Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

Hardy-type inequalities on the cones of monotone functions

We establish necessary and sufficient conditions for various Hardy-type inequalities on the cones of monotone functions.     

Norm inequalities related to operator monotone functions

Let A,B be positive semidefinite matrices and any unitarily invariant norm on the space of matrices. We show for any non-negative operator monotone function f(t) on , and for non-negative increasing function g(t) on with g(0) = 0 and , whose inverse function is operator monotone. Received: 1 February 1999     

The modular inequalities for a class of convolution operators on monotone functions

This paper is devoted to the study of modular inequality

where and is a class of Volterra convolution operators restricted to the monotone functions. When with and the kernel , our results will extend those for the Hardy operator on monotone functions on Lebesgue spaces.

     

Constants in inequalities for mean values of some periodic arithmetic functions

Constants in estimates of short Gaussian sums and sums of products of Legendre symbols over sequences of natural numbers shifted by various numbers are refined.     

Stable monotone variational inequalities

Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate to a stability concept involving a natural parameter. These include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity. The monotone complementarity problem is covered as a special case, and the results are sharpened for linear monotone complementarity and for generalized linear programming.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 at the University of Wisconsin - Madison and by the National Science Foundation under Grant No. DMS-8405179 at the University of Illinois at Urbana-Champaign.     

An explicit algorithm for monotone variational inequalities

We introduce a fully explicit method for solving monotone variational inequalities in Hilbert spaces, where orthogonal projections onto the feasible set are replaced by projections onto suitable hyperplanes. We prove weak convergence of the whole generated sequence to a solution of the problem, under only the assumptions of continuity and monotonicity of the operator and existence of solutions.     

A continuation method for monotone variational inequalities

This paper presents a continuation method for monotone variational inequality problems based on a new smooth equation formulation. The existence, uniqueness and limiting behavior of the path generated by the method are analyzed.This work was supported by the National Science Foundation Presidential Young Investigator Award ECE-8552773 and by a grant from the Burlington Northern Railroad.     

Strong convergence result for monotone variational inequalities

Our aim in this paper is to study strong convergence results for L-Lipschitz continuous monotone variational inequality but L is unknown using a combination of subgradient extra-gradient method and viscosity approximation method with adoption of Armijo-like step size rule in infinite dimensional real Hilbert spaces. Our results are obtained under mild conditions on the iterative parameters. We apply our result to nonlinear Hammerstein integral equations and finally provide some numerical experiments to illustrate our proposed algorithm.     

Optimality conditions for strongly monotone variational inequalities

Necessary conditions for optimal controls have been obtained for strongly monotone variational inequalities by the penalty method, Ekeland's Variational Principle, and lower-semicontinuity of set-valued mappings. It has been shown that these conditions are easy to apply and can imply some known necessary conditions. They also yield new optimality conditions.     

Interior projection-like methods for monotone variational inequalities

We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.The work of this author was partially supported by the United States–Israel Binational Science Foundation, BSF Grant No. 2002-2010.     

Rearrangement inequalities for functionals with monotone integrands

The inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u₁,…,u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives ∂_i∂_jF for all i≠j. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nondegeneracy conditions, all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant (i.e., measure-preserving maps for the Hardy-Littlewood inequality, translations for the Riesz inequality). The proofs rely on monotone changes of variables in the spirit of Sklar's theorem.     

Iterative algorithms for generalized monotone variational inequalities

We propose some new iterative methods for solving the generalized variational inequalities where the underlying operator T is monotone. These methods may be viewed as projection-type methods. Convergence of these methods requires that the operator T is only monotone. The methods and the proof of the convergence are very simple. The results proved in this paper also represent a significant improvement and refinement of the known results.     

Interior-point algorithms for monotone affine variational inequalities

Given ann×n matrixM, a vectorq in ⁿ , a polyhedral convex setX={x|Axb, Bx=d}, whereA is anm×n matrix andB is ap×n matrix, the affinne variational inequality problem is to findxX such that (Mx+q) ^T (y–x)0 for allyX. IfM is positive semidefinite (not necessarily symmetric), the affine variational inequality can be transformeo to a generalized complementarity problem, which can be solved in polynomial time using interior-point algorithms due to Kojima et al. We develop interior-point algorithms that exploit the particular structure of the problem, rather than direictly reducing the problem to a standard linear complemntarity problem.This work was partially supported by the Air Force Office of Scientific Research, Grant AFOSR-89-0410 and the National Science Foundation, Grant CCR-91-57632.The authors acknowledge Professor Osman Güler for pointing out the valoidity of Theorem 2.1 without further assumptions and the proof to that effect. They are also grateful for his comments to improve the presentation of this paper.     

A descent algorithm for solving monotone variational inequalities and monotone complementarity problems

The paper provides a descent algorithm for solving certain monotone variational inequalities and shows how this algorithm may be used for solving certain monotone complementarity problems. Convergence is proved under natural monotonicity and smoothness conditions; neither symmetry nor strict monotonicity is required.The author is grateful to two anonymous referees for their very valuable comments on an earlier draft of this paper.     

On inequalities for the norms of intermediate derivatives of multiply monotone functions defined on a finite segment

We study the following modification of the Landau?CKolmogorov problem: Let k; r ?? ?, 1 ?? k ?? r ? 1, and p, q, s ?? [1,??]. Also let MM ^m , m ?? ?; be the class of nonnegative functions defined on the segment [0, 1] whose derivatives of orders 1, 2,??,m are nonnegative almost everywhere on [0, 1]. For every ?? > 0, find the exact value of the quantity $$ \omega_{p,q,s}^{k,r}\left( {\delta; M{M^m}} \right): = \sup \left\{ {{{\left\| {{x^{(k)}}} \right\|}_q}:x \in M{M^m},{{\left\| x \right\|}_p} \leqslant \delta, {{\left\| {{x^{(k)}}} \right\|}_s} \leqslant 1} \right\}. $$ We determine the quantity $ \omega_{p,q,s}^{k,r}\left( {\delta; M{M^m}} \right) $ in the case where s = ?? and m ?? {r, r ? 1, r ? 2}. In addition, we consider certain generalizations of the above-stated modification of the Landau?CKolmogorov problem.