Showing papers on "Monotone polygon published in 1983"
TL;DR: A general convergence theorem is provided for algorithms of this type including the calculation of fixed points of contraction and monotone mappings arising in linear and nonlinear systems of equations, optimization problems, shortest path problems, and dynamic programming.
Abstract: We present an algorithmic model for distributed computation of fixed points whereby several processors participate simultaneously in the calculations while exchanging information via communication links. We place essentially no assumptions on the ordering of computation and communication between processors thereby allowing for completely uncoordinated execution. We provide a general convergence theorem for algorithms of this type, and demonstrate its applicability to several classes of problems including the calculation of fixed points of contraction and monotone mappings arising in linear and nonlinear systems of equations, optimization problems, shortest path problems, and dynamic programming.
257 citations
TL;DR: The method of partial inverses (MOP) as mentioned in this paper is a method for solving problems in which the object is to findx ∈ A andy ∈ a closed subspace such that y ∈ T(x).
Abstract: ForT a maximal monotone operator on a Hilbert spaceH andA a closed subspace ofH, the “partial inverse”T
A
ofT with respect toA is introduced.T
A
is maximal monotone. The proximal point algorithm, as it applies toT
A
, results in a simple procedure, the “method of partial inverses”, for solving problems in which the object is to findx ∈ A andy ∈ A
⊥ such thaty ∈ T(x). This method specializes to give new algorithms for solving numerous optimization and equilibrium problems.
212 citations
01 May 1983-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: The optimal algorithm for finding the visibility polygon of a point inside or outside a simple polygon is given, which uses only one stack, instead of three as used in a previously known algorithm by El-Gindy and Avis.
Abstract: The hidden line problem in the plane is studied and an optimal algorithm for finding the visibility polygon of a point inside or outside a simple polygon is given. The algorithm uses only one stack, instead of three as used in a previously known algorithm by El-Gindy and Avis (J. Algor. 2, 1981, 186–197). As a result, the algorithm is simpler to implement and easier to understand and its correctness can be easily verified.
198 citations
TL;DR: In this article, it was shown that standard monotone difference schemes, satisfying a fairly unrestrictive CFL condition, converge to the "correct" physical solution even in the case when a non-uniform spatial mesh is employed.
Abstract: Monotone finite difference schemes used to approximate solutions of scalar conservation laws have the advantage that these approximations can be proved to converge to the proper solution as the mesh size tends to zero. The greatest disadvantage in using such approximating schemes is the computational expense encountered since monotone schemes can have at best first order accuracy. Computation savings and effective accuracy could be gained if the spatial mesh were refined in regions of expected rapid solution variation. In this paper we prove that standard monotone difference schemes, (satisfying a fairly unrestrictive CFL condition), converge to the "correct" physical solution even in the case when a nonuniform spatial mesh is employed.
148 citations
TL;DR: In this paper, the authors considered the assignment problem when there are junction interactions and gave an objective function which measures the extent to which a traffic distribution departs from equilibrium, and an algorithm which (under certain conditions) calculates equilibria by steadily reducing the objective function to zero.
Abstract: The paper considers the assignment problem when there are junction interactions. We give an objective function which measures the extent to which a traffic distribution departs from equilibrium, and an algorithm which (under certain conditions) calculates equilibria by steadily reducing the objective function to zero. It is shown that the algorithm certainly works if the network cost-flow function is monotone and continuously differentiable, and a boundary condition is satisfied.
111 citations
TL;DR: A linear time algorithm for finding the convex hull of a simple polygon that requires only one stack, instead of two, and runs more efficiently than the result of McCallum and Avis.
Abstract: In this paper we present a linear time algorithm for finding the convex hull of a simple polygon. Compared to the result of McCallum and Avis, our algorithm requires only one stack, instead of two, and runs more efficiently.
93 citations
TL;DR: In this article, it was shown that every linear admissible feedback control stabilizes the control system under a controllability assumption, using this fact, a procedure to construct a monotone decreasing sequence of quadratic forms which converges to the optimal value of the control problem is discussed.
Abstract: In the first part of the paper linear–quadratic control problem under indenendent random perturbations is considered. Under a controllability assumption it is shown that every linear admissible feedback control stabilizes the control system. Using this fact,a procedure to construct a monotone decreasing sequence of quadratic forms which converges to the optimal value of the control problem is discussed. Similar results for the linear–quadratic control problem under jump Markov perturbations are also obtained. In the last part of the paper, some necessary conditions for the stabilizability of a class of linear discrete–time control systems with independent random perturbations are derived
77 citations
TL;DR: In this paper, a set of data consisting of measurements ofn objects with respect top variables displayed in ann ×p matrix is considered, and two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model.
Abstract: Consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann ×p matrix. A monotone transformation of the values in each column, represented as a linear combination of integrated basis splines, is assumed determined by a linear combination of a new set of values characterizing each row object. Two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model. Examples for artificial and real data are presented. The results indicate that both methods are helpful in choosing dimensionality and that the Eckart-Young model is also helpful in displaying the relationships among the objects and the variables. Also, results suggest that the resulting transformations are themselves illuminating.
61 citations
TL;DR: A classical degree function is constructed for pseudomonotone mappings from a reflexive Banach space to its dual, using Galerkin approximations to yield a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form.
Abstract: A classical degree function is constructed for pseudomonotone mappings from a reflexive Banach space to its dual, using Galerkin approximations. This generalizes the Leray-Schauder degree when the Banach space is a Hilbert space and yields a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form.
53 citations
TL;DR: In this paper, the authors employ the LyapunovSchmidt method to obtain the minimal and maximal solutions as limits of monotone iterates for second-order boundary value problems.
Abstract: IN RECENT years, there has been an extensive study of the existence of periodic solutions [l, S-11, 14, 151. In [8, 111, the existence of solutions of first and second order PBVP (period boundary value problems) has been obtained by a novel approach of combining the classical method of lower and upper solutions and the method of alternative problems (LyapunovSchmidt method), which provide conditions that are easily verifiable and which covers previous known results of other authors. As a constructive method for obtaining extremal solutions of initial and boundary value problems, the monotone iterative procedure has been employed by several researchers [5-7, 11-13, 151. The objective of this paper is to employ this useful technique for second order PBVP to obtain the minimal and maximal solutions as limits of monotone iterates. Our method can be used to study semilinear parabolic initial boundary value problems and other problems at resonance.
53 citations
01 Dec 1983
TL;DR: It is proved that any such circuit or formula for detecting the existence of cliques in an N-node graph has at least 2Ω(Nε) gates for some ε > 0 independent of N.
Abstract: In this paper we consider monotone Boolean circuits with three alternations, in the order “or”, “and”, “or.” Whenever the number of alternations is limited to a fixed constant the formula and circuit size measures are polynomially related to each other. We shall therefore refer to this measure interchangeably as ΣπΣ-formula size or ΣπΣ-circuit size. We shall prove that any such circuit or formula for detecting the existence of cliques in an N-node graph has at least 2Ω(Ne) gates for some e > 0 independent of N.
TL;DR: If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ϱ−nn−32 (n → ∞) are obtained for several classes of tree structures.
TL;DR: In this paper, a distribution theory for liklihood ratio tests of homogeneity of a collection of normal means when the collection is "decreasing on the average" and for testing 'decreases on average' as a null hypothesis, is presented.
Abstract: : In certain problems, it may be expected that a regression function has a substantial overall tendency to be monotone and yet we may not be certain that all of the restrictions imposed by a simple order are satisfied. Distribution theory for liklihood ratio tests of homogeneity of a collection of normal means when the collection is 'decreasing on the average' and for testing 'decreasing on the average' as a null hypothesis, is presented. The restriction 'decreasing on the average' is less restrictive than the usual monotone restriction and allows the data to give rise to 'reversals' over short ranges of values of the parameter set. It is closely related to the 'starshaped ordering' restriction.
TL;DR: In this article, the existence result for maximal monotone maps with images not necessarily convex is given, where A is a maximal nonconvex map and F is a set-valued map.
Abstract: We give an existence result for
$$\dot x \in -- Ax + F(x)$$
whereA is a maximal monotone map andF is a set-valued map, with images not necessarily convex.
TL;DR: This paper shows that the anO(n) algorithm, presented for solving the two-dimensional hidden-line problem in ann-sided simple polygon, can be used to solve other geometric problems.
Abstract: Recently ElGindy and Avis (EA) presented anO(n) algorithm for solving the two-dimensional hidden-line problem in ann-sided simple polygon. In this paper we show that their algorithm can be used to solve other geometric problems. In particular, triangulating anL-convex polygon and finding the convex hull of a simple polygon can be accomplished inO(n) time, whereas testing a simple polygon forL-convexity can be done inO(n
2) time.
TL;DR: The classical degree function constructed earlier for pseudomonotone mappings has been used to develop a broader degree theory of classical type for the sum of a maximal monotone map from a reflexive Banach space to its dual together with a bounded pseudomonOTone map.
Abstract: The classical degree function constructed earlier for pseudomonotone mappings has been used to develop a broader degree theory of classical type for the sum of a maximal monotone map from a reflexive Banach space to its dual together with a bounded pseudomonotone map. The proof uses the generalized Yosida approximation of the maximal monotone mapping.
TL;DR: In this paper, a detailed study of properties of monotone mappings of a finite-dimensional space is presented, where the authors use the concept of Monotone Mapping (MOM) as a generalization of Galerkin's method to infinite-dimensional spaces.
Abstract: In the theories of ordinary differential equations, I to 's stochastic differential equations, and equations in Banach spaces one frequently uses the concept of monotone mapping, introduced in [ 1 ]. This concept is used not only in studying equations with monotone operators, but also in studying other equations (cf. [2-6]). The concept of monotonici ty turned out to be convenient, in the first place, because it is easy to pass to the limit under the sign of a monotone operator, and the investigation of certain equations or others in infinite-dimensional spaces with the help of the method of Galerkin usually reduces to the investigation of equations in finite-dimensional spaces. To the author 's mind there is interest in the detailed s tudy of propert ies of monotone mappings of a finite-dimensional space.
TL;DR: In this article, it was shown that fp converges uniformly as p → ∞ to a best L∞-approximation to f by elements of a closed convex cone comprised of non-decreasing functions.
01 Apr 1983
TL;DR: In this article, a criterion for the existence of at least one solution to the nonlinear problem u" + u = g(u) + h with w(0) = m(jt) = 0, where h E L2(0, w) and g is continuous monotone nonincreasing.
Abstract: A criterion is proved for the existence of at least one solution to the equation u" + u = g(u) + h with w(0) = m(jt) = 0, where h E L2(0, w) and g is continuous monotone nonincreasing. In a number of papers (5-7) we have considered the problem of the periodic solutions (harmonics) of Lienard systems, and we have given sufficient conditions for the existence of such solutions in a number of rather general situations. This we have done by using variants of the alternative method (cf. also (1,2,3,4,8)). We are interested here in the solutions of the nonlinear problem (1) u" + u = g(u) + h, u(0) = m(w) = 0, \g(u)\
TL;DR: The classical degree function developed earlier for pseudomonotone operators from a reflexive Banach space to its dual is extended for the special case of Sobolev spaces to the sum of a pseudomonOTone mapping and a strongly nonlinear lower order term.
Abstract: The classical degree function developed earlier for pseudomonotone operators from a reflexive Banach space to its dual is extended for the special case of Sobolev spaces to the sum of a pseudomonotone mapping (representing a nonlinear elliptic operator) and a strongly nonlinear lower order term.
01 Jan 1983
TL;DR: In this article, first-order equations in "conservation laws'' form perturbed by a semilinear nonlinearity of monotone type, and the known existence and uniqueness results for conservation laws are easily adapted to this situation and some qualitative properties of the solutions are discussed.
Abstract: The authors consider first-order equations in "conservation laws'' form perturbed by a semilinear nonlinearity of monotone type. The known existence and uniqueness results for conservation laws—results due to Kruzhkov—are easily adapted to this situation and some qualitative properties of the solutions are discussed.
TL;DR: For some classes K of mappings, this paper showed that the condition that all mappings between factor spaces of two given inverse systems are in K implies that the limit mapping between the inverse limit spaces is in K, for the classes of monotone, confluent and confluent mappings of compact spaces.
TL;DR: A new adaptive finite-difference scheme for scalar hyperbolic conservation laws is introduced and it is shown that the scheme is $L^1 $-stable in the sense of Kuznetsov, and that it generates convergent approximations for linear problems.
Abstract: A new adaptive finite-difference scheme for scalar hyperbolic conservation laws is introduced. A key aspect of the method is a new automatic mesh selection algorithm for problems with shocks. We show that the scheme is $L^1 $-stable in the sense of Kuznetsov, and that it generates convergent approximations for linear problems. Numerical evidence is presented that indicates that if an error of size $\varepsilon $ is required, our scheme takes at most $O(\varepsilon ^{ - 3} )$ operations. Standard monotone difference schemes can take up to $O(\varepsilon ^{ - 4} )$ calculations for the same problems.
TL;DR: In this article, an iterative method is presented which starting from a lower or from an upper periodic solution, provides a monotone sequence converging to a periodic solution of (1).
Abstract: An iterative method is presented which starting from a lower or from an upper periodic solution, provides a monotone sequence converging to a periodic solution of (1). With some restrictions on the growth off, the method extends to functional differential equations of type (1?). Two numerical examples with an "a posteriori" error analysis are given.
TL;DR: In this article, the problem of finding an extremal estimate of the probability of some event under restrictions on the probabilities of other events is investigated, and it is proved to be equivalent to combinatorial problems on monotone classes of -graphs.
Abstract: The problem of finding an extremal estimate of the probability of some event under restrictions on the probabilities of other events is investigated, and it is proved to be equivalent to combinatorial problems on monotone classes of -graphs. New probability inequalities are obtained as specific results. Bibliography: 22 titles.
01 Jan 1983
TL;DR: In this article, implicit approximate factorization algorithms are modified to use the monotonic switch in the type of finite-differencing that was developed by Godunov for the Euler equations.
Abstract: Numerical calculations of transonic flows by potential equations typically use algorithms that change the method of calculation for regions of subsonic and supersonic flow. In this paper, implicit approximate-factorization algorithms are modified to use the monotonic switch in the type of finite-differencing that was developed by Godunov for the Euler equations. Calculations of flows over airfoils by these algorithms are compared with calculations by other methods that are in common usage. For the small-disturbance potential equation, comparisons are made with the Murman-Cole method and the monotone method of Engquist and Osher for both steady and unsteady flows. For the full potential equation, comparisons are made with the methods of Jameson and of Holst and Ballhaus for steady flows. The comparisons show that the monotone methods are more stable. For steady flows, solutions are obtained for cases where the Murman-Cole switch requires a time step over ten times smaller in order for the calculations to remain stable. These improvements are achieved with no increase in computer storage and only minor modifications in current codes.
TL;DR: In this paper, a class of empirical Bayes estimators (EBE's) is proposed for estimating the natural parameter of a one-parameter exponential family, which possess the nice property of being monotone by construction.
Abstract: A class of empirical Bayes estimators (EBE's) is proposed for estimating the natural parameter of a one-parameter exponential family. In contrast to related EBE's proposed and investigated until now, the EBE's presented in this paper possess the nice property of being monotone by construction. Based on an arbitrary reasonable estimator of the underlying marginal density, a simple algorithm is given to construct a monotone EBE. Two representations of these EBE's are given, one of which serves as a tool in establishing asymptotic results, while the other one, related with isotonic regression, proves useful in the actual computation.