3,788 results
Search Results
2. Alternative KKT conditions for (semi)infinite convex optimization.
- Author
-
Correa, Rafael, Hantoute, Abderrahim, and López, Marco A.
- Subjects
- *
SUBDIFFERENTIALS , *CONVEX programming , *CONVEX functions - Abstract
This paper is intended to provide an updated survey of recent optimality theory for infinite-dimensional convex programming. It aims at establishing theoretical support for algorithmic developments. Two alternative strategies inspire the approaches presented in the paper. The first one consists of replacing the family of constraints by a single one, appealing to the supremum function, and is based on various characterizations of the subdifferential of the pointwise supremum of convex functions. The second one uses appropriate characterizations of affine consequent inequalities of the constraint system exploiting ad hoc constraint qualifications. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Sharp conditions for the existence of infinitely many positive solutions to $ q $-$ k $-Hessian equation and systems.
- Author
-
Wan, Haitao and Shi, Yongxiu
- Subjects
- *
HESSIAN matrices , *EXISTENCE theorems , *CONVEX functions , *MATHEMATICAL bounds , *GENERALIZATION - Abstract
In this paper, only under the q - k -Keller–Osserman conditions, we consider the existence and global estimates of innumerable radial q - k -convex positive solutions to the q - k -Hessian equation and systems. Our conditions are strictly weaker than those in previous papers. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Constrained minimum variance and covariance steering based on affine disturbance feedback control parameterization.
- Author
-
Balci, Isin M. and Bakolas, Efstathios
- Subjects
- *
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
- View/download PDF
5. Insight into the gas–liquid transition from the Berthelot model.
- Author
-
Mi, Li-Qin, Li, Dandan, Li, Shanshan, and Li, Zhong-Heng
- Subjects
- *
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
- Full Text
- View/download PDF
6. On Approximate Variational Inequalities and Bilevel Programming Problems.
- Author
-
Upadhyay, Balendu Bhooshan, Stancu-Minasian, Ioan, Poddar, Subham, and Mishra, Priyanka
- Subjects
- *
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
- Full Text
- View/download PDF
7. Further Hermite–Hadamard-Type Inequalities for Fractional Integrals with Exponential Kernels.
- Author
-
Li, Hong, Meftah, Badreddine, Saleh, Wedad, Xu, Hongyan, Kiliçman, Adem, and Lakhdari, Abdelghani
- Subjects
- *
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
- Full Text
- View/download PDF
8. Panoptic Segmentation with Convex Object Representation.
- Author
-
Yao, Zhicheng, Wang, Sa, Zhu, Jinbin, and Bao, Yungang
- Subjects
- *
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
9. A subclass of analytic functions with negative coefficient defined by generalizing Srivastava-Attiya operator.
- Author
-
Hamaad, Suha J., Juma, Abdul Rahman S., and Ebrahim, Hassan H.
- Subjects
- *
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
- Full Text
- View/download PDF
10. A necessary and sufficient conditions for the global existence of solutions to fractional reaction-diffusion equations on RN.
- Author
-
Chung, Soon-Yeong and Hwang, Jaeho
- Subjects
- *
CONVEX functions , *CONTINUOUS functions , *EQUATIONS , *REACTION-diffusion equations - Abstract
A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations u t = Δ α u + ψ (t) f (u) , in R N × (0 , ∞) , u (· , 0) = u 0 ≥ 0 , in R N , has not been known and remained as an open problem for a few decades, where N ≥ 2 , Δ α = - - Δ α / 2 denotes the fractional Laplace operator with 0 < α ≤ 2 , ψ is a nonnegative and continuous function, and f is a convex function. The purpose of this paper is to resolve this problem completely as follows: There is a global solution to the equation if and only if ∫ 1 ∞ ψ (t) t N α f ϵ t - N α d t < ∞ , for some ϵ > 0. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
11. Feasibility problems via paramonotone operators in a convex setting.
- Author
-
Camacho, J., Cánovas, M.J., Martínez-Legaz, J.E., and Parra, J.
- Subjects
- *
CONVEX sets , *HILBERT space , *BANACH spaces , *CONVEX functions , *LINEAR systems - Abstract
This paper is focussed on some properties of paramonotone operators on Banach spaces and their application to certain feasibility problems for convex sets in a Hilbert space and convex systems in the Euclidean space. In particular, it shows that operators that are simultaneously paramonotone and bimonotone are constant on their domains, and this fact is applied to tackle two particular situations. The first one, closely related to simultaneous projections, deals with a finite amount of convex sets with an empty intersection and tackles the problem of finding the smallest perturbations (in the sense of translations) of these sets to reach a nonempty intersection. The second is focussed on the distance to feasibility; specifically, given an inconsistent convex inequality system, our goal is to compute/estimate the smallest right-hand side perturbations that reach feasibility. We advance that this work derives lower and upper estimates of such a distance, which become the exact value when confined to linear systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Existence and general decay rate estimates of a coupled Lamé system only with viscoelastic dampings.
- Author
-
Feng, Baowei, Hajjej, Zayd, and Balegh, Mohamed
- Subjects
- *
CONVEX functions , *MEMORY - Abstract
In this paper, we consider a coupled Lamé system only with viscoealstic dampings. By assuming a more general of relaxation functions and by using some properties of convex functions, we establish optimal explicit and general energy decay results to the system. This result improves previous results in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Generalized Jensen and Jensen–Mercer inequalities for strongly convex functions with applications.
- Author
-
Ivelić Bradanović, Slavica and Lovričević, Neda
- Subjects
- *
JENSEN'S inequality , *UNCERTAINTY (Information theory) , *CONVEX functions , *GENERALIZATION - Abstract
Strongly convex functions as a subclass of convex functions, still equipped with stronger properties, are employed through several generalizations and improvements of the Jensen inequality and the Jensen–Mercer inequality. This paper additionally provides applications of obtained main results in the form of new estimates for so-called strong f-divergences: the concept of the Csiszár f-divergence for strongly convex functions f, together with particular cases (Kullback–Leibler divergence, χ 2 -divergence, Hellinger divergence, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence.) Furthermore, new estimates for the Shannon entropy are obtained, and new Chebyshev-type inequalities are derived. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. Optimum Achievable Rates in Two Random Number Generation Problems with f -Divergences Using Smooth Rényi Entropy †.
- Author
-
Nomura, Ryo and Yagi, Hideki
- Subjects
- *
RENYI'S entropy , *INFORMATION theory , *CONVEX functions , *DISTRIBUTION (Probability theory) - Abstract
Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
15. On Quasiconvex Multiobjective Optimization and Variational Inequalities Using Greenberg–Pierskalla Based Generalized Subdifferentials.
- Author
-
Mishra, Shashi Kant, Laha, Vivek, and Hassan, Mohd
- Subjects
- *
NONSMOOTH optimization , *SUBDIFFERENTIALS , *VARIATIONAL principles , *CONVEX functions - Abstract
In this paper, we first characterize generalized convex functions introduced by Linh and Penot Optimization (62: 943–959, 2013) by using generalized monotonicity of the generalized subdifferentials. We use vector variational inequalities in terms of generalized subdifferentials to identify efficient solutions of a multiobjective optimization problem involving quasiconvex functions. We also establish the Minty variational principle by utilizing the mean value theorem established by Kabgani and Soleimani-damaneh (Numer. Funct. Anal. Optim 38: 1548–1563, 2017) for quasiconvex functions in terms of Greenberg–Pierskalla subdifferentials. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
16. On Global Error Bounds for Convex Inequalities Systems.
- Author
-
Long, Vo Si Trong
- Subjects
- *
CONVEX functions , *LINEAR systems , *SET functions , *CONTINUOUS functions , *SUBDIFFERENTIALS - Abstract
In this paper, we first present necessary and sufficient conditions for the existence of global error bounds for a convex function without additional conditions on the function or the solution set. In particular, we obtain characterizations of such global error bounds in Euclidean spaces, which are often simple to check. Second, we prove that under a suitable assumption the subdifferential of the supremum function of an arbitrary family of convex continuous functions coincides with the convex hull of the subdifferentials of functions corresponding to the active indices at given points. As applications, we study the existence of global error bounds for infinite systems of linear and convex inequalities. Several examples are provided as well to explain the advantages of our results with existing ones in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Globalized distributionally robust optimization problems under the moment-based framework.
- Author
-
Ding, Ke-wei, Huang, Nan-jing, and Wang, Lei
- Subjects
- *
CONVEX functions , *SPECIAL functions , *AMBIGUITY , *ROBUST optimization - Abstract
This paper is devoted to reduce the conservativeness of distributionally robust optimization with moments information. Since the optimal solution of distributionally robust optimization is required to be feasible for all uncertain distributions in a given ambiguity distribution set and so the conservativeness of the optimal solution is inevitable. To address this issue, we introduce the globalized distributionally robust counterpart (GDRC) which allows constraint violations controlled by functional distance of the true distribution to the inner distribution set. We obtain the deterministic equivalent forms for several GDRCs under the moment-based framework. To be specific, we show the deterministic equivalent systems of inequalities for GDRCs under second order moment information with a separable convex distance function and a special jointly convex function, respectively. We also obtain the deterministic equivalent inequality for GDRC under first order moment and support information. The computationally tractable examples are presented for these GDRCs. Numerical tests of a portfolio optimization problem are given to show the effectiveness of our method and the results demonstrate that the globalized distributionally robust solution is non-conservative and flexible compared to the distributionally robust solution. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. Optimal gradient tracking for decentralized optimization.
- Author
-
Song, Zhuoqing, Shi, Lei, Pu, Shi, and Yan, Ming
- Subjects
- *
UNDIRECTED graphs , *CONVEX functions , *ALGORITHMS , *DISTRIBUTED algorithms - Abstract
In this paper, we focus on solving the decentralized optimization problem of minimizing the sum of n objective functions over a multi-agent network. The agents are embedded in an undirected graph where they can only send/receive information directly to/from their immediate neighbors. Assuming smooth and strongly convex objective functions, we propose an Optimal Gradient Tracking (OGT) method that achieves the optimal gradient computation complexity O κ log 1 ϵ and the optimal communication complexity O κ θ log 1 ϵ simultaneously, where κ and 1 θ denote the condition numbers related to the objective functions and the communication graph, respectively. To our best knowledge, OGT is the first single-loop decentralized gradient-type method that is optimal in both gradient computation and communication complexities. The development of OGT involves two building blocks that are also of independent interest. The first one is another new decentralized gradient tracking method termed "Snapshot" Gradient Tracking (SS-GT), which achieves the gradient computation and communication complexities of O κ log 1 ϵ and O κ θ log 1 ϵ , respectively. SS - G T can be potentially extended to more general settings compared to OGT. The second one is a technique termed Loopless Chebyshev Acceleration (LCA), which can be implemented "looplessly" but achieves a similar effect by adding multiple inner loops of Chebyshev acceleration in the algorithm. In addition to SS - G T , this LCA technique can accelerate many other gradient tracking based methods with respect to the graph condition number 1 θ . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. A family of quadrature formulas with their error bounds for convex functions and applications.
- Author
-
Toseef, Muhammad, Mateen, Abdul, Aamir Ali, Muhammad, and Zhang, Zhiyue
- Subjects
- *
DEFINITE integrals , *CONVEX functions , *INTEGRAL inequalities , *NUMERICAL analysis - Abstract
In numerical analysis, the quadrature formulas serve as a pivotal tool for approximating definite integrals. In this paper, we introduce a family of quadrature formulas and analyze their associated error bounds for convex functions. The main advantage of these new error bounds is that from these error bounds, we can find the error bounds of different quadrature formulas. This work extends the traditional quadrature formulas such as the midpoint formula, trapezoidal formula, Simpson's formula, and Boole's formula. We also use the power mean and Hölder's integral inequalities to find more general and sharp bounds. Furthermore, we give numerical example and applications to quadrature formulas of the newly established inequalities. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. Symmetry of solutions to a class of geometric equations for hypersurfaces in [formula omitted].
- Author
-
Chen, Shibing, Li, Qi-Rui, and Xu, Liang
- Subjects
- *
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
- View/download PDF
21. Integral inequalities of h‐superquadratic functions and their fractional perspective with applications.
- Author
-
Butt, Saad Ihsan and Khan, Dawood
- Subjects
- *
INTEGRAL operators , *FRACTIONAL integrals , *PROBABILITY density function , *BESSEL functions , *OPERATOR functions - Abstract
The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of h$$ h $$‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of h$$ h $$‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. A Minimax-Program-Based Approach for Robust Fractional Multi-Objective Optimization.
- Author
-
Li, Henan, Hong, Zhe, and Kim, Do Sang
- Subjects
- *
FRACTIONAL programming , *CONVEX functions - Abstract
In this paper, by making use of some advanced tools from variational analysis and generalized differentiation, we establish necessary optimality conditions for a class of robust fractional minimax programming problems. Sufficient optimality conditions for the considered problem are also obtained by means of generalized convex functions. Additionally, we formulate a dual problem to the primal one and examine duality relations between them. In our results, by using the obtained results, we obtain necessary and sufficient optimality conditions for a class of robust fractional multi-objective optimization problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Extension of Milne-type inequalities to Katugampola fractional integrals.
- Author
-
Lakhdari, Abdelghani, Budak, Hüseyin, Awan, Muhammad Uzair, and Meftah, Badreddine
- Subjects
- *
FRACTIONAL integrals , *FRACTIONAL calculus , *INTEGRAL operators , *CONVEX functions , *APPLIED sciences - 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
- View/download PDF
24. Fractional Newton‐type integral inequalities by means of various function classes.
- Author
-
Hezenci, Fatih and Budak, Hüseyin
- Subjects
- *
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
- Full Text
- View/download PDF
25. New Improvements of the Jensen–Mercer Inequality for Strongly Convex Functions with Applications.
- Author
-
Adil Khan, Muhammad, Ivelić Bradanović, Slavica, and Mahmoud, Haitham Abbas
- Subjects
- *
CONVEX functions , *JENSEN'S inequality , *INFORMATION theory - Abstract
In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for χ -divergence, Kullback–Leibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence. Additionally, we prove some applications to Mercer-type power means at the end. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Some Classical Inequalities Associated with Generic Identity and Applications.
- Author
-
Javed, Muhammad Zakria, Awan, Muhammad Uzair, Bin-Mohsin, Bandar, Budak, Hüseyin, and Dragomir, Silvestru Sever
- Subjects
- *
CONVEX functions , *DIFFERENTIABLE functions , *SPECIAL functions , *INTEGRAL inequalities , *EXPLANATION - Abstract
In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski's inequality, trapezoidal inequality, midpoint inequality, Simpson's inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of γ and parameter ξ. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Exterior-Point Optimization for Sparse and Low-Rank Optimization.
- Author
-
Das Gupta, Shuvomoy, Stellato, Bartolomeo, and Van Parys, Bart P. G.
- Subjects
- *
CONVEX functions , *MACHINE learning , *PROBLEM solving , *DATA science , *ALGORITHMS - Abstract
Many problems of substantial current interest in machine learning, statistics, and data science can be formulated as sparse and low-rank optimization problems. In this paper, we present the nonconvex exterior-point optimization solver (NExOS)—a first-order algorithm tailored to sparse and low-rank optimization problems. We consider the problem of minimizing a convex function over a nonconvex constraint set, where the set can be decomposed as the intersection of a compact convex set and a nonconvex set involving sparse or low-rank constraints. Unlike the convex relaxation approaches, NExOS finds a locally optimal point of the original problem by solving a sequence of penalized problems with strictly decreasing penalty parameters by exploiting the nonconvex geometry. NExOS solves each penalized problem by applying a first-order algorithm, which converges linearly to a local minimum of the corresponding penalized formulation under regularity conditions. Furthermore, the local minima of the penalized problems converge to a local minimum of the original problem as the penalty parameter goes to zero. We then implement and test NExOS on many instances from a wide variety of sparse and low-rank optimization problems, empirically demonstrating that our algorithm outperforms specialized methods. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. A Bregman proximal subgradient algorithm for nonconvex and nonsmooth fractional optimization problems.
- Author
-
Long, Xian Jun, Wang, Xiao Ting, Li, Gao Xi, and Li, Geng Hua
- Subjects
- *
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
- View/download PDF
29. Estimation of the neighborhood of metric regularity for quadratic functions.
- Author
-
Xu, Wending
- Subjects
- *
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
- View/download PDF
30. On Fenchel c-conjugate dual problems for DC optimization: characterizing weak, strong and stable strong duality.
- Author
-
Fajardo, M. D. and Vidal, J.
- Subjects
- *
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
- View/download PDF
31. On optimality conditions and duality for multiobjective fractional optimization problem with vanishing constraints.
- Author
-
Wang, Haijun, Kang, Gege, and Zhang, Ruifang
- Subjects
- *
SUBDIFFERENTIALS , *GENERALIZATION , *LIPSCHITZ spaces , *DUALITY theory (Mathematics) , *CONVEX functions - Abstract
The aim of this paper is to investigate the optimality conditions for a class of nonsmooth multiobjective fractional optimization problems subject to vanishing constraints. In particular, necessary and sufficient conditions for (weak) Pareto solution are presented in terms of the Clark subdifferential. Furthermore, we construct Wolfe and Mond–Weir-type dual models and derive some duality theorems by using generalized quasiconvexity assumptions. Some examples to show the validity of our conclusions are provided. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. On New Generalized Hermite–Hadamard–Mercer-Type Inequalities for Raina Functions.
- Author
-
Çiftci, Zeynep, Coşkun, Merve, Yildiz, Çetin, Cotîrlă, Luminiţa-Ioana, and Breaz, Daniel
- Subjects
- *
FRACTIONAL integrals , *INTEGRAL operators , *CONVEX functions , *DEFINITIONS - Abstract
In this research, we demonstrate novel Hermite–Hadamard–Mercer fractional integral inequalities using a wide class of fractional integral operators (the Raina fractional operator). Moreover, a new lemma of this type is proved, and new identities are obtained using the definition of convex function. In addition to a detailed derivation of a few special situations, certain known findings are summarized. We also point out that some results in this study, in some special cases, such as setting α = 0 = φ , γ = 1 , and w = 0 , σ (0) = 1 , λ = 1 , are more reasonable than those obtained. Finally, it is believed that the technique presented in this paper will encourage additional study in this field. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Subclasses of Bi-Univalent Functions Connected with Caputo-Type Fractional Derivatives Based upon Lucas Polynomial.
- Author
-
Alsager, Kholood M., Murugusundaramoorthy, Gangadharan, Breaz, Daniel, and El-Deeb, Sheza M.
- Subjects
- *
STAR-like functions , *ANALYTIC functions , *CONVEX functions , *POLYNOMIALS , *UNIVALENT functions - Abstract
In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients a 2 and a 3 for functions in these subclasses. Using the values of a 2 and a 3 , we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. A convex level-set method with multiplicative-additive model for image segmentation.
- Author
-
Li, Zhixiang, Tang, Shaojie, Sun, Tianyu, Yang, Fuqiang, Ye, Wenguang, Ding, Wenyu, and Huang, Kuidong
- Subjects
- *
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
35. On strongly generalized convex stochastic processes.
- Author
-
Sharma, Nidhi, Mishra, Rohan, and Hamdi, Abdelouahed
- Subjects
- *
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
- Full Text
- View/download PDF
36. Analytical and geometrical approach to the generalized Bessel function.
- Author
-
Bulboacă, Teodor and Zayed, Hanaa M.
- Subjects
- *
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
- Full Text
- View/download PDF
37. A linearized approach for solving differentiable vector optimization problems with vanishing constraints.
- Author
-
Antczak, Tadeusz
- Subjects
- *
CONVEX functions , *MULTI-objective optimization - 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
- View/download PDF
38. NEW ITERATIVE SCHEMES FOR GENERAL HARMONIC VARIATIONAL INEQUALITIES.
- Author
-
NOOR, MUHAMMAD ASLAM and NOOR, KHALIDA INAYAT
- Subjects
- *
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
- View/download PDF
39. 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
40. 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
41. 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
42. A trust-region scheme for constrained multi-objective optimization problems with superlinear convergence property.
- Author
-
Bisui, Nantu Kumar and Panda, Geetanjali
- Subjects
- *
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
43. L2-Maximal Functions on Graded Lie Groups.
- Author
-
Cardona, Duván
- Subjects
- *
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
- View/download PDF
44. On the complexity of a quadratic regularization algorithm for minimizing nonsmooth and nonconvex functions.
- Author
-
Amaral, V. S., Lopes, J. O., Santos, P. S. M., and Silva, G. N.
- Subjects
- *
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
- View/download PDF
45. Certain Geometric Study Involving the Barnes–Mittag-Leffler Function.
- Author
-
Alenazi, Abdulaziz and Mehrez, Khaled
- Subjects
- *
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
46. Some New Approaches to Fractional Euler–Maclaurin-Type Inequalities via Various Function Classes.
- Author
-
Gümüş, Mehmet, Hezenci, Fatih, and Budak, Hüseyin
- Subjects
- *
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
47. Smoothing algorithms for nonsmooth optimization over the Stiefel manifold with applications to the graph Fourier basis problem.
- Author
-
Zhu, Jinlai, Huang, Jianfeng, Yang, Lihua, and Li, Qia
- Subjects
- *
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
48. An adaptive fractional-order regularization primal-dual image denoising algorithm based on non-convex function.
- Author
-
Li, Minmin, Bi, Shaojiu, and Cai, Guangcheng
- Subjects
- *
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
49. On the local dominance properties in single machine scheduling problems.
- Author
-
Jorquera-Bravo, Natalia and Vásquez, Óscar C.
- Subjects
- *
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
50. Algorithmic complexity of triple Roman dominating functions on graphs.
- Author
-
Poureidi, Abolfazl and Fathali, Jafar
- Subjects
- *
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
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.