Showing papers on "Monotone polygon published in 1998"

Global attractivity in delayed Hopfield neural network models

Abstract: Two different approaches are employed to investigate the global attractivity of delayed Hopfield neural network models. Without assuming the monotonicity and differentiability of the activation functions, Liapunov functionals and functions (combined with the Razumikhin technique) are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays. In the case of monotone and smooth activation functions, the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model. Such criteria depend on the magnitude of delays and show that self-inhibitory connections can contribute to the global convergence.

Estimating smooth monotone functions

Abstract: Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D 2 f = w Df, where w is an unconstrained coefficient function, comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C 0 + C 1 D -1 {exp(D -1 w)}, where C 0 and C 1 are arbitrary constants and D -1 is the partial integration operator. A basis for expanding w is suggested that permits explicit integration in the expression of f. In fitting data, it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f. Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.

Minimax and monotonicity

Abstract: Functional analytic preliminaries.- Multifunctions.- A digression into convex analysis.- General monotone multifunctions.- The sum problem for reflexive spaces.- Special maximal monotone multifunctions.- Subdifferentials.- Discontinuous positive linear operators.- The sum problem for general banach spaces.- Open problems.

A Globally Convergent Inexact Newton Method for Systems of Monotone Equations

Abstract: We propose an algorithm for solving systems of monotone equations which combines Newton, proximal point, and projection methodologies. An important property of the algorithm is that the whole sequence of iterates is always globally convergent to a solution of the system without any additional regularity assumptions. Moreover, under standard assumptions the local superlinear rate of convergence is achieved. As opposed to classical globalization strategies for Newton methods, for computing the stepsize we do not use linesearch aimed at decreasing the value of some merit function. Instead, linesearch in the approximate Newton direction is used to construct an appropriate hyperplane which separates the current iterate from the solution set. This step is followed by projecting the current iterate onto this hyperplane, which ensures global convergence of the algorithm. Computational cost of each iteration of our method is of the same order as that of the classical damped Newton method. The crucial advantage is that our method is truly globally convergent. In particular, it cannot get trapped in a stationary point of a merit function. The presented algorithm is motivated by the hybrid projection-proximal point method proposed in [25].

Testing monotonicity

Abstract: We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f: {0, 1}/sup n/-{0, 1} at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is /spl epsiv/-far from being monotone (i.e., every monotone function differs from f on more than an /spl epsiv/ fraction of the domain). The complexity of the test is poly(n//spl epsiv/). The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. We also consider the problem of testing monotonicity based only on random examples labeled by the function. We show an /spl Omega/(/spl radic/2/sup n///spl epsiv/) lower bound on the number of required examples, and provide a matching upper bound (via an algorithm).

Monotone B-Spline Smoothing

Abstract: Estimation of growth curves or item response curves often involves monotone data smoothing. Methods that have been studied in the literature tend to be either less flexible or more difficult to compute when constraints such as monotonicity are incorporated. Built on the ideas of Koenker, Ng, and Portnoy and Ramsay, we propose monotone B-spline smoothing based on L 1 optimization. This method inherits the desirable properties of spline approximations and the computational efficiency of linear programs. The constrained fit is similar to the unconstrained estimate in terms of computational complexity and asymptotic rate of convergence. Through applications to some real and simulated data, we show that the method is useful in a variety of applications. The basic ideas utilized in monotone smoothing can be useful in some other constrained function estimation problems.

A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space

Abstract: We consider a generalized proximal point method for solving variational inequality problems with monotone operators in a Hilbert space. It differs from the classical proximal point method (as discussed by Rockafellar for the problem of finding zeroes of monotone operators) in the use of generalized distances, called Bregman distances, instead of the Euclidean one. These distances play not only a regularization role but also a penalization one, forcing the sequence generated by the method to remain in the interior of the feasible set so that the method becomes an interior point one. Under appropriate assumptions on the Bregman distance and the monotone operator we prove that the sequence converges (weakly) if and only if the problem has solutions, in which case the weak limit is a solution. If the problem does not have solutions, then the sequence is unbounded. We extend similar previous results for the proximal point method with Bregman distances which dealt only with the finite dimensional case and which applied only to convex optimization problems or to finding zeroes of monotone operators, which are particular cases of variational inequality problems.

Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces

Abstract: We use the relative modular operator to define a generalized relative entropy for any convex operator function g on the positive real line satisfying g(1) = 0. We show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy, a unique family (parameterized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including the familiar logarithmic relative entropy. The relative entropies, Riemannian metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.

Convergence analysis and applications of the Glowinski-Le Tallec splitting method for finding a zero of the sum of two maximal monotone operators

Abstract: Many problems of convex programming can be reduced to finding a zero of the sum of two maximal monotone operators For solving this problem, there exists a variety of methods such as the forward–backward method, the Peaceman–Rachford method, the Douglas–Rachford method, and more recently the θ-scheme This last method has been presented without general convergence analysis by Glowinski and Le Tallec and seems to give good numerical results The purpose of this paper is first to present convergence results and an estimation of the rate of convergence for this recent method, and then to apply it to variational inequalities and structured convex programming problems to get new parallel decomposition algorithms

Polygon representation in an integrated circuit layout

Abstract: An approach for representing polygons in an integrated circuit (IC) layout is provided. Polygons are represented by one or more wires, which in turn are each represented by one or more wire segments. Each wire segment is represented by a pair of directed line segments. A data structure hierarchy includes polygon data, wire data, wire segment data and branch data. The polygon data represents a set of IC devices to be represented in the IC layout. The wire data represents the wires that represent the polygons and specifies the associated wire segments and associated polygons. The wire segment data represents the wire segments and specifies the associated directed line segments for each wire segment that represent the wires and references the wire data. The branch data specifies connections between wires by specifying the connecting wire segments in the wires. A spacing check between a first polygon and a second polygon involves determining the canonical direction from the first polygon to the second polygon and testing the two closest faces between the polygons. To satisfy a spacing violation, an exclusion zone is constructed around the first polygon and the second polygon is moved a distance outside the exclusion zone which causes the minimum spacing required by a set of predetermined spacing criteria to be satisfied.

Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control

Abstract: This article applies splitting techniques developed for set-valued maximal monotone operators to monotone affine variational inequalities, including, as a special case, the classical linear complementarity problem. We give a unified presentation of several splitting algorithms for monotone operators, and then apply these results to obtain two classes of algorithms for affine variational inequalities. The second class resembles classical matrix splitting, but has a novel "under-relaxation" step, and converges under more general conditions. In particular, the convergence proofs do not require the affine operator to be symmetric. We specialize our matrix-splitting-like method to discrete-time optimal control problems formulated as extended linear-quadratic programs in the manner advocated by Rockafellar and Wets. The result is a highly parallel algorithm, which we implement and test on the Connection Machine CM-5 computer family.

ε-Enlargements of Maximal Monotone Operators: Theory and Applications

Abstract: Given a maximal monotone operator T,we consider a certain e-enlargement T e , playing the role of the e-subdifferential in nonsmooth optimization. We establish some theoretical properties of T e , including a transportation formula, its Lipschitz continuity, and a result generalizing Bronsted & Rockafellar’s theorem. Then we make use of the e-enlargement to define an algorithm for finding a zero of T.

Polygon Area Decomposition for Multiple-Robot Workspace Division

Abstract: A new polygon decomposition problem, the anchored area partition problem, which has applications to a multiple-robot terrain-covering problem is presented. This problem concerns dividing a given polygon P into n polygonal pieces, each of a specified area and each containing a certain point (site) on its boundary or in its interior. First the algorithm for the case when P is convex and contains no holes is presented. Then the generalized version that handles nonconvex and nonsimply connected polygons is presented. The algorithm uses sweep-line and divide-and-conquer techniques to construct the polygon partition. The input polygon P is assumed to have been divided into a set of p convex pieces (p = 1 when P is convex), which can be done in O(vPloglog vP) time, where vP is the number of vertices of P and p = O(vP), using algorithms presented elsewhere in the literature. Assuming this convex decomposition, the running time of the algorithm presented here is O(pn2+vn), where v is the sum of the number of vertices of the convex pieces.

Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions

Abstract: In this paper, we introduce a new class of generalized nonlinear implicit quasi-variational inclusions and prove its equivalence with a class of fixed point problems by making use of the properties of maximal monotone. We also prove the existence of solutions for this generalized nonlinear implicit quasi-variational inclusions and the convergence of iterative sequences generated by the perturbed algorithms.

Monotonicity and convergence results in order-preserving systems in the presence of symmetry

Abstract: This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.

The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: Easy and hard cases

Abstract: This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge—Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge—Toeplitz QAP: (Pl) The “Turbine Problem”, i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrixB that lead to a simple solution for the Anti-Monge—Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge—Toeplitz QAP is NP-hard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Strange functions in real analysis

Abstract: Introduction: basic concepts Cantor and Peano type functions Functions of first Baire class New! Semicontinuous functions that are not countably continuous New! Singular monotone functions Everywhere differentiable nowhere monotone functions Nowhere approximately differentiable functions Blumberg's theorem and Sierpinski-Zygmund functions Lebesgue nonmeasurable functions and functions without the Baire property Hamel basis and Cauchy functional equation Luzin sets, Sierpinski sets, and their applications Absolutely nonmeasurable additive functions New! Egorov type theorems Sierpinski's partition of the Euclidean plane Bad functions defined on second category sets New! Sup-measurable and weakly sup-measurable functions Generalized step-functions and superposition operators New! Ordinary differential equations with bad right-hand sides Nondifferentiable functions from the point of view of category and measure Bibliography Subject Index

Existence of equilibria for monotone multivalued mappings

Abstract: Using a particular kind of pseudomonotonicity for multivalued mappings, an existence result is proved for equilibria, variational inequalities, and a combination of both.

Truncation approximations of invariant measures for Markov chains

Abstract: Let P be the transition matrix of a positive recurrent Markov chain on the integers, with invariant distribution π. If (n) P denotes the n x n ‘northwest truncation’ of P, it is known that approximations to π(j)/π(0) can be constructed from (n) P, but these are known to converge to the probability distribution itself in special cases only. We show that such convergence always occurs for three further general classes of chains, geometrically ergodic chains, stochastically monotone chains, and those dominated by stochastically monotone chains. We show that all ‘finite’ perturbations of stochastically monotone chains can be considered to be dominated by such chains, and thus the results hold for a much wider class than is first apparent. In the cases of uniformly ergodic chains, and chains dominated by irreducible stochastically monotone chains, we find practical bounds on the accuracy of the approximations.

A non-linear hierarchy for discrete event dynamical systems

Abstract: nonexpansive maps, fixed points, cycle time Dynamical systems of monotone homogeneous functions appear in Markov decision theory, in discrete event systems and in Perron-Frobenius theory. We consider the case when these functions are given by finite algebraic expressions involving the operations max, min, convex hull, translations and an infinite family of binary operations, of which max and min are limit cases. We set up a hierarchy of monotone homogeneous functions that reflects the complexity of their defining algebraic expressions. For two classes of this hierarchy, we show that the trajectories of the corresponding dynamical systems admit a linear growth rate (cycle time). The first class generalizes the min-max functions considered previously in the literature. The second class generalizes both max-plus linear maps and ordinary non-negative linear maps.

Methods for manipulating curves constrained to unparameterized surfaces

Abstract: A computer-implemented method for drawing and manipulating curves on polygon mesh surfaces possesses a built in surface constraint that provides a better intuition for the shape and position of a curve than conventional unconstrained three dimensional space curves. The method includes storing the following graphical data structures in a memory: a 2-D polygon mesh embedded in a 3-D ambient space, a 1-D face point curve v embedded in the 2-D polygon mesh, a first-order curve weighting function α, and a second-order curve weighting function β. The method also comprises computing a displacement dv of the face point curve v to produce a displaced face point curve v′=v+dv, wherein v′ is embedded in the 2-D polygon mesh, and wherein the displacement dv is computed in dependence upon α and β. A rendered representation of the polygon mesh and the displaced face point curve are then displayed. In a preferred embodiment of the invention, computing the displacement dv comprises calculating a set of displacement vectors corresponding to a set of face points of the curve v, wherein each vector in the set of displacement vectors is tangent to the polygon mesh at a face point corresponding to the vector. The displacement dv is computed in dependence upon a function S defined on the polygon mesh. The function S may be a function determined in part from user input, from an external force, and/or from a curvature of the polygon mesh. The external force may be determined from a coloring of the polygon mesh, and/or from a user manipulating the curve. The computing and displaying steps can be repeated to produce a displayed relaxation of the curve.

A characterization of span program size and improved lower bounds for monotone span programs

Abstract: We give a characterization of span program size by a combinatorial-algebraic measure. The measure we consider is a generalization of a measure on covers which has been used to prove lower bounds on formula size and has also been studied with respect to communication complexity.In the monotone case our new methods yield nΩ(log n) lower bounds for the monotone span program complexity of explicit Boolean functions in n variables over arbitrary fields, improving the previous lower bounds on monotone span program size. Our characterization of span program size implies that any matrix with superpolynomial separation between its rank and cover number can be used to obtain superpolynomial lower bounds on monotone span program size. We also identify a property of bipartite graphs that is sufficient for constructing Boolean functions with large monotone span program complexity.

Some basic properties of the MLE's for a multivariate normal distribution with monotone missing data

Abstract: SYNOPTIC ABSTRACTIn this paper we study some basic properties of the MLE's and for a p-variate normal population Np(μ, Σ), based on a k-step monotone sample. For k = 2 and 3, we obtain the exact means and variances of and the exact bias of . Asymptotic expansions of the distributions of these estimators are also obtained in the situation when the sample size is large. For a general k, the MLE's of μ and the usual transformed matrix Δ of Σ are given in explicit forms. Some analogous properties are also obtained.

On Monotone and Geometric Convergence of Schwarz Methods for Two-Sided Obstacle Problems

Abstract: Convergence of Schwarz methods is discussed for certain discretizations of two-sided obstacle problems. Monotone and especially geometrical convergence are established if the stiffness matrix is a kind of M-matrix, and an h-independent convergence rate is proved for uniformly overlapping decomposition.

Object structure graph generation and data conversion using the same

Abstract: Shape expressions in CAD or CG have often been carried out in polygon data. In polygon representations, the amount of data becomes very large if precision is pursued. Another shape representation utilizing the existing polygon data asset is proposed. Polygon data showing the shape of an object is first obtained. Topological information of the object is extracted from the polygon data. Based on the information, the polygon data is converted into topological data. The inversion is carried out upon necessity.

Mapping properties of integral averaging operators

Abstract: Characterizations are obtained for those pairs of weight functions u and v for which the operators Tf(x) = R b(x) a(x) f(t)dt with a and b certain non-negative functions are bounded from L p(0;1) to L q(0;1), 0 < p;q <1, p 1. Sucient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing dierences and

Convergence Acceleration via Combined Nonlinear-Condensation Transformations

Abstract: A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound state calculations.