A Generalization of Szász–Mirakyan Operators Based on α Non-Negative Parameter

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity.

On the Rate of Convergence of the Generalized Szász Type Operators for Functions of Bounded Variation

On the approximation of convex functions by Bernstein-type operators

Results concerning local boundedness of operators have a long history. In 1994, Vesel´y connected the concept of approximate monotonicity of an operator with local boundedness of that. It is our desire in this note to characterize an approximate monotone operator. Actually, we show that a well-known property of monotone operators, namely representing by convex functions, remains valid for the larger subclass of operators. In this general framework we establish the similar results by Fitzpatrick. Also, celebrated results of Mart´ inez-Legaz and Th´era inspired us to prove that the set of maximal e -monotone operators between a normed linear space X and its continuous dual X ∗ can be identified as some subset of convex functions on X × X ∗ .

We introduce Kantorovich variant of Stancu-Lupas? operators and study convergence and qstatistical convergence properties using Korovkin theorem. Rate of convergence is analyzed in terms of modulus of continuity, elements of Lipschitz class and Peetre?s K-functional. Direct theorems are proved and Voronovskaja type theorem is established. Graphical analysis of convergence and error estimations are presented with the help of MATLAB.

In the present article, we construct p , q -Szász-Mirakjan-Kantorovich-Stancu operators with three parameters λ , α , β . First, the moments and central moments are estimated. Then, local approximation properties of these operators are established via K -functionals and Steklov mean in means of modulus of continuity. Also, a Voronovskaja-type theorem is presented. Finally, the pointwise estimates, rate of convergence, and weighted approximation of these operators are studied.

In the present paper, we propose the q analogue of Sz a′sz-Beta-Stancu operators. By estimate the moments, we establish direct results in terms of the modulus of smoothness. Investigate the rate of point-wise convergence and weighted approximation properties of the q operators. Voronovskaja type theorem is also obtained.Our results generalize and supplement some convergence results of the q-Sz a′sz-Beta operators, thus they improve the existing results.

This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

In this paper we provide the asymptotic theory of the general of φ-divergences measures, which include the most common divergence measures : R´enyi and Tsallis families and the Kullback-Leibler measure. We are interested in divergence measures in the discrete case. One sided and two-sided statistical tests are derived as well as symmetrized estimators. Almost sure rates of convergence and asymptotic normality theorem are obtained in the general case, and next particularized for the R´enyi and Tsallis families and for the Kullback-Leibler measure as well. Our theoretical results are validated by simulations.

Relying on fixed point techniques, Mahey, Oualibouch and Tao introduced in a 1995 paper the scaled proximal decomposition on the graph of a maximal monotone operator (SPDG) algorithm and analysed its performance on inclusions for strongly monotone and Lipschitz continuous operators. The SPDG algorithm generalizes the Spingarn’s partial inverse method by allowing scaling factors, a key strategy to speed up the convergence of numerical algorithms. In this note, we show that the SPDG algorithm can alternatively be analysed by means of the original Spingarn’s partial inverse framework, tracing back to the 1983 Spingarn’s paper. We simply show that under the assumptions considered in by Mahey, Oualibouch and Tao, the Spingarn’s partial inverse of the underlying maximal monotone operator is strongly monotone, which allows one to employ recent results on the convergence and iteration-complexity of proximal point-type methods for strongly monotone operators. By doing this, we additionally obtain a potentially faster convergence for the SPDG algorithm and a more accurate upper bound on the number of iterations needed to achieve prescribed tolerances, specially on ill-conditioned problems.

In this article, we investigate the possibilities of accelerating the double smoothing (DS) technique when solving unconstrained nondifferentiable convex optimization problems. This approach relies on the regularization in two steps of the Fenchel dual problem associated with the problem to be solved into an optimization problem having a differentiable strongly convex objective function with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method. The aim of this article is to show how the properties of the functions in the objective of the primal problem influence the implementation of the DS approach and its rate of convergence. The theoretical results are applied to linear inverse problems by making use of different regularization functionals.

In this paper, we present a forward---backward splitting algorithm with additional inertial term for solving a strongly convex optimization problem of a certain type. The strongly convex objective function is assumed to be a sum of a non-smooth convex and a smooth convex function. This additional knowledge is used for deriving a worst-case convergence rate for the proposed algorithm. It is proved to be an optimal algorithm with linear rate of convergence. For certain problems this linear rate of convergence is better than the provably optimal worst-case rate of convergence for smooth strongly convex functions. We demonstrate the efficiency of the proposed algorithm in numerical experiments and examples from image processing.

Advances in optimization theory have been made systematically by the desire to solve more and more complicated geometric structures that are realised in contemporary applications. This is a rigorous investigation of monotone vector fields and proximal algorithms in the deep geometrical setting of generalized metric spaces (G-metric spaces). Our study fills a general deficiency in the literature by generalizing classical monotonicity principles and proximal point algorithms to support the complex three-point distance structure of G-metric spaces.In this way, by conductinga strict theoretical study, we prove the existence and uniqueness of solutions in the concept of monotone inclusion, are able to develop effective proximal algorithms with guaranteed convergence rates, and illustrate their successful application in different areas of practice. Theoretical contributions that we have made include: (1) the extension of monotonicity theory in all its forms to G-metric spaces with complete characterizations, (2) the construction of strongly convergent proximalpoint algorithms that are explicit in rate of convergence, and (3) its application to variational inequalities and multi-objective optimization problems in non-standard geometries, where the old metric structures are no longer applicable. Our findings create new opportunities to deal with optimization problems in complex networks, social systems, and the present-day machine learning paradigms. Keywords: G-metric spaces, monotone operators, proximal algorithms,

Ritt operators and convergence in the method of alternating projections

Subgradient methods (method A and method B) are considered for finding the minimum point of a convex function for the known optimal value of the function. Both methods use known Polyak’s step, which depends on scalar parameter m ≥ 1 and guarantees monotonous decrease of the distance to the minimum point. If m = 1 the methods A and B are applicable to arbitrary convex function. Parameter m > 1 permits to take into account special classes of convex functions – convex quadratic functions (m = 2), differentiable homogeneous of degree σ (m = σ), etc. For both methods theorems are proven about the convergence rate O 1/ √ k for arbitrary convex function and convergence with geometric progression rate for a convex function with acute minimum. Method A is a subgradient method with Polyak’s step in original space of variables. The parameter m = 1 corresponds to classical Polyak’s step for arbitrary convex function. Parameter m = 2 can be used for quadratic functions minimization, for which the value of Polyak’s step is doubled comparing to the classical Polyak’s step with m = 1. For onedimentional functions method A finds the minimum point in one step from the arbitrary start point, which corresponds to one iteration of the Newton’s method. Proof of theorems about convergence rate of the method A is based on convex function characteristics in original space of variables, among which a constant c1 plays a crucial role – it bounds a subgradient norm. Method B is a subgradient method with space transformation, where the Polyak’s step is calculated in space transformed by linear operator. The method is defined by nonsingular matrix B. The parameter m = 1 corresponds to classical Polyak’s step for arbitrary convex function in transformed space of variables. If m = 2, for one-dimentional quadratic function, the method B finds the minimum point in one step with both arbitrary matrix B and start point. Proof of theorems about convergence rate of the method B is based on function characteristics in transformed space of variables, which are analogous to the method A characteristics in original space.