Convex analysis and optimization

For variational problems of the form we propose a dual method which decouples the difficulties relative to the functionals f and g from the possible ill-conditioning effects of the linear operator A. The approach is based on the use of an Augmented Lagrangian functional and leads to an efficient and simply implementable algorithm. We study also the finite element approximation of such problems, compatible with the use of our algorithm. The method is finally applied to solve several problems of continuum mechanics. Many problems, in Physics, Mechanics, and Mathematical Economics, can be formulated as the following variational problem: (9") $f,{f(Av)+g(v)J. V and Y are twd Hilbert spaces, endowed with the norm topology; A is a continuous linear operator from V into Y; f and g are convex functions defined respectively on Y and V, and taking their values in (-z, +m]. This formulation, first used by Rockafellar[l] to extend Fenchel's duality theory, includes, in particular, the ordinary problems of convex programming. We call v * the solution of ( ), when it exists. Let A natural approach to solve (9) consists in searching directly for the minimum of 4(v) over V. If 4 is differentiable iterative techniques based on the use of its gradient are available to construct a sequence of points in V, converging to v *. But the speed of convergence is slow if the operator A is ill-conditioned [2]. This difficulty may disappear if we use the following device: in a first time, we introduce the additional variable y E Y, linked to the original variable v E V by the constraint Av -y = 0, and consider the constrained problem (yC) on V x Y, obviously equivalent to (9): (PC) Inf{f(y)+g(v)l(v,~)E Vx Y,Av-Y =O>. We then eliminate the artificial constraint, just introduced, by a dual approach to solve (PC).

The well-known Jensen inequality and Hermite-Hadamard inequality were extended using iterated integrals by Z. Retkes in 2008 and then by P. Kórus in 2019. In this paper, we consider analytical convex (concave) functions in order to obtain new refinements of Jensen's inequality. We apply the main result to the classical HM-GM-AM, AM-RMS, triangle inequalities and present an application to the geometric series. We also give Mercer type variants of Jensen's inequality.

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-norm minimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature. Further, the recovery error from noisy data is within a constant of three targets: 1) the minimax risk, 2) an 'oracle' error that would be available if the column space of the matrix were known, and 3) a more adaptive 'oracle' error which would be available with the knowledge of the column space corresponding to the part of the matrix that stands above the noise. Lastly, the error bounds regarding low-rank matrices are extended to provide an error bound when the matrix has full rank with decaying singular values. The analysis in this paper is based on the restricted isometry property (RIP) introduced in [6] for vectors, and in for matrices.

We explore the error of the weighted quadrature formulae and give the sufficient and necessary conditions for this type of quadrature formula to have Schur-convexity property. Some special cases of the weigted quadrature formulae are considered.

Matrix Direct Learning (MDL) is introduced as a novel paradigm for training artificial intelligence models by directly solving matrix equations rather than using iterative gradient-based optimization, representing a potential fundamental shift in AI training. This paper develops a rigorous mathematical foundation for MDL, presenting formal definitions and proofs, and derives algorithms that leverage linear algebra techniques (e.g., pseudoinverse solutions, SVD-based optimizations) to train models in closed-form. We compare MDL analytically and empirically with traditional backpropagation-based deep learning, highlighting faster convergence, theoretical guarantees, and competitive accuracy. Through case studies in finance, genomics, medical imaging, climate modeling and other domains, we demonstrate the potential of MDL to efficiently tackle domainspecific learning problems. We also discuss scalability of MDL on modern hardware (GPUs, TPUs, and specialized matrix processors) and provide a practical implementation guide using Python/NumPy and MATLAB. The results indicate that MDL offers a mathematically elegant and computationally powerful alternative to gradient descent, opening new avenues for AI training and data-driven modeling.

This study focuses on construction of a new explicit iterative scheme for approximation of zeros of nonlinear mappings in reflexive real Banach space with uniformly Gateaux differentiable norm. In the study, strong convergence of the proposed iterative scheme is proved under mild conditions on the iterative parameters. The scheme does not involve resolvent of the mappings under consideration. Furthermore, applications of results obtained to Dirichlet and Neumann problems are given. Our Theorems improve, extend and unify most of the results that had been proved for this class of mappings.

Concave functions on intervals of real numbers are used in the analysis of "cake eating problems". In this note we present the proof of the existence of both left-hand and right hand derivatives for concave functions defined on open intervals and some interesting consequences that follow from this result.

We develop a parallel theory to that concerning the concept of integral mean value of a function, by replacing the additive framework with a multiplicative one. Particularly, we prove results which are multiplicative analogues of the Jensen and Hermite-Hadamard inequalities.

This study focuses on MATLAB code programs of the entire stages of solving Stochastic Transportation Linear Programming Problems with Fuzzy Uncertainty Information on Probability Distribution Space (STLPPFI) with its algorithm outlines. A MATLAB code program of STLPPFI problem solver with algorithm outlines are proposed to solve STLPPFI model problems, and it utilizes many concepts as Alpha-Cut technique, Truth Degrees technique, Linear Fuzzy Membership Function (LFMF), Trapezoidal Fuzzy Number , Triangular Fuzzy Number , Linear Fuzzy Ranking Function (LFRF), Expectation Weighted Summation technique (EWS) and analyzing cases via second condition test of alpha-cut technique. The STLPPFI problem solver is utlized to convert STLPPFI into its corresponding equivalent Deterministic Transportation Linear Programming Problem (DTLPP) via defuzzifying from fuzziness on probability distribution space and derandomization randomness of problem formulation respectively. In addition, Dual-Simplex algorithm method with Vogel Approximation Algorithm Method (VAM) are used to obtain optimal solution from DTLPP. All MATLAB code programs with their proposed algorithm outlines are new except Dual-Simplex and VAM. The MATLAB code program of STLPPFI problem solver are more efficient along with a numerical example on electricity field illustrating practicability of this proposed MATLAB code program with its algorithm. Finally, the solution procedure illustrates the MATLAB code program of the proposed method is practical and applicable in the fields of energy and industry as it facilitates the method of transforming the energy at the lowest cost, least time running and is commercially applicable. Comparative comments are provided between Dual-Simplex and VAM in solution process.

is remembered today as one of the key architects of the Italian school of algebraic geometry in the first half of the twentieth century. (For a brief discussion of some of his most important contributions to algebraic geometry, see the appendix.) His eagerness to serve the Fascist regime after Mussolini became dictator in 1925 has cast a long shadow over his name. Even though Severi survived multiple investigations in Italy, rediscovered his Catholic faith, and emphatically denied repeated accusations that he was anti-Semitic, the stigma has persisted. Among the first, if not the first, to come to Severi's defense in the post-World War II era was Beniamino Segre, who had lost his own academic chair in Bologna in 1938 in the wake of the government's anti-Jewish legislation. Relieved, at Severi's behest, of his duties as an editor of Italy's oldest scientific journal in the wake of the regime's racial laws, Segre nevertheless insisted that rumors of Severi's "supposed anti-Semitism" were "flimsy" and based on "some misunderstanding"

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher and the authors accept no legal responsibility for any damage caused by improper use of the instructions and programs contained in this book and the CD. Although the software has been tested with extreme care, errors in the software cannot be excluded.

A geometric approach to the improvement of Blundon's inequalites given in [11] is presented. If φ = min{|A − B|,|B −C|,|C − A|}, then we proved the inequality − cos φ cos ION cos φ , where O is the circumcenter, I is the incenter, and N is the Nagel point of triangle ABC. As a direct consequence, we obtain a sharp version to Gerretsen's inequalities [7].

We discuss the existence of a strengthening of Hermite-Hadamard inequality in the case of log-convex functions. Unlike the classical case, which belongs to the …eld of linear functional analysis, this analogue involves nonlinear means such as the geometric mean and the logarithmic mean.

The aim of this paper is to show that Jensen's Inequality and an extension of Chebyshev's Inequality complement one another, so that they both can be formulated in a pairing form, including a second inequality, that provides an estimate for the classical one.

The article aims to answer whether Gottlob Frege's letter to Adolph Mayer, dated 8 July 1896, could help German mathematicians get acquainted with Giuseppe Peano's mathematical work, including his mathematical logic. It is the fi rst publication of this letter in English. At the beginning, I present the main characters of this story. Next, I refer to the letters concerning Peano and his mathematical results. Thus, I show the background of Frege's letter to Mayer. In the last part, I collect information about Peano's contacts with German mathematicians-where he was quoted and by whom, who was quoted by Peano, and in which period of his life. I conclude that Peano was known in Germany before Frege wrote to Mayer in 1896. However, the letter could have helped publish fi ve of Peano's articles in Germany, where Peano's mathematical logic was hardly known then. Undoubtedly, the letter promoted that knowledge.

Some new oscillation criteria are given for second-order nonlinear differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literatures. Some examples are considered to illustrate the main results.

Most of the inequalities that we encounter in mathematics are based on a monotonicity or convexity argument. The functions that are constructed during a proof are monotone or convex (concave) throughout their domains. However, there are functions that change their monotonicity and convexity on their domains. In this case, a chopping of the domain into intervals on which a function is monotone or convex (concave) is necessary. Two inequalities about pairs of Hölder conjugate numbers are presented. One follows very elegantly from Young inequality, and the other requires chopping the domain into three subintervals, and proving the inequality differently on each of them.

Jensen's inequality induces different forms of functionals which enables refinements for many classic inequalities ([5]). Several refinements of Jensen's inequalities were given in [4]. In this paper we refine Jensen's inequality by separating a discrete domain of it. At the end, we give some applications.

Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A: E → E∗ be lipschitz and strongly monotone such that A−1(0)≠∅. For an arbitrary ({x1}, {y1})∈E, we define the sequences {xn} and {yn} byyn = xn − θnJ−1(Axn), n≥1xn+1 = yn − λnJ−1(Ayn), n≥1where λn and θn are positive real number and J is the duality mapping of E. Letting (λn, θn)∈(0, 1), then xn and yn converges strongly to ρ∗, a unique solution of the equation Ax = 0. We also applied our algorithm in convex minimization and also proved the convergence of it in Lp, lp or Wm,p. At the end we proposed the algorithm of it in Lp(Ω) and its inverse Lq(Ω).

We consider the nonlinear dynamic system x Δ t a t g y t , y Δ t −f t, x σ t. We establish some necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions with special asymptotic properties for the system. We generalize the known results in the literature. Some examples are included to illustrate the results.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher and the authors accept no legal responsibility for any damage caused by improper use of the instructions and programs contained in this book and the CD. Although the software has been tested with extreme care, errors in the software cannot be excluded.

Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A : E → E∗
be lipschitz and strongly monotone such that A− 1(0) 6= ∅. For an arbitrary ({x1}, {y1}) ∈ E, we
define the sequences {xn} and {yn} by
( yxn +1= =xny−n −θnJλ−nJ1(−Ax1(Ayn), nn), n ≥ 1
where λn and θn are positive real number and J is the duality mapping of E. Letting (λn,θn) ∈ (0, 1)
, then xn and yn converges strongly to ρ∗, a unique solution of the equation Ax = 0. We also applied
our algorithm in convex minimization and also proved the convergence of it in L p,`p or Wm,p. At the
end we proposed the algorithm of it in L p(Ω) and its inverse L q(Ω).

Softcover reprint of the hardcover 1st edition 2000 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhiiuser Boston, clo Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

In this paper the asymptotic behavior for all nonoscillatory solutions of third order nonlinear neutral differential equations have been investigated, where some necessary and sufficient conditions are obtained to guarantee the convergence of these solutions to zero or tends to infinity as t → ∞. We introduced Lemma 2.1 and Lemma 2.2 which are a generalization of Lemma 1.5.

Some new oscillation criteria are given for second-order nonlinear differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literatures. Some examples are considered to illustrate the main results.

Some new oscillation criteria are given for second-order nonlinear differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literatures. Some examples are considered to illustrate the main results.

The first part of the paper proves the conjectures on an inequalities in the Schatten p-quasi-norm of matrices. The second part of the paper uses the inequalities for proving a sufficient condition when the Schatten p-quasi-norm minimization can be used for low rank matrix recovery. More precisely, when the restricted isometry constant δ 2s < 1, there exists a real number p 0 < 1 such that any solution of the p minimization is the minimal rank solution for any p ≤ p 0. In addition, in the noisy setting, the estimate of the difference X − X 0 is also given which is useful for applications.

Grassmann’s powerful but largely undefined approach to affine and projective geometries has roots in Möbius’ barycentric calculus [1]. In his appraisal of the former’s work, Peano showed explicitly the relation of barycentric coordinates with Cartesian coordinates [2]. In our Treatise ([3] p. 33) we went beyond Peano’s work by displaying the advantage of barycentric coordinates in dealing with the main theorems of projective geometry (Desargues, Pappus, etc.). By giving the equations of lines and planes with barycentric coordinates, we explain the geometric duality in a purely algebraic way ([3] p. 43, [4]). The generalization of the barycentric coordinates leads to projective frames and coordinates, which allows us to work with the whole projective geometry of an n-dimensional space without defining the projective space PRn as projection of an n+1 dimensional space. We will also display the advantages of expressing the equations of quadrics with projective coordinates of the three-...

The aim of this paper is to study G. Fano’s (1871-1952) personal collection of volumes and offprints, currently preserved in the Special Mathematical Library “G. Peano” of the University of Turin. This type of investigation allows to examine some aspects connected to Fano’s international relationships, as well as to give a glimpse of his personal trajectory. Furthermore, the analysis performed through the digital humanities software Palladio provides an interesting overview on the cultural roots of his research activity. Finally, following the works on C. Segre’s and A. Terracini’s library heritages, this case-study adds another piece to the reconstruction of the cultural heritage of the Italian School of algebraic geometry

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type:x″(t)+p(t)|x(τ(t))|α-1x(τ(t))=0,t≠θk;Δx&#39;(t)|t=θk+qk|x(τ(θk))|α-1x(τ(θk))=0;Δx(t)|t=θk=0,  (0

Detecting the validity of an assumed multivariate linear regression is an important part in a serious regression analysis before the fitted model is further applied in the practice. In this work an asymptotic method for testing the validity of the model based on the moving sums processes of the observation vectors is established. While the common test has been investigated for the usual regular lattice design of experiment, in this study the observations are sampled according to a design generated by a probability measure. In this paper we define a reasonable test statistic as an integral function of the moving sums processes of the observation vectors. Under mild condition we show the limiting distributions of the statistic under H 1 as well as under H 0. It is shown that the limit of the test statistics is expressed as a functional of the well-known multivariate Slepian field. The finite sample size behavior of the test is investigated by Monte Carlo simulation. The simulation shows a strong evidence that our approach gives good approximation. The application of the test procedure to a real data obtained form agriculture is also demonstrated.

In this note we give a recipe which describes upper and lower bounds for the Jensen functional under superquadraticity conditions. Some results involve the Chebychev functional. We give a more general definition of these functionals and establish the analogue results.

We study the phase synchronization problem with noisy measurements Y = z * z * H + σW ∈ C n×n , where z * is an n-dimensional complex unit-modulus vector and W is a complex-valued Gaussian random matrix. It is assumed that each entry Y jk is observed with probability p. We prove that an SDP relaxation of the MLE achieves the error bound (1 + o(1)) σ 2 2np under a normalized squared ℓ 2 loss. This result matches the minimax lower bound of the problem, and even the leading constant is sharp. The analysis of the SDP is based on an equivalent non-convex programming whose solution can be characterized as a fixed point of the generalized power iteration lifted to a higher dimensional space. This viewpoint unifies the proofs of the statistical optimality of three different methods: MLE, SDP, and generalized power method. The technique is also applied to the analysis of the SDP for Z 2 synchronization, and we achieve the minimax optimal error exp −(1 − o(1)) np 2σ 2 with a sharp constant in the exponent.

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argumentx‴(t)+q(t)x′(t)+r(t)f(x(φ(t)))=0. Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operatorℒx=x‴+q(t)x′is oscillatory.