Showing papers on "Strongly monotone published in 1995"
Journal Article•
TL;DR: It is shown that this is tight in the sense that for any subexponential time algorithm there is a monotone Boolean function for which this algorithm cannot approximate with error better than O(1/√n).
Abstract: In this paper, monotone Boolean functions are studied using harmonic analysis on the cube. The main result is that any monotone Boolean function has most of its power spectrum on its Fourier coefficients of degree at most O(√n) under any product distribution. This is similar to a result of Linial et al. [1993], which showed that AC 0 functions have almost all of their power spectrum on the coefficients of degree, at most (log n) O(1) , under the uniform distribution. As a consequence of the main result, the following two corollaries are obtained : -For any e > 0, monotone Boolean functions are PAC learnable with error e under product distributions in time 2 O((1/e) √ n) . -Any monotone Boolean function can be approximated within error e under product distributions by a non-monotone Boolean circuit of size 2 O(1/e √ n) and depth O(1/e √n). The learning algorithm runs in time subexponential as long as the required error is Ω(1/(√n log n)). It is shown that this is tight in the sense that for any subexponential time algorithm there is a monotone Boolean function for which this algorithm cannot approximate with error better than O(1/√n). The main result is also applied to other problems in learning and complexity theory. In learning theory, several polynomial-time algorithms for learning some classes of monotone Boolean functions, such as Boolean functions with O(log 2 n/log log n) relevant variables, are presented. In complexity theory, some questions regarding monotone NP-complete problems are addressed.
79 citations
TL;DR: A proof of convergence that does not use the concept of partial inverse is given and how to choose a scaling factor to accelerate the convergence in the strongly monotone case is shown.
Abstract: We present an algorithm to solve: Find $( x,y ) \in A \times A^ \bot $ such that $y \in Tx$, where A is a subspace and T is a maximal monotone operator. The algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover Spingarn’s decomposition method. We give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case. Numerical results performed on quadratic problems confirm the robust behaviour of the algorithm.
66 citations
TL;DR: It is shown that themonotone analogue of logspace computation is more powerful than monotone log-depth circuits: monot one bounded fanin circuits for a certain function in monotones logspace require depth ?
34 citations
TL;DR: In this article, the authors investigated the relation between maximal monotone operators of dense type and maximal monotonous operators of non-densemble type. But they focused on the non-reflexive case.
Abstract: Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.
33 citations
Book•
29 Sep 1995TL;DR: In this article, the theory of generalised topological degree for densely defined A-proper operators is described and applied to boundary value problems of nonlinear ordinary and partial differential equations, which are intractable under any other existing theory.
Abstract: This book describes many new results and extensions of the theory of generalised topological degree for densely defined A-proper operators and presents important applications, particularly to boundary value problems of non-linear ordinary and partial differential equations, which are intractable under any other existing theory. A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation. This theory subsumes classical theory involving compact vector fields, as well as the more recent theories of condensing vector-fields, strongly monotone and strongly accretive maps. Researchers and graduate students in mathematics, applied mathematics and physics who make use of non-linear analysis will find this an important resource for new techniques.
24 citations
TL;DR: In this article, Ozawa et al. describe self-duality and C*reflexivity of Hilbert A-modules over monotone complete C*-algebras by the completeness of the unit ball of ℳ with respect to two types of convergence being defined, and by a structural criterion.
Abstract: The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ℳ over monotone complete C*-algebras A by the completeness of the unit ball of ℳ with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182, 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ℳ can be continued to an A-valued inner product on it's A-dual Banach A-module ℳ' turning ℳ' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End′(ℳ) on self-dual Hilbert A-modules ℳ over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.
19 citations
TL;DR: In this paper, a unified theory for the solvability of the inclusions S c R(A + B), intS C R (A+B), where A is either maximal monotone or maccretive, and B is either compact or pseudo-monotone, is presented.
Abstract: A more comprehensive and unified theory is developed for the solvability of the inclusions S c R(A + B), intS C R(A + B), where A: X D D(A) -+ 2y, B: X D D(B) -+ Y and S c X. Here, X is a real Banach space and Y = X or Y = X*. Mainly, A is either maximal monotone or maccretive, and B is either pseudo-monotone or compact. Cases are also considered where A has compact resolvents and B is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators A and B. Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbations of maximal monotone operators, and the Leray-Schauder degree theory.
17 citations
TL;DR: These algorithms are obtained as a combination of known algorithms for possibly multi-valued maximal monotone operators with appropriate decompositions of the domain as well as a method due to two of the authors in the single-valued and linear case.
Abstract: This paper presents general algorithms for the parallel solution of finite element problems associated with maximal monotone operators of local type. The latter concept, which is also introduced here, is well suited to capture the idea that the given operator is the discretization of a differential operator that may involve nonlinearities and/or constraints as long as those are of a local nature. Our algorithms are obtained as a combination of known algorithms for possibly multi-valued maximal monotone operators with appropriate decompositions of the domain. This work extends a method due to two of the authors in the single-valued and linear case.
14 citations
14 citations
01 Aug 1995
TL;DR: In this article, it was shown that for continuous linear monotone operators, all these notions coincide and are equivalent to the monotonicity of the conjugate operator, and the latter condition is analyzed and illustrated by several examples.
Abstract: The concept of a monotone operator --- which covers both linear positive semi-definite operators and subdifferentials of convex functions --- has turned out to be very powerful in various branches of mathematics Over the last few decades, several new notions of monotonicity have been introduced: Gossez' maximal monotone of type (D), Simons' monotone of type (WD) and of type (NI), Fitzpatrick and Phelps' locally maximal monotone While these monotonicities are automatic for maximal monotone operators in reflexive Banach spaces and for subdifferentials of convex functions, their precise relationship is largely unknown In view of the origin of the theory of monotone operators, it is very natural to investigate linear monotone (ie positive semi-definite) operators Here, it is shown --- within the beautiful framework of Convex Analysis --- that for continuous linear monotone operators, {\em all these notions coincide and are equivalent to the monotonicity of the conjugate operator} The latter condition is analyzed and illustrated by several examples Some nonlinear results on regularizations conclude the paper
11 citations
TL;DR: In this article, it was shown that the ordered group of all relatively bounded homomorphisms from A to E is a monotone-complete dimension group, and this property can supply the necessary amount of completeness in several kinds of problems.
Abstract: We show that whenever A is a monotone -complete dimension group, then A + ({1} is countably equationally compact, and we show how this property can supply the necessary amount of completeness in several kinds of problems. In particular, if A is a countable dimension group and E is a monotone -complete dimension group, then the ordered group of all relatively bounded homomorphisms from A to E is a monotone -complete dimension group.
TL;DR: It is proved that there is a monotone function in AC40 which requires exponential-sizemonotone perceptrons of depth 3, which solves the monotones version of a problem which, in the general case, would imply an oracle separation of PPPH.
Abstract: It is proved that there is a monotone function in AC40 which requires exponential-size monotone perceptrons of depth 3 This solves the monotone version of a problem which, in the general case, would imply an oracle separation of PPPH
TL;DR: In this article, the authors consider a certain inequality that arises in the study of iterative methods for solving equations in a Hilbert space, and give equivalent characterizations of the inequality, and show that the inequality is satisfied by the members of a large class of networks of monotone (possibly dynamic) two-terminal elements.
Abstract: We consider a certain inequality that arises in the study of iterative methods for solving equations in a Hilbert space, and give equivalent characterizations of the inequality. We then show that the inequality is satisfied by the members of a large class of networks of monotone (possibly dynamic) two-terminal elements. This establishes the applicability of a simple algorithm that, for a large class of monotone resistive networks, will converge to a solution of the network equations whenever a solution exists, and that will generate an unbounded sequence of iterates if no solution exists. >
01 Jan 1995
TL;DR: Based on the demiregular convergence theory of the operator equations involving strongly monotone operators by Anselone and Lei (1986), this article studied approximation solvability of the nonlinear equations with strongly accretive operators.
Abstract: Based on the demiregular convergence theory of the operator equations involving strongly monotone operators by Anselone and Lei (1986), we study approximation solvability of the nonlinear equations involving strongly accretive operators.
01 Nov 1995
04 Dec 1995
TL;DR: A threshold query has the form ” f(x) ≥ y?” for a pair (x, y) for an unknown monotone discrete function f with domain and range of size m and n, respectively, where m ≤ n.
Abstract: Let us be given an unknown monotone discrete function f with domain and range of size m and n, respectively, where m ≤ n. A threshold query has the form ” f(x) ≥ y?” for a pair (x, y).
TL;DR: In this paper, the authors studied the solvability of the problem of finding a maximal monotone operator in a subspace of a real Hilbert space, where the subspace is a Hilbert space.
Abstract: The aim of this note consists in studying the solvability of the following problem find x ∈ A; y ∈ A⊥ such that y ∈ T(x) T is a maximal monotone operator and A a subspace of a real Hilbert space H.
TL;DR: A new ball algorithm for bounding a zero point of a nonlinear quasi-strongly monotone operator in Hilbert spaces is presented and it is shown that the algorithm converges much more rapidly than the existing ball algorithms for the given problems.
Abstract: A new ball algorithm for bounding a zero point of a nonlinear quasi-strongly monotone operator in Hilbert spaces is presented. It is shown that the algorithm converges much more rapidly than the existing ball algorithms for the given problems. Numerical comparisons are made to support the conclusion.
TL;DR: In this paper, the Riesz convexity theorem for semi-linear operators on monotone functions is extended to weighted norm inequalities for non-decreasing functions f and 1 ≦ p < ∞.
Abstract: Weighted norm inequalities are investigated by giving an extension of the Riesz convexity theorem to semi-linear operators on monotone functions. Several properties of the classes B(p, n) and C(p, n) introduced by Neugebauer in [13] are given. In particular, we characterize the weight pairs w, v for which for nondecreasing functions f and 1 ≦ p < ∞.