oscillation criteria for a class of second order emden fowler delay dynamic equations on time scales abstract by means of riccati transformation technique we establish some new oscillation criteria for the second order emden fowler delay dynamic equations x t p t x t 0 on a time scale t here is a quotient of odd positive integers with p t real valued positive rd continuous functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales our results in this paper not only extend the results given in r p agarwal m bohner s h saker oscillation of second order delay dynamic equations can appl math q 13 1 2005 1 18 but also unify the oscillation of the second order emden fowler delay differential equation and the second order emden fowler delay difference equation

iteration complexity of block decomposition algorithms and the alternating direction method of multipliers in this paper we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point to set maximal monotone operator with a separable two block structure and introduce a framework of block decomposition prox type algorithms for solving it which allows for each one of the single block proximal subproblems to be solved in an approximate sense moreover by showing that any method in this framework is also a special instance of the hybrid proximal extragradient hpe method introduced by solodov and svaiter we derive corresponding convergence rate results we also describe some instances of the framework based on specific and inexpensive schemes for solving the single block proximal subproblems finally we consider some applications of our methodology to establish for the first time i the iteration complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and as a consequence the ergodic iteration complexity of the douglas

majorization relations for hadamard products abstract we settle affirmatively the conjecture of johnson and bapat on the hadamard product a b of positive definite matrices a b i k n i a b ab k 1 2 n

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

prox method with rate of convergence o 1 t for variational inequalities with lipschitz continuous monotone operators and smooth convex concave saddle point problems we propose a prox type method with efficiency estimate o epsilon 1 for approximating saddle points of convex concave c 1 1 functions and solutions of variational inequalities with monotone lipschitz continuous operators application examples include matrix games eigenvalue minimization and computing the lovasz capacity number of a graph and these are illustrated by numerical experiments with large scale matrix games and lovasz capacity problems

complexity of variants of tseng s modified f b splitting and korpelevich s methods for hemivariational inequalities with applications to saddle point and convex optimization problems in this paper we consider both a variant of tseng s modified forward backward splitting method and an extension of korpelevich s method for solving hemivariational inequalities with lipschitz continuous operators by showing that these methods are special cases of the hybrid proximal extragradient method introduced by solodov and svaiter we derive iteration complexity bounds for them to obtain different types of approximate solutions in the context of saddle point problems we also derive complexity bounds for these methods to obtain another type of an approximate solution namely that of an approximate saddle point finally we illustrate the usefulness of the above results by applying them to a large class of linearly constrained convex programming problems including for example cone programming and problems whose objective functions converge to infinity as the boundaries of their effective domains are approached

epsilon proximal decomposition method we propose a modification of the proximal decomposition method investigated by spingarn 30 and mahey et al 19 for minimizing a convex function on a subspace for the method to be favorable from a computational point of view particular importance is the introduction of approximations in the proximal step first we couple decomposition on the graph of the epsilon subdifferential mapping and cutting plane approximations to get an algorithmic pattern that falls in the general framework of rockafellar inexact proximal point algorithms 26 recently solodov and svaiter 27 proposed a new proximal point like algorithm that uses improved error criteria and an enlargement of the maximal monotone operator defining the problem we combine their idea with bundle mecanism to devise an inexact proximal decomposition method with error condition which is not hard to satisfy in practice then we present some applications favorable to our development first we give a new regularized version of benders decomposition method in convex programming called the proximal convex benders decomposition algorithm second we derive a new algorithm for nonlinear multicommodity flow problems among which the message routing problem in telecommunications data networks

on an eigenvalue inequality involving the hadamard product abstract let a b be n n positive definite matrices then for 0 t 1 i k n i a b i k n i a t b a 1 t b i k n i a b k 1 n this gives a weighted extension of a result of ando 1 the case where the eigenvalues i are replaced with the singular values i is also considered

improved pointwise iteration complexity of a regularized admm and of a regularized non euclidean hpe framework this paper describes a regularized variant of the alternating direction method of multipliers admm for solving linearly constrained convex programs it is shown that the pointwise iteration complexity of the new variant is better than the corresponding one for the standard admm method and that up to a logarithmic term is identical to the ergodic iteration complexity of the latter method our analysis is based on first presenting and establishing the pointwise iteration complexity of a regularized non euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error it is then shown that the new admm variant is a special instance of the latter framework where the sequence of summable errors is identically zero when the admm stepsize is less than one or a nontrivial sequence when the stepsize is in the interval 1 1 sqrt 5 2

primal dual first order methods with mathcal o 1 epsilon iteration complexity for cone programming in this paper we consider the general cone programming problem and propose primal dual convex smooth and or nonsmooth minimization reformulations for it we then discuss first order methods suitable for solving these reformulations namely nesterov s optimal method nesterov in doklady an sssr 269 543 547 1983 math program 103 127 152 2005 nesterov s smooth approximation scheme nesterov in math program 103 127 152 2005 and nemirovski s prox method nemirovski in siam j opt 15 229 251 2005 and propose a variant of nesterov s optimal method which has outperformed the latter one in our computational experiments we also derive iteration complexity bounds for these first order methods applied to the proposed primal dual reformulations of the cone programming problem the performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances we also compare the approach based on the variant of nesterov s optimal method with the low rank method proposed by burer and monteiro math program ser b 95 329 357 2003 math program 103 427 444 2005 for solving a set of randomly generated sdp instances

general projective splitting methods for sums of maximal monotone operators we describe a general projective framework for finding a zero of the sum of n maximal monotone operators over a real hilbert space unlike prior methods for this problem we neither assume n 2 nor first reduce the problem to the case n 2 our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator at the cost of a single computationally simple projection step this framework gives rise to a family of splitting methods of unprecedented flexibility numerous parameters including the proximal stepsize may vary by iteration and by operator the order of operator evaluation may vary by iteration and may be either serial or parallel the analysis essentially generalizes our prior results for the case n 2 we also include a relative error criterion for approximately evaluating resolvents which was not present in our earlier work

spectral inequalities for kubo ando operator means abstract eigenvalue and singular value inequalities involving kubo ando operator connections or means are established previous results from lemos and soares in 2018 are generalized or complemented some log majorizations are included as a consequence a refinement of an independent result by ando and by visick in 1995 on the eigenvalues of the hadamard product is derived some singular value inequalities by zou in 2017 are further extended

asymptotic behavior of second order dynamic equations we prove several growth theorems for second order dynamic equations on time scales these theorems contain as special cases results for second order differential equations difference equations and q difference equations

oscillation and non oscillation of euler type half linear differential equations abstract we investigate oscillatory properties of second order euler type half linear differential equations whose coefficients are given by periodic functions and functions having mean values we prove the conditional oscillation of these equations in addition we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case the presented results are new for linear equations as well

some log majorizations and an extension of a determinantal inequality abstract an eigenvalue inequality involving a matrix connection and its dual is established and some log majorization type results are obtained in particular some eigenvalues inequalities considered by f hiai and m lin 9 an associated conjecture and a singular values inequality by l zou 20 are revisited a reformulation of the inequality det a u b det a b for positive semidefinite matrices a b with u a unitary matrix that appears in the polar decomposition of b a is also extended using some known norm inequalities associated to furuta inequality and araki cordes inequality

on a family of operator means involving the power difference means abstract a new family of operator means is introduced it interpolates the arithmetic geometric harmonic and logarithmic means moreover it includes some special operator means for example the power difference stolarsky and identric means continuously then order relations among them are obtained

oscillation and non oscillation criteria for linear and half linear difference equations abstract we find very general classes of linear and half linear difference equations which are conditionally oscillatory we identify the critical oscillation constant whose value implies the oscillation or non oscillation of studied equations our results are divided into oscillatory and non oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values in addition our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values we point out that the obtained results are new even for linear equations with periodic coefficients such consequences are formulated at the end of this paper

modified prufer angle and conditional oscillation of perturbed linear and half linear differential equations abstract the research and results described in this paper belong to the qualitative theory of differential equations more precisely the partial differential equations with the one dimensional p laplacian using a method whose core is formed by the prufer technique we identify a borderline case between oscillatory and non oscillatory equations moreover we are able to decide whether the studied equations are oscillatory or not even in the so called critical i e the borderline case the advantage of our approach is the fact that we obtain new and strong results for linear and half linear equations i e the equations with the one dimensional p laplacian at the same time in addition we are able to work with equations whose coefficients are non constant and non periodic the novelty of our results is documented by examples and corollaries

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

dual extrapolation and its applications to solving variational inequalities and related problems in this paper we suggest new dual methods for solving variational inequalities with monotone operators we show that with an appropriate step size strategy our method is optimal both for lipschitz continuous operators o 1 over epsilon iterations and for the operators with bounded variations o 1 over epsilon 2 iterations our technique can be applied for solving non smooth convex minimization problems with known structure in this case the worst case complexity bound is o 1 over epsilon iterations

on the complexity of the hybrid proximal extragradient method for the iterates and the ergodic mean in this paper we analyze the iteration complexity of the hybrid proximal extragradient hpe method for finding a zero of a maximal monotone operator recently proposed by solodov and svaiter one of the key points of our analysis is the use of new termination criteria based on the varepsilon enlargement of a maximal monotone operator the advantage of using these termination criteria is that their definition do not depend on the boundedness of the domain of the operator we then show that korpelevich s extragradient method for solving monotone variational inequalities falls in the framework of the hpe method as a consequence using the complexity analysis of the hpe method we obtain new complexity bounds for korpelevich s extragradient method which do not require the feasible set to be bounded as assumed in a recent paper by nemirovski another feature of our analysis is that the derived iteration complexity bounds are proportional to the distance of the initial point to the solution set the hpe framework is also used to obtain the first iteration complexity result for tseng s modified forward backward splitting method for finding a zero of the sum of a monotone lipschitz continuous map with an arbitrary maximal monotone operator whose resolvent is assumed to be easily computable also using the framework of the hpe method we study the complexity of a variant of a newton type extragradient algorithm proposed by solodov and svaiter for finding a zero of a smooth monotone function with lipschitz continuous jacobian

critical oscillation constant for euler type dynamic equations on time scales in this paper we study the second order dynamic equation on the time scale t of the form r t y d email protected t email protected t y s 0 wherer qare positive rd continuous periodic functions with inf r t email protected t 0 and c is an arbitrary real constant this equation corresponds to euler type differential resp euler type difference equation for continuous resp discrete case our aim is to prove that this equation is conditionally oscillatory i e there exists a constant c 0 such that studied equation is oscillatory for c c and non oscillatory for c c

accelerating block decomposition first order methods for solving composite saddle point and two player nash equilibrium problems this article considers the two player composite nash equilibrium cne problem with a separable nonsmooth part which is known to include the composite saddle point csp problem as a special case due to its two block structure this problem can be solved by any algorithm belonging to the block decomposition hybrid proximal extragradient bd hpe framework proposed in r d c monteiro and b f svaiter siam j optim 23 2013 pp 475 507 the framework consists of a family of inexact proximal point methods for solving a more general two block structured monotone inclusion problem which at every iteration solves two prox subinclusions according to a certain relative error criterion by exploiting the fact that the two prox subinclusions in the context of the cne problem are equivalent to two composite convex programs this article proposes a new instance of the bd hpe framework that approximately solves them using an accelerated gradient method it is shown that the new instance is able to tak

an arithmetic geometric mean inequality for singular values and its applications abstract in this short note we give a new equivalent form of the arithmetic geometric mean inequality for singular values as applications of our result we give a new proof of an inequality due to bhatia and davis 1993 4 and we obtain a singular value inequality for matrix means which is similar to one proved by drury 2012 9 finally we present a log majorization inequality for singular values

a family of projective splitting methods for the sum of two maximal monotone operators a splitting method for two monotone operators a and b is an algorithm that attempts to converge to a zero of the sum a b by solving a sequence of subproblems each of which involves only the operator a or only the operator b prior algorithms of this type can all in essence be categorized into three main classes the douglas peaceman rachford class the forward backward class and the little used double backward class through a certain extended solution set in a product space we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in hilbert space our algorithms are essentially standard projection methods using splitting decomposition to construct separators we prove convergence through fejer monotonicity techniques but showing fejer convergence of a different sequence to a different set than in earlier splitting methods our projective algorithms converge under more general conditions than prior splitting methods allowing the proximal parameter to vary from iteration to iteration and even from operator to operator while retaining convergence for essentially arbitrary pairs of operators the new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases

nonoscillation criteria for half linear second order difference equations abstract nonoscillation criteria for the half linear second order difference equation are established these criteria are derived using the riccati and variational technique