
In continuation of Zayed and Bulboacă work in (J. Inequal. Appl. 2022:158, 2022), this paper discusses the geometric characterization of the normalized form of the generalized Bessel function defined by V ρ , r (z) : = z + ∑ k = 1 ∞ (− r) k 4 k (1) k (ρ) k z k + 1 , z ∈ U , for ρ , r ∈ C ∗ : = C ∖ { 0 } . Precisely, we will use a sharp estimate for the Pochhammer symbol, that is, Γ (a + n) / Γ (a + 1) > (a + α) n − 1 , or equivalently (a) n > a (a + α) n − 1 , that was firstly proved by Baricz and Ponnusamy for n ∈ N ∖ { 1 , 2 } , a > 0 and α ∈ [ 0 , 1.302775637 ... ] in (Integral Transforms Spec. Funct. 21(9):641–653, 2010), and then proved in our paper by another method to improve it using the partial derivatives and the two-variable functions' extremum technique for n ∈ N ∖ { 1 , 2 } , a > 0 and 0 ≤ α ≤ 2 , and used to investigate the orders of starlikeness and convexity. We provide the reader with some examples to illustrate the efficiency of our theory. Our results improve, complement, and generalize some well-known (nonsharp) estimates, as seen in the Concluding Remarks and Outlook section. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. A linearized approach for solving differentiable vector optimization problems with vanishing constraints.
- Author
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Antczak, Tadeusz
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CONVEX functions - Abstract
In this paper, two mathematical methods are used for solving a complex multicriteria optimization problem as the considered convex differentiable vector optimization problem with vanishing constraints. First of them is the linearized approach in which, for the original vector optimization problem with vanishing constraints, its associated multiobjective programming problem is constructed at the given feasible solution. Since the aforesaid multiobjective programming problem constructed in the linearized method is linear, one of the existing methods for solving linear vector optimization problems is applied for solving it. Thus, the procedure for solving the considered differentiable vector optimization problems with vanishing constraints is presented in the paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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11. NEW ITERATIVE SCHEMES FOR GENERAL HARMONIC VARIATIONAL INEQUALITIES.
- Author
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NOOR, MUHAMMAD ASLAM and NOOR, KHALIDA INAYAT
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CONVEX sets , *HARMONIC functions , *VARIATIONAL inequalities (Mathematics) , *CONVEX functions , *SET functions - Abstract
Some new classes of general harmonic convex sets and convex functions are introduced and studied in this paper. The optimality criteria of the differentiable general harmonic functions is characterized by the general harmonic variational inequalities. Special cases are also pointed out as applications of the new concepts. Auxiliary principle technique involving an arbitrary operator is applied to suggest and analysis several inertial type methods are suggested. Convergence criteria is investigated of the proposed methods under weaker conditions. The results obtained in this paper may inspire further research along with implementable numerical methods for solving the general harmonic variational inequalities and related optimization problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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12. Symmetry of solutions to a class of geometric equations for hypersurfaces in [formula omitted].
- Author
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Chen, Shibing, Li, Qi-Rui, and Xu, Liang
- Subjects
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HYPERSURFACES , *SYMMETRY , *CONVEX functions , *EQUATIONS , *CONVEX bodies , *INAPPROPRIATE prescribing (Medicine) - Abstract
In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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13. Extension of Milne-type inequalities to Katugampola fractional integrals.
- Author
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Lakhdari, Abdelghani, Budak, Hüseyin, Awan, Muhammad Uzair, and Meftah, Badreddine
- Abstract
This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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14. Fractional Newton‐type integral inequalities by means of various function classes.
- Author
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Hezenci, Fatih and Budak, Hüseyin
- Subjects
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FRACTIONAL integrals , *CONVEX functions , *INTEGRAL functions , *INTEGRAL inequalities - Abstract
The authors of the paper present a method to examine some Newton‐type inequalities for various function classes using Riemann‐Liouville fractional integrals. Namely, some fractional Newton‐type inequalities are established by using convex functions. In addition, several fractional Newton‐type inequalities are proved by using bounded functions by fractional integrals. Moreover, we construct some fractional Newton‐type inequalities for Lipschitzian functions. Furthermore, several Newton‐type inequalities are acquired by fractional integrals of bounded variation. Finally, we provide our results by using special cases of obtained theorems and examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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15. Estimation of the neighborhood of metric regularity for quadratic functions.
- Author
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Xu, Wending
- Subjects
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MATHEMATICAL optimization , *CONVEX functions , *NEIGHBORHOODS - Abstract
Metric regularity is widely concerned since its important applications in optimization and control theory. For promoting the application of metric regularity, it is valuable to study the estimation of the neighborhood which makes the regularity hold. However, it seems that no result has been established about this issue. This paper investigates the estimation of the neighborhood of metric regularity for quadratic functions. The main result gives the expression of the neighborhood of metric regularity for a kind of convex quadratic functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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16. On Fenchel c-conjugate dual problems for DC optimization: characterizing weak, strong and stable strong duality.
- Author
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Fajardo, M. D. and Vidal, J.
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CONVEX functions - Abstract
In this paper we present two Fenchel-type dual problems for a DC (difference of convex functions) optimization primal one. They have been built by means of the c-conjugation scheme, a pattern of conjugation which has been shown to be suitable for evenly convex functions. We study characterizations of weak, strong and stable strong duality for both pairs of primal–dual problems. We also give conditions which relate the existence of strong and stable strong duality for both pairs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. A convex level-set method with multiplicative-additive model for image segmentation.
- Author
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Li, Zhixiang, Tang, Shaojie, Sun, Tianyu, Yang, Fuqiang, Ye, Wenguang, Ding, Wenyu, and Huang, Kuidong
- Subjects
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SMOOTHNESS of functions , *CONVEX functions , *KERNEL functions , *ENERGY function , *IMAGE segmentation - Abstract
• Double bias fields are introduced into fidelity term to approximate image intensity inhomogeneity. • The proposed energy function is strictly convex, and allows flexible initialization. • A TV (total variation) regularization term is introduced to keep convex level-set function smooth. • The proposed method is robust against to noise and intensity inhomogeneity. The existing active contour models (ACMs) based on bias field (BF) correction mostly rely on a single BF assumption and lack in-depth discussion on the convexity of the energy functional, often leading to the problem of local minima. To address this issue, this paper introduces a dual BF and proposes a convex level-set (LS) method based on multiplicative-additive (MA) model to achieve global minima. Firstly, a MA model is adopted as the fidelity term, and a kernel function is introduced to adjust the size of the intensity inhomogeneous neighborhood, enhancing the adaptability to intensity inhomogeneity. Then, the convex LS function is embedded in the variational framework to ensure convexity of each variable in the energy functional. This transformation turns the segmentation problem into a convex optimization problem. By introducing the total variation regularization term to smooth the LS function, the model's resistance to noise is effectively enhanced. Finally, by minimizing the proposed energy functional, image segmentation and BF correction are successfully achieved. Experimental results validate the global minima property of our model, while also demonstrating good flexibility in the initial contour. The proposed model achieves superior segmentation results compared to other classical ACMs on various types of images. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. A Bregman proximal subgradient algorithm for nonconvex and nonsmooth fractional optimization problems.
- Author
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Long, Xian Jun, Wang, Xiao Ting, Li, Gao Xi, and Li, Geng Hua
- Subjects
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NONSMOOTH optimization , *SMOOTHNESS of functions , *ALGORITHMS , *CONVEX functions - Abstract
In this paper, we study a class of nonconvex and nonsmooth fractional optimization problem, where the numerator of which is the sum of a nonsmooth and nonconvex function and a relative smooth nonconvex function, while the denominator is relative weakly convex nonsmooth function. We propose a Bregman proximal subgradient algorithm for solving this type of fractional optimization problems. Under moderate conditions, we prove that the subsequence generated by the proposed algorithm converges to a critical point, and the generated sequence globally converges to a critical point when the objective function satisfies the Kurdyka-Łojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, two numerical experiments illustrate the effectiveness and superiority of the algorithm. Our results give a positive answer to an open problem proposed by Bot et al. [14]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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19. A trust-region scheme for constrained multi-objective optimization problems with superlinear convergence property.
- Author
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Bisui, Nantu Kumar and Panda, Geetanjali
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CONVEX functions , *CONSTRAINED optimization , *ALGORITHMS - Abstract
In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable $ l_{\infty } $ l∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. L2-Maximal Functions on Graded Lie Groups.
- Author
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Cardona, Duván
- Subjects
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LIE groups , *CONVEX bodies , *CONVEX functions , *KERNEL functions , *FOURIER transforms , *MAXIMAL functions - Abstract
Bourgain in his seminal paper of 1986 about the analysis of maximal functions associated to convex bodies has estimated in a sharp way the |$L^{2}$| -operator norm of the maximal function associated to a kernel |$K\in L^{1},$| with differentiable Fourier transform |$\widehat{K}.$| We formulate the extension to Bourgain's |$L^{2}$| -estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the |$L^{p}$| -boundedness of maximal functions on graded Lie groups. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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21. On the complexity of a quadratic regularization algorithm for minimizing nonsmooth and nonconvex functions.
- Author
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Amaral, V. S., Lopes, J. O., Santos, P. S. M., and Silva, G. N.
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NONSMOOTH optimization , *HOLDER spaces , *ALGORITHMS , *CONVEX functions - Abstract
In this paper, we consider the problem of minimizing the function $ f(x)=g_1(x)+g_2(x)-h(x) $ f(x)=g1(x)+g2(x)−h(x) over $ \mathbb {R}^n $ Rn, where $ g_1 $ g1 is a proper and lower semicontinuous function, $ g_2 $ g2 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 $ g2 and a nonconvex regularization term. We show that the number of iterations required to reach our stopping criterion is $ \mathcal {O}(\max \{\epsilon ^{-\frac {\beta +1}{\beta }},\eta ^\frac {2}{\beta } \epsilon ^{-\frac {2(\beta +1)}{\beta }}\}) $ O(max{ϵ−β+1β,η2βϵ−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. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
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22. Certain Geometric Study Involving the Barnes–Mittag-Leffler Function.
- Author
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Alenazi, Abdulaziz and Mehrez, Khaled
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GAMMA functions , *STAR-like functions , *UNIVALENT functions , *CONVEX functions , *ANALYTIC functions - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
23. Some New Approaches to Fractional Euler–Maclaurin-Type Inequalities via Various Function Classes.
- Author
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Gümüş, Mehmet, Hezenci, Fatih, and Budak, Hüseyin
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FRACTIONAL integrals , *FRACTIONAL calculus , *CONVEX functions , *INTEGRAL functions - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
24. On the local dominance properties in single machine scheduling problems.
- Author
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Jorquera-Bravo, Natalia and Vásquez, Óscar C.
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SOCIAL dominance , *MATHEMATICAL inequalities , *SCHEDULING , *COMPUTATIONAL complexity , *CONVEX functions - Abstract
We consider a non-preemptive single machine scheduling problem for a non-negative penalty function f, where an optimal schedule satisfies the left-shifted property, i.e. in any optimal sequence all executions happen without idle time with a starting time t 0 ≥ 0 . For this problem, every job j has a priority weight w j and a processing time p j , and the goal is to find an order on the given jobs that minimizes ∑ w j f (C j) , where C j is the completion time of job j. This paper explores local dominance properties, which provide a powerful theoretical tool to better describe the structure of optimal solutions by identifying rules that at least one optimal solution must satisfy. We propose a novel approach, which allows to prove that the number of sequences that respect the local dominance property among three jobs is only two, not three, reducing the search space from n! to n ! / 3 ⌈ n / 3 ⌉ schedules. In addition, we define some non-trivial cases for the problem with a strictly convex penalty function that admits an optimal schedule, where the jobs are ordered in non-increasing weight. Finally, we provide some insights into three future research directions based on our results (i) to reduce the number of steps required by an exact exponential algorithm to solve the problem, (ii) to incorporate the dominance properties as valid inequalities in a mathematical formulation to speed up implicit enumeration methods, and (iii) to show the computational complexity of the problem of minimizing the sum of weighted mean squared deviation of the completion times with respect to a common due date for jobs with arbitrary weights, whose status is still open. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Algorithmic complexity of triple Roman dominating functions on graphs.
- Author
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Poureidi, Abolfazl and Fathali, Jafar
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DOMINATING set , *GRAPH theory , *BIPARTITE graphs , *APPROXIMATION algorithms , *CONVEX functions - Abstract
Given a graph G = (V, E), a function f: V → {0, 1, 2, 3, 4} is a triple Roman dominating function (TRDF) of G, for each vertex v ∈ V, (i) if f (v) = 0, then v must have either one neighbour in V4, or either two neighbours in V2 ∪ V3 (one neighbour in V3) or either three neighbours in V2, (ii) if f (v) = 1, then v must have either one neighbour in V3 ∪ V4 or either two neighbours in V2, and if f (v) = 2, then v must have one neighbour in V2 ∪ V3 ∪ V4. The triple Roman domination number of G is the minimum weight of an TRDF f of G, where the weight of f is v V f (v). The triple Roman domination problem is to compute the triple Roman domination number of a given graph. In this paper, we study the triple Roman domination problem. We show that the problem is NP-complete for the star convex bipartite and the comb convex bipartite graphs and is APX-complete for graphs of degree at most 4. We propose a linear-time algorithm for computing the triple Roman domination number of proper interval graphs. We also give an (2H (Δ(G) + 1) - 1)-approximation algorithm for solving the problem for any graph G, where Δ(G) is the maximum degree of G and H (d) denotes the first d terms of the harmonic series. In addition, we prove that for any ε > 0 there is no (1/4 - ε) ln |V |-approximation polynomial-time algorithm for solving the problem on bipartite and split graphs, unless NP ⊆ DTIME (|V |O(log log |V |)). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Optimal energy decay for a viscoelastic Kirchhoff equation with distributed delay acting on nonlinear frictional damping.
- Author
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Mohammed, Aili and Khemmoudj, Ammar
- Subjects
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WAVE equation , *CONVEX functions , *EQUATIONS , *DELAY differential equations - Abstract
In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. An adaptive fractional-order regularization primal-dual image denoising algorithm based on non-convex function.
- Author
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Li, Minmin, Bi, Shaojiu, and Cai, Guangcheng
- Subjects
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IMAGE denoising , *REGULARIZATION parameter , *ALGORITHMS , *DIFFUSION coefficients , *CONVEX functions , *MATHEMATICAL regularization , *QUASI-Newton methods - Abstract
In this paper, a novel non-convex fractional-order image denoising model is proposed to suppress the staircase effect produced by the TV model while maintaining a neat contour. The model combines ℓ q (0 < q < 1) quasi-norm and fractional-order regularization, and employs a diffusion coefficient with a faster convergence rate to preserve more image edges and details. Additionally, an adaptive regularization parameter is designed to adjust the denoising performance of the algorithm. To obtain the optimal approximate solution of the model, an enhanced primal-dual algorithm is adopted and the complexity and convergence of the algorithm are theoretically analyzed. Finally, the effectiveness of the proposed method is demonstrated through numerical experiments. • A new non-convex FOTV model is proposed. • A new diffusion coefficient is introduced to retain more edge details of the image. • The existing primal-dual algorithm is improved, and the convergence of this algorithm is analyzed. • An adaptive regularization parameter is designed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Smoothing algorithms for nonsmooth optimization over the Stiefel manifold with applications to the graph Fourier basis problem.
- Author
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Zhu, Jinlai, Huang, Jianfeng, Yang, Lihua, and Li, Qia
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SMOOTHING (Numerical analysis) , *NONSMOOTH optimization , *SMOOTHNESS of functions , *CONVEX functions , *CONTINUOUS functions , *LINEAR operators , *PROBLEM solving - Abstract
In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. On generalization of Hermite–Hadamard–Mercer inequalities for interval-valued functions with generalized geometric–arithmetic convexity.
- Author
-
Fahad, Asfand, Qian, Youhua, Ali, Zammad, and Younus, Awais
- Subjects
- *
CONVEXITY spaces , *MATHEMATICAL inequalities , *JENSEN'S inequality , *CONVEX functions , *EXPONENTIAL functions , *ARITHMETIC functions , *GENERALIZATION - Abstract
In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-h-convex functions (abbreviated as GA-Cr-h-CFs) and explore its properties. The family of GA-Cr-h-CFs simultaneously covers the family of GA-CFs, GA-h-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), h-CFs, Cr-CFs and Cr-h-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-h-CFs via two corresponding real-valued GA-h-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-h-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-h-CFs studied in the recent literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. The Multipoint-based Hermite-Hadamard Inequalities for Fractional Integrals with Exponential Kernels.
- Author
-
Zhengrong Yuan and Tingsong Du
- Subjects
- *
INTEGRAL inequalities , *FRACTIONAL integrals , *CONVEX functions , *ABSOLUTE value , *TRAPEZOIDS - Abstract
This paper concentrates on addressing fractional inequalities for exponential type convex functions. By means of exponential-type convexity, we firstly establish Hermite-Hadamard (HH) type inequalities for fractional integrals with exponential kernels. Secondly, based on the discovered fractional identity by separating [a; b] to n equal subintervals, and the fact that the twice derivative in absolute value is exponential type convex, we present multipoint-based HH inequalities, which cover the trapezoid- and Bullen-type inequalities for n = 1 and 2, correspondingly. During the period, some numerical examples with graphs are provided to show the validity of the deduced inequalities. [ABSTRACT FROM AUTHOR]
- Published
- 2024
31. Some Simpson- and Ostrowski-Type Integral Inequalities for Generalized Convex Functions in Multiplicative Calculus with Their Computational Analysis.
- Author
-
Zhan, Xinlin, Mateen, Abdul, Toseef, Muhammad, and Aamir Ali, Muhammad
- Subjects
- *
CONVEX functions , *CALCULUS , *GENERALIZED integrals , *INTEGRAL inequalities , *DIFFERENTIABLE functions , *NUMERICAL integration - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
32. Sufficiency Conditions for a Class of Convex Functions Connected with Tangent Functions Associated with the Combination of Babalola Operators and Binomial Series.
- Author
-
El-Deeb, Sheza M. and Cotîrlă, Luminita-Ioana
- Subjects
- *
TANGENT function , *CONVEX functions , *LOGARITHMIC functions - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
33. Generalized Euclidean operator radius.
- Author
-
Alomari, Mohammad W., Sababheh, Mohammad, Conde, Cristian, and Moradi, Hamid Reza
- Subjects
- *
HILBERT space , *CONVEX functions , *GENERALIZATION , *RADIUS (Geometry) - Abstract
In this paper, we introduce the f-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Generalized strongly n-polynomial convex functions and related inequalities.
- Author
-
Özcan, Serap, Kadakal, Mahir, İşcan, İmdat, and Kadakal, Huriye
- Subjects
- *
INTEGRAL inequalities , *CONVEX functions , *LITERATURE - Abstract
This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. On Minty variational principle for nonsmooth multiobjective optimization problems on Hadamard manifolds.
- Author
-
Bhooshan Upadhyay, Balendu, Treanţă, Savin, and Mishra, Priyanka
- Subjects
- *
VARIATIONAL principles , *NONSMOOTH optimization , *VARIATIONAL inequalities (Mathematics) , *CONVEX functions , *GEODESICS - Abstract
In this paper, we consider classes of approximate Minty and Stampacchia type vector variational inequalities using Clarke subdifferential on Hadamard manifolds and a class of nonsmooth multiobjective optimization problems. We investigate the relationship between the solution of these approximate vector variational inequalities and the solution of nonsmooth multiobjective optimization problems involving geodesic approximately convex functions. The results presented in this paper extend and generalize some existing results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Weighted Simpson-like type inequalities for quasi-convex functions.
- Author
-
Ayed, Hamida and Meftah, Badreddine
- Subjects
- *
HOLDER spaces , *CONVEX functions - Abstract
In this paper, by considering the identity established by Luo et al. in [C. Luo, T.-S. Du, M. Kunt and Y. Zhang, Certain new bounds considering the weighted Simpson-like type inequality and applications, J. Inequal. Appl. 2018 2018, Paper No. 332] and under the assumption of the quasi-convexity of the first derivative, we establish some new error estimates of the Simpson-like type inequalities. We also discuss the case where the first derivative satisfies the Hölder condition. At the end, we provide some applications to special means. The obtained results represent a continuation of the above-mentioned paper. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. Automatic setting of learning rate and mini-batch size in momentum and AdaM stochastic gradient methods.
- Author
-
Franchini, Giorgia and Porta, Federica
- Subjects
- *
RATE setting , *CONVEX functions , *MACHINE learning , *STOCHASTIC learning models - Abstract
The effectiveness of stochastic gradient methods strongly depends on a suitable selection of the hyperparameters which define them. Particularly, in the context of large-scale optimization problems often arising in machine learning applications, to properly fix both the learning rate and the mini-batch size in the definition of the stochastic directions is crucial for obtaining fast and efficient learning procedures. In a recent paper [1], the authors propose to define these hyperparameters by combining an adaptive subsampling strategy and a line search scheme. The aim of this work is to adapt this idea to both the stochastic gradient algorithm with momentum and the AdaM method in order to exploit the good numerical behaviour of the momentum-like stochastic gradient methods and the automatic technique to select the hyperparameters discussed in [1]. An extensive numerical experimentation carried out on convex functions, with different data sets, highlights that such combined hyperparameters technique makes the tuning of the hyperparameters computationally less expensive than the selection of suitable constant learning rate and mini-batch size and this is significant from the perspective of GreenAI. Furthermore, the proposed versions of the stochastic gradient method with momentum and AdaM have promising convergence behaviour compared to the original counterparts. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. On the Fading-Paper Achievable Region of the Fading MIMO Broadcast Channel.
- Author
-
Bennatan, Amir and Burshtein, David
- Subjects
- *
BROADCASTING industry , *MIMO systems , *WIRELESS communications , *TELECOMMUNICATION transmitters & transmission , *CONVEX functions , *GAUSSIAN distribution - Abstract
We consider transmission over the ergodic fading multiple-antenna broadcast (MIMO-BC) channel with partial channel state information at the transmitter and full information at the receiver. Over the equivalent non-fading channel, capacity has recently been shown to be achievable using transmission schemes that were designed for the "dirty paper" channel. We focus on a similar "fading paper" model. The evaluation of the fading paper capacity is difficult to obtain. We confine ourselves to the linear-assignment capacity, which we define, and use convex analysis methods to prove that its maximizing distribution is Gaussian. We compare our fading-paper transmission to an application of dirty paper coding that ignores the partial state information and assumes the channel is fixed at the average fade. We show that a gain is easily achieved by appropriately exploiting the information. We also consider a cooperative upper bound on the sum-rate capacity as suggested by Sato. We present a numeric example that indicates that our scheme is capable of realizing much of this upper bound. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
39. New version of midpoint-type inequalities for co-ordinated convex functions via generalized conformable integrals.
- Author
-
Kiriş, Mehmet Eyüp, Vivas-Cortez, Miguel, Uzun, Tuğba Yalçin, Bayrak, Gözde, and Budak, Hüseyin
- Subjects
- *
GENERALIZED integrals , *FRACTIONAL integrals , *RIEMANN integral , *INTEGRAL inequalities , *CONVEX functions - Abstract
In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. New error bounds for Newton's formula associated with tempered fractional integrals.
- Author
-
Hezenci, Fatih and Budak, Hüseyin
- Subjects
- *
INTEGRAL calculus , *CONVEX functions , *DIFFERENTIABLE functions , *FRACTIONAL integrals , *INTEGRAL inequalities , *GAUSSIAN quadrature formulas , *FRACTIONAL calculus - Abstract
In this paper, we first construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson's second formula, namely Newton–Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. A randomized sparse Kaczmarz solver for sparse signal recovery via minimax-concave penalty.
- Author
-
Yu-Qi Niu and Bing Zheng
- Subjects
- *
SIGNAL reconstruction , *CONVEX functions , *MATHEMATICAL regularization , *LINEAR systems - Abstract
The randomized sparse Kaczmarz (RSK) method is an algorithm used to calculate sparse solutions for the basis pursuit problem. In this paper, we propose an algorithm framework for computing sparse solutions of linear systems, which includes the sparse Kaczmarz and sparse block Kaczmarz algorithms. In order to overcome the limitations of the l1 penalty, we design an effective and new randomized sparse Kaczmarz algorithm (RSK-MCP) based on the non-convex minimax-concave penalty (MCP) in sparse signal reconstruction. Additionally, we prove that the RSK-MCP algorithm is equivalent to the randomized coordinate descent method for the corresponding dual problem. Based on this result, we demonstrate that the RSK-MCP algorithm exhibits linear convergence, meaning it converges to a sparse solution of the MCP model when the regularization of MCP is a strongly convex function. Numerical experiments indicate that the RSK-MCP algorithm outperforms RSK-L1 in terms of both efficiency and accuracy [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. One-Rank Linear Transformations and Fejer-Type Methods: An Overview.
- Author
-
Semenov, Volodymyr, Stetsyuk, Petro, Stovba, Viktor, and Velarde Cantú, José Manuel
- Subjects
- *
SUBGRADIENT methods , *CONVEX functions , *CONVEX programming - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
43. ON UPPER BOUNDS OF H2,1(f) AND H2,2(f) HANKEL DETERMINANTS FOR A SUBCLASS OF ANALYTIC FUNCTIONS.
- Author
-
KAMALI, MUHAMMET
- Subjects
- *
ANALYTIC functions , *HANKEL functions , *STAR-like functions , *UNIVALENT functions , *CONVEX functions - Abstract
In this paper, we give upper bounds of the Hankel determinants H2,1(f) and H2,2(f) for the classes S*(λ,n), where f is analytic in the open unit disk Δ = {z ∈ C: |z| < 1g and normalized so that f(z) = z+a2z²+a3z³+... [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Novel Estimations of Hadamard-Type Integral Inequalities for Raina's Fractional Operators.
- Author
-
Coşkun, Merve, Yildiz, Çetin, and Cotîrlă, Luminiţa-Ioana
- Subjects
- *
FRACTIONAL integrals , *INTEGRAL operators , *CONVEX functions , *JENSEN'S inequality , *INTEGRAL inequalities - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
45. Differential Stability Properties of Convex Optimization and Optimal Control Problems.
- Author
-
Toan, Nguyen Thi and Thuy, Le Quang
- Subjects
- *
BANACH spaces , *CONVEX functions , *EQUATIONS of state , *LINEAR equations - Abstract
This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the ε -subdifferential of the optimal value function are derived without qualification conditions. We first calculate the ε -subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the ε -subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Simpson Type Inequalities for Twice-differentiable Functions Arising from Tempered Fractional Integral Operators.
- Author
-
Jieyin Cai, Bin Wang, and Tingsong Du
- Subjects
- *
FRACTIONAL integrals , *INTEGRAL operators , *ABSOLUTE value , *CONVEX functions , *DIFFERENTIABLE functions - Abstract
Simpson inequalities for first-order differentiable convex functions and various fractional integrals have been studied extensively. However, Simpson type inequalities for twice-differentiable functions are researched slightly. Therefore, in the present paper, we endeavor to study fractional inequalities of Simpson type for twice-differentiable convex functions. To achieve this goal, we establish a new twicedifferentiable Simpson's identity by using tempered fractional integral operators. Based upon it, we prove several fractional Simpson type inequalities whose second derivatives in absolute value are convex. Finally, we give some examples to illustrate the correctness of the obtained results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
47. Some new properties of geometrically-convex functions.
- Author
-
Furuichi, Shigeru, Minculete, Nicuşor, Moradi, Hamid Reza, and Sababheh, Mohammad
- Subjects
- *
HYPERBOLIC functions , *EXPONENTIAL functions , *CONVEX functions , *INTEGRAL inequalities - Abstract
The class of geometrically convex functions is a rich class that contains some important functions. In this paper, we further explore this class and present many interesting new properties, including fundamental inequalities, supermultiplicative type inequalities, Jensen-Mercer inequality, integral inequalities, and refined forms. The obtained results extend some celebrated results from the context of convexity to geometric convexity, with interesting applications to numerical inequalities for the hyperbolic and exponential functions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Sharp anisotropic singular Trudinger–Moser inequalities in the entire space.
- Author
-
Guo, Kaiwen and Liu, Yanjun
- Subjects
- *
CONVEX functions - Abstract
In this paper, we investigate sharp singular Trudinger–Moser inequalities involving the anisotropic Dirichlet norm ∫ R N F N (∇ u) d x 1 N in Sobolev-type space D N , q (R N) , N ≥ 2 , q ≥ 1 . Here F : R N → [ 0 , + ∞) is a convex function of class C 2 (R N \ { 0 }) , which is even and positively homogeneous of degree 1. Combing with the connection between convex symmetrization and Schwarz symmetrization, we will establish anisotropic singular Trudinger–Moser inequalities and discuss their sharpness under different situations, including the case ‖ F (∇ u) ‖ N ≤ 1 , the case ‖ F (∇ u) ‖ N a + ‖ u ‖ q b ≤ 1 , and whether they are associated with exact growth. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations.
- Author
-
Sabir, Pishtiwan Othman
- Subjects
- *
CONVEX functions , *UNIVALENT functions , *SYMMETRIC functions , *ANALYTIC functions , *SCHWARZ function , *STAR-like functions - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
50. Stochastic Variance Reduction for DR-Submodular Maximization.
- Author
-
Lian, Yuefang, Du, Donglei, Wang, Xiao, Xu, Dachuan, and Zhou, Yang
- Subjects
- *
OPTIMIZATION algorithms , *SUBMODULAR functions , *STOCHASTIC approximation , *APPROXIMATION algorithms , *CONVEX functions - Abstract
Stochastic optimization has experienced significant growth in recent decades, with the increasing prevalence of variance reduction techniques in stochastic optimization algorithms to enhance computational efficiency. In this paper, we introduce two projection-free stochastic approximation algorithms for maximizing diminishing return (DR) submodular functions over convex constraints, building upon the Stochastic Path Integrated Differential EstimatoR (SPIDER) and its variants. Firstly, we present a SPIDER Continuous Greedy (SPIDER-CG) algorithm for the monotone case that guarantees a (1 - e - 1) OPT - ε approximation after O (ε - 1) iterations and O (ε - 2) stochastic gradient computations under the mean-squared smoothness assumption. For the non-monotone case, we develop a SPIDER Frank–Wolfe (SPIDER-FW) algorithm that guarantees a 1 4 (1 - min x ∈ C ‖ x ‖ ∞) OPT - ε approximation with O (ε - 1) iterations and O (ε - 2) stochastic gradient estimates. To address the practical challenge associated with a large number of samples per iteration, we introduce a modified gradient estimator based on SPIDER, leading to a Hybrid SPIDER-FW (Hybrid SPIDER-CG) algorithm, which achieves the same approximation guarantee as SPIDER-FW (SPIDER-CG) algorithm with only O (1) samples per iteration. Numerical experiments on both simulated and real data demonstrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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