Showing papers on "Strongly monotone published in 1976"

Monotone Operators and the Proximal Point Algorithm

Abstract: For the problem of minimizing a lower semicontinuous proper convex function f on a Hilbert space, the proximal point algorithm in exact form generates a sequence $\{ z^k \} $ by taking $z^{k + 1} $ to be the minimizes of $f(z) + ({1 / {2c_k }})\| {z - z^k } \|^2 $, where $c_k > 0$. This algorithm is of interest for several reasons, but especially because of its role in certain computational methods based on duality, such as the Hestenes-Powell method of multipliers in nonlinear programming. It is investigated here in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferential $\partial f$ is replaced by an arbitrary maximal monotone operator T. Convergence is established under several criteria amenable to implementation. The rate of convergence is shown to be “typically” linear with an arbitrarily good modulus if $c_k $ stays large enough, in fact superlinear if $c_k \to \infty $. The case of $T = \partial f$ is treated in extra detail. Applicati...

On a convexity property of the range of a maximal monotone operator

Abstract: An example is given which shows that the closure of the range of a maximal monotone operator from a (nonreflexive) Banach space into its dual is not necessarily convex. Introduction. Let A be a real Banach space with dual X* and let T: X -* 2X be a maximal monotone operator with domain D{T) and range R{T). In general R{T) is not a convex set (cf. [4]) but it is known that when X is reflexive, the (norm) closure of R{T) is convex (cf. [5]). Without reflexivity, the convexity of cl R{T) is still true when T is the subdifferential of a lower semicontinuous proper convex function (cf. [1]), or more generally, when the associated monotone operator Tx: X" —> 2X is maximal (cf. [2] where the proof is given under a slightly stronger assumption). Here Tx denotes the operator whose graph is defined by gr7f = {(x",x*) E X"xX*;3 a net {Xj,x*) E grFwith Xj bounded,xt -* x" weak" and x-> x* in norm}. {X is identified as usual to a subspace of its bidual X**.) The question was raised some years ago as to whether or not the convexity of cl R{T) holds in general. In this note we answer this question negatively. We exhibit a (everywhere defined and coercive) maximal monotone operator from /' to 2/0° whose range has not a convex closure. Our construction is based on a result of [3]. Example. Let A: lx -> /°° be the bounded linear operator defined by 00 {Ax)n = 2l xmamn m=\ for x = {xx,x2,...) E /', where a■ = 0 if m = n, amn = -1 if n > m and am„ = +1 if n 2R is given by s{t) = -I if t 0. For A > 0, the mapping XJ + A is clearly maxReceived by the editors March 31, 1975. A MS (MOS) subject classifications (1970). Primary 47H05; Secondary 46B10, 35J60.

On nowhere monotone functions

Abstract: The existence of everywhere differentiable but nowhere monotone functions is established using the Baire Category Theorem, and the relatively easy fact that there are nontrivial bounded derivatives with a dense set of zeros. Interest in everywhere differentiable, nowhere monotone functions was revived by Katznelson and Stromberg in [4] where they gave a construction of such a function which is considerably simpler than the original one due to Kopeke or the one in the book by Hobson [3, pp. 412-421]. This work was followed up by Goffman in [2] where a much shorter construction is given but which uses a deep theorem due to Zahorski. Here the existence of such functions is established using the Baire Category Theorem. Let R denote the real line and let D = {f: R —> R: fis bounded and there is a function F such that F'(x) = f(x) for all x in 7?}, and endow D with the metric d(fg)= sup \f(x) g(x)\. xBR This is the metric of uniform convergence, and by a standard advanced calculus theorem, a uniform limit of a sequence of bounded derivatives is a bounded derivative. Hence D is a complete metric space. Let D0 = {/ G D: [x: f(x) = 0) is dense in 7?}, and give to D0 the metric of D. Then D0 itself is complete for if {fk} is a sequence in D0 converging in metric to f G D, then for each k, Zk = {x: fk(x) = 0} is a dense Gs set and hence Z = D£L i Zk is dense in R. But Z C [x:f(x) = 0}. Thus/ G D0. It is not hard to show that D0 contains more than just the zero function (see [1, p. 27] or [5]). The existence of such a function and the fact that Z>0 is closed Received by the editors October 8, 1975. A MS (MOS) subject classifications (1970). Primary 26A24, 26A21. © American Mathematical Society 1976 388 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON NOWHERE MONOTONE FUNCTIONS 389 Then clearly under addition will be used below. The proof of the latter is like the completeness of D0 only easier. Theorem. Let E = {f G Dq : there is an interval on which f is unsigned}. Then E is of the first category in D0. Proof. Let {/„} be an ordering of the collection of all closed intervals having rational endpoints. Let En = {/ G DQ:fix) > 0 for all x G /„} and F„ = {fG D0: fix) < 0 for all x G /„}. 00 E= U(E„ U F„); n=\ so it suffices to prove that En and Fn are closed and contain no spheres. The argument will be carried out for En. A similar procedure works for Fn. That En is closed is immediate. To prove that En contains no sphere suppose f G Dq and e > 0. Since f G D0 there is an x G /„ such that /(x) = 0. Since there are bounded derivatives having a dense set of zeros that are not identically zero, by pushing and crushing it is not hard to prove that there is a function h G D0 such that /i(x) < 0 and sup/eÄ|/i(y)| < e. Then g = / + h belongs to D0, difg) < e, and g £ En since g(x) = fix) + /i(x) = /i(x) < 0 and x G In. Thus the sphere of radius e about/is not contained in E„. References 1. A. Bruckner and J. L. Leonard, Derivatives, Amer. Math. Monthly 73 (1966), no. 4, part II, 24-56. MR 33 #5797. 2. Casper Goffman, Everywhere differentiable functions and the density topology, Proc. Amer. Math. Soc. 51 (1975), 250. 3. E. W. Hobson, Theory of functions of a real variable and the theory of Fourier series. Vol. 2, Dover, New York, 1958. MR 29, 1166. 4. Y. Katznelson and K. Stromberg, Everywhere differentiable, nowhere monotone functions, Amer. Math. Monthly 81 (1974), 349-354. MR 49 #481. 5. D. Pompeiu, Sur les fonctions dérivées, Math. Ann. 63 (1907), 326-332. Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Singular Hammerstein equations and maximal monotone operators

Abstract: the operator KF is defined on all of X, and singular otherwise. In some recent papers (summarized in [2]), the writers have studied the existence theory for regular Hammerstein equations in L(|3) with 1

Extensions of monotone operator functions

Abstract: It is shown that a monotone operator function f defined on an open subset A of the real numbers may be extended to a monotone operator function on the convex hull of A Let f be a bounded real-valued Borel-measurable function defined on a Borel subset A of the real numbers Let A be a bounded selfadjoint operator on a separable complex Hilbert space H such that the spectrum a(A) of A is contained in A Then A(A) is the selfadjoint operator on H defined by ff(X)K , 0, p e H, where E is the resolution of the identity corresponding to A The functionfis said to be a monotone operator function on A if f(A) < f(B) whenever A and B are bounded selfadjoint operators on a Hilbert space H such that a(A) C A, a(B) C A, and A < B If f satisfies this monotonicity condition for the totality of finite-dimensional complex Hilbert spaces, then f is said to be a monotone matrix function on A The following characterization of monotone operator functions is essentially due to Loewner [5] Loewner proved the theorem for monotone matrix functions; Bendat and Sherman [1] proved that a function f is monotone matrix on an open interval (a,b) if and only if f is monotone operator on (a,b) THEOREM 1 A real-valued function f defined on an open interval (a,b) is a monotone operator function on (a,b) if and only if f is analytic on (a,b), can be analytically continued onto the upper half-plane, and represents there a holomorphic function with nonnegative imaginary part It will be shown that there is a similar characterization of monotone operator functions defined on an arbitrary open subset A of the real numbers Let f be a monotone operator function on an open subset A of the real numbers By Theorem 1, f is continuously differentiable on A; see also [4, p 73] Associated with f is the kernel K, defined on A x A by Loewne madeeK(xtei fv ue f() (t k Ar)e K(i,s ) = f mt() Loewner made extensive use of this kernel in his study of monotone matrix Received by the editors May 21, 1975 AMS (MOS) subject classifications (1970) Primary 47A60, 47B 15; Secondary 30A14, 30A76

Existence theory for the complex nonlinear complementarity problem

Abstract: The main result in this paper is an existence theorem for the following complex nonlinear complementarity problem: find z such that where S is a polyhedral cone in C n , S * the polar cone, and g is a mapping from C n into itself. It is shown that the above problem has a unique solution if the mapping g is continuous and strongly monotone on the polyhedral cone S .

Moduli of Monotonicity with Applications to Monotone Polynomial Approximation

Abstract: This article introduces new concepts called the moduli of monotonicity of a real function defined on an interval. They are a one-sided analogue of the well-known modulus of continuity, and are a measure of the extent by which a given function fails to be monotone. It is shown that they naturally arise in the process of approximating a real function by nondecreasing polynomials. Upper and lower bounds on the “degree of approximation” by monotone polynomials are derived in terms of these moduli.

The realization of monotone Boolean functions (Preliminary Version)

Abstract: In this paper we study the complexity of realizing a monotone but otherwise arbitrary Boolean function. We consider realizations by means of networks and formulae. In both cases the possibility exists that although a monotone function can always be realized in terms of monotone basis functions, a more economical realization may be possible if basis functions that are not themselves monotone are used. Thus, we have four cases, namely:1. The cost of realizing a monotone function with a network over a universal basis.2. The cost of realizing a monotone function with a network over a monotone basis.3. The cost of realizing a monotone function with a formula over a universal basis.4. The cost of realizing a monotone function with a formula over a monotone basis.For the first case, we obtain a complete solution to the problem. For the other three cases, we obtain improvements over previous results and come within a logarithmic factor or two of a complete solution.

Negative theorems on monotone approximation

Abstract: In this paper we show that for f continuous on [-1, + 1] and satisfying (f(x2) f(x1 ))/(x2 xI) _ 0 > 0, it is possible to have infinitely many of the polynomials of best uniform approximation tof not increasing on [-1, +1].

Tietze-type theorems on monotone increasing sets

Abstract: The Tietze theorem on convex sets is generalized to monotone increasing sets and strictly monotone increasing sets, which include convex sets as a special case The main theorem is that a closed connected set in E2 is monotone increasing if and only if it is locally monotone increasing A similar result is proved for strictly monotone increasing sets