Search

This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases. [ABSTRACT FROM AUTHOR]

In this paper, using the results of a recent paper by the author, we give a new method for proving refinements of inequalities related to convex functions on intervals. In many cases, the proof is simpler and more transparent than using the usual techniques, and the essence of the refinement is clearer. This is illustrated by two refinements of the Jensen’s inequality and one refinement of the Lah-Ribarič inequality. As an application we generalize a recent result for strongly convex functions. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f || 1/6 [f(A) x 1 + 4f - A - 1 + 1 B 2 - + 1 f(B) - - Z 1 0 f((1 - k)A < 1 + k1 B)dk || 5\36 1 B-A 1 f'I,+8. [ABSTRACT FROM AUTHOR]

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]

In this paper we show that inequalities for functions convex on the coordinates can be derived from inequalities for convex functions defined on real intervals, and essentially only this method works. As applications, we show how our result works for the Jensen's and Hermite- Hadamard inequalities for functions convex on the coordinates. Finally, we extend the classical notion of f -divergence functional to functions convex on the coordinates, and as a further application of our main result, we study the refinement of a basic inequality corresponding to the new divergence. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert space have been obtained. The recent progression of the Hilbert space inequalities following the definition of the convex operator inequality has lead researchers to explore the concept of Hilbert space inequalities even further. The motivation for this paper stems from the recent development in the theory of tensorial and Hilbert space inequalities. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f... 1 6 A01 + 10 B 2) -exp(A) 0 1 + 4 exp +10exp(B) 1-4 + ∣ exp((--~-a 0 1 1 C + k} 1 0 b) k-1 dk ∕ exp-1 -- A 0 1 + k 1 0 (1 -- k)- 2dk-|| 47 ≤ "1 0 B -- A 0 1∣∣2 (Hexp(A)H + l∣exp(B)∣∣)∙ 360 Tensorial version of a Lemma given by Hezenci is derived and utilized to obtain the desired inequalities. in the introduction section is given a brief history of the inequalities, while in the preliminary section we give necessary Lemmas and results in order to understand the paper. Structure and novelty of the paper are discussed at the end of the introduction section. [ABSTRACT FROM AUTHOR]

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]

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]

In this paper, by using Poisson distribution probability, some characteristics of analytic multivalent q -symmetric starlike and q -symmetric convex functions of order η are examined. Then, by utilizing the Poisson distribution and the concept of the q -analogue Salagean integral operator, the p -valent convergence polynomial was introduced. Furthermore, a number of subclasses of analytic symmetric p -valent functions linked to novel polynomials are also deduced. After that, specific coefficient constraints are determined and symmetric δ , q -neighborhoods for p -valent functions are defined. In relation to symmetric δ , q -neighborhoods of q -symmetric p -valent functions formed by Poisson distributions, this paper presents new inclusion results. In addition, a detailed discussion of certain q -symmetric inequalities of analytic functions with negative coefficients is also provided. [ABSTRACT FROM AUTHOR]

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]

In this paper, the asynchronous control problem is investigated and a multiple convex Lyapunov functions (MCLF) approach is introduced for a class of discrete-time switched linear systems under the Φ-dependent integrated dwell time (ΦDIDT) switching strategy. For the problem of asynchronous switching, this paper considers that Lyapunov functions may jump when the subsystem switches or the controller changes. Thus, the constructed MCLF is dependent on both the asynchronous interval and the synchronous interval, and the synchronous interval is divided into the convex interval and non-convex interval parts. Some sufficient conditions of stability with Linear matrix inequality (LMI) forms are obtained, and the asynchronous controller is designed to guarantee the globally uniform exponential stability of the system under study. In addition, the proposed method can degenerate to the existing methods to deal with the asynchronous control problem. Finally, a numerical example illustrates the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we prove an identity for differentiable convex functions related to conformable fractional integrals. Moreover, some parameterized inequalities are established by using conformable fractional integrals. To be more precise, parameterized inequalities are obtained by taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Furthermore, previous and new results are presented by using special cases of the obtained theorems. [ABSTRACT FROM AUTHOR]

A previously introduced operator defined by applying the Riemann-Liouville fractional integral to the convex combination of well-known Ruscheweyh and Salagean differential operators is used for defining a new fuzzy subclass. The authors suggest that the operator introduced here can be utilized to define other classes of analytic functions or to generalize other types of differential operators. The new operator defined in this paper can be used to introduce other specific subclasses of analytic functions, and quantum calculus can be also investigated in future studies. The fractional differential operator and the Mittag-Leffler functions are combined to formulate and arrange a new operator of fractional calculus. [Extracted from the article]

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]

DC (difference of convex functions) programming and DC algorithm (DCA) are powerful tools for nonsmooth nonconvex optimization. This field was created in 1985 by Pham Dinh Tao in its preliminary state, then the intensive research of the authors of this paper has led to decisive developments since 1993, and has now become classic and increasingly popular worldwide. For 35 years from their birthday, these theoretical and algorithmic tools have been greatly enriched, thanks to a lot of their applications, by researchers and practitioners in the world, to model and solve nonconvex programs from many fields of applied sciences. This paper is devoted to key open issues, recent advances and trends in the development of these tools to meet the growing need for nonconvex programming and global optimization. We first give an outline in foundations of DC programming and DCA which permits us to highlight the philosophy of these tools, discuss key issues, formulate open problems, and bring relevant answers. After outlining key open issues that require deeper and more appropriate investigations, we will present recent advances and ongoing works in these issues. They turn around novel solution techniques in order to improve DCA's efficiency and scalability, a new generation of algorithms beyond the standard framework of DC programming and DCA for large-dimensional DC programs and DC learning with Big data, as well as for broader classes of nonconvex problems beyond DC programs. [ABSTRACT FROM AUTHOR]

It is well known that the recursive least squares (RLS) algorithm is renowned for its rapid convergence and excellent tracking capability. However, its performance is significantly compromised when the system is sparse or when the input signals are contaminated by impulse noise. Therefore, in this paper, the minimum error entropy (MEE) criterion is introduced into the cost function of the RLS algorithm in this paper, with the aim of counteracting the interference from impulse noise. To address the sparse characteristics of the system, we employ a universally applicable convex function to regularize the cost function. The resulting new algorithm is named the convex regularization recursive minimum error entropy (CR-RMEE) algorithm. Simulation results indicate that the performance of the CR-RMEE algorithm surpasses that of other similar algorithms, and the new algorithm excels not only in scenarios with sparse systems but also demonstrates strong robustness against pulse noise. [ABSTRACT FROM AUTHOR]

For the adjacency matrix A(G) and diagonal matrix of the vertex degrees D(G) of a simple graph G, the A(G) matrix is the convex combinations of D(G) and A(G), and is defined as A(G) = D(G)+(1)A(G), for 0 n be the eigenvalues of A(G) (which we call -adjacency eigenvalues of the graph G). The generalized adjacency energy also called -adjacency energy of the graph G is defined as EA (G) = is the average vertex degree, m is the size and n is the order of G. The -adjacency energy of a graph G merges the theory of energy (adjacency energy) and the signless Laplacian energy, as EA0 (G) = E (G) and 2E A 12 (G) = QE(G), where E (G) is the energy and QE(G) is the signless Laplacian energy of G. In this paper, we obtain some new upper and lower bounds for the generalized adjacency energy of a graph, in terms of different graph parameters like the vertex covering number, the Zagreb index, the number of edges, the number of vertices, etc. We characterize the extremal graphs attained these bounds. [ABSTRACT FROM AUTHOR]

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]

In this paper we obtain several operator inequalities providing upper bounds for the Davis-Choi-Jensen's Difference Φ(f (A)) -- f (Φ (A)) for any convex function f : I → R, any selfadjoint operator A in H with the spectrum Sp (A) ⊂ I and any linear, positive and normalized map Φ : B (H) → B (K), where H and K are Hilbert spaces. Some examples of convex and operator convex functions are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we estimate sharp bounds for certain Hankel determinants, H2,3(f), H3,1(f) and Zalcman functional |a²3 -- a5| for the class of convex function with respect to symmetric points, hence proving the recent conjecture made by Virendra et al., that affirms the sharp bound for the third Hankel determinant in the classes of convex functions with respect to symmetric points is 4/135. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities of the form.. are obtained with variations due to the convexity properties of the mapping f. [ABSTRACT FROM AUTHOR]

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type. [ABSTRACT FROM AUTHOR]

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]

In this paper, we propose an adaptive algorithm with multi-step inertia for solving a convex bilevel optimization problem. Under suitable parameter conditions, we prove that our algorithm converges strongly to some solution of the problem, which is the unique solution to some variational inequality problem. The effectiveness of the proposed algorithm is verified by numerical experiments and compared with other algorithms. [ABSTRACT FROM AUTHOR]

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]

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]

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]

To solve a distributed optimal resource allocation problem, a collective neurodynamic approach based on recurrent neural networks (RNNs) is proposed in this paper. Multiple RNNs cooperatively solve a global constrained optimization problem in which the objective function is a total of local non-smooth convex functions and is subject to local convex sets and a global equality constraint. Different from the projection dynamics to deal with local convex sets in the existing work, an internal dynamics with projection output is designed in the algorithm to relax the Slater's condition satisfied by the optimal solution. To overcome continuous-time communication in a group of RNNs, an aperiodic communication scheme, called the event-triggered scheme, is presented to alleviate communication burden. It is analyzed that the convergence of the designed collective neurodynamic approach based on the event-triggered communication does not rely on global information. Furthermore, it is proved the freeness of the Zeno behavior in the event-triggered scheme. Two examples are presented to illustrate the obtained results [ABSTRACT FROM AUTHOR]

In the present paper, we consider a subclass of starlike functions introduced by Silverman. It is defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the sharp bounds of coefficient problems for the inverse of functions in this class. We derive the upper bounds of some initial coefficients, the Fekete–Szegö type inequality and the second Hankel determinant for . On the third Hankel determinant , we give a bound on the inverse of . All the results are proved to be sharp. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm. [ABSTRACT FROM AUTHOR]

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]

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]

In this paper, we investigate the distributed approximate subgradient-type method for minimizing a sum of differentiable and non-differentiable convex functions subject to nondifferentiable convex functional constraints in a Euclidean space. We establish the convergence of the sequence generated by our method to an optimal solution of the problem under consideration. Moreover, we derive a convergence rate of order O(N1−a) for the objective function values, where a ∈ (0.5, 1). Finally, we provide a numerical example illustrating the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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]

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]

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]

In this paper, we derive some new results for the separation of two not necessarily convex cones by a (convex) cone / conical surface in real (reflexive) normed spaces. In essence, we follow the nonlinear and nonsymmetric separation approach developed by Kasimbeyli (2010, SIAM J. Optim. 20), which is based on augmented dual cones and Bishop-Phelps type (normlinear) separating functions. Compared to Kasimbeyli's separation theorem, we formulate our theorems for the separation of two cones under weaker conditions (concerning convexity and closedness requirements) with respect to the involved cones. By a new characterization of the algebraic interior of augmented dual cones in real normed spaces, we are able to establish relationships between our cone separation results and the results derived by Kasimbeyli (2010, SIAM J. Optim. 20) and by García-Castaño, Melguizo-Padial and Parzanese (2023, Math. Meth. Oper. Res. 97). [ABSTRACT FROM AUTHOR]

In this paper, we consider an inexact projected gradient method equipped with a nonmonotone line search rule for smooth constrained multiobjective optimization. In this method, a new nonmonotone line search technique proposed here is employed and the relative errors on the search direction is admitted. We demonstrate that this method is well-defined. Then, we prove that each accumulation point of the sequence generated by this method is Pareto stationary and analyze the convergence rate of the algorithm. When the objective function is convex, the convergence of the sequence to a weak Pareto optimal point of the problem is established. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of quasi-point-separable topological vector spaces, which has the following important properties: (1) in general, the conditions for a topological vector space to be quasi-point-separable is not difficult to verify; (2) the class of quasi-point-separable topological vector spaces is large and includes locally convex topological vector spaces and pseudonorm adjoint topological vector spaces as special cases; (3) every quasi-point-separable Housdorrf topological vector space has the fixed point property (that is, every continuous self-mapping on any given nonempty closed and convex subset has a fixed point), which is the result of the main theorem of this paper. Finally, we provide some concrete examples of quasi-point-separable topological vector spaces, which are not locally convex. [ABSTRACT FROM AUTHOR]

Extreme learning machine (ELM) is a machine learning technique with simple structure, fast learning speed, and excellent generalization ability, which has received a lot of attention since it was proposed. In order to further improve the sparsity of output weights and the robustness of the model, this paper proposes a sparse and robust ELM based on zero-norm regularization and a non-convex quadratic loss function. The zero-norm regularization obtains sparse hidden nodes automatically, and the introduced non-convex quadratic loss function enhances the robustness by setting constant penalties to outliers. The optimization problem can be formulated as the difference of convex functions (DC) programming. This DC programming is solved by using the DC algorithm (DCA) in this paper. The experiments on the artificial and Benchmark datasets verify that the proposed method has promising robustness while reducing the number of hidden nodes, especially on the datasets with higher outliers level. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of approximate convexity to setvalued maps and obtain some relations between approximate convexity and approximate monotonicity of their normal subdifferential. [ABSTRACT FROM AUTHOR]