Showing papers on "Monotone polygon published in 1997"

Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations

Abstract: We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.

Interior-Point Methods for the Monotone Semidefinite Linear Complementarity Problem in Symmetric Matrices

Abstract: The SDLCP (semidefinite linear complementarity problem) in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization. It is defined as the problem of finding a pair $(\X,\Y)$ of $n \times n$ symmetric positive semidefinite matrices which lies in a given $n(n+1)/2$ dimensional affine subspace $\FC$ of $\SC^2$ and satisfies the complementarity condition $\X \bullet \Y = 0$, where $\SC$ denotes the $n(n+1)/2$-dimensional linear space of symmetric matrices and $\X \bullet \Y$ the inner product of $\X$ and $\Y$. The problem enjoys a close analogy with the LCP in the Euclidean space. In particular, the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace $\FC$. The aim of this paper is to establish a theoretical basis of interior-point methods with the use of Newton directions toward the central trajectory for the monotone SDLCP.

Solution of monotone complementarity problems with locally Lipschitzian functions

Abstract: The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsź(x)=0 or as the problem of minimizing the merit functionź=1/2źźź22, we extend results which hold for sufficiently smooth functionsF to the nonsmooth case. In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 źźź(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesź is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.

A Class of Projection and Contraction Methods for Monotone Variational-Inequalities

Abstract: In this paper we introduce a new class of iterative methods for solving the monotone variational inequalities $$u* \in \Omega , (u - u*)^T F(u*) \geqslant 0, \forall u \in \Omega $$ Each iteration of the methods presented consists essentially only of the computation ofF(u), a projection to Ω,v:=P Ω[u-F(u)], and the mappingF(v) The distance of the iterates to the solution set monotonically converges to zero Both the methods and the convergence proof are quite simple

A variant of korpelevich’s method for variational inequalities with a new search strategy

Abstract: We present a variant of Korpelevich's method for variational inequality problems with monotone operators. Instead of a fixed and exogenously given stepsize, possible only when a Lipschitz constant for the operator exists and is known beforehand, we find an appropriate stepsize in each iteration through an Armijo-type search. Differently from other similar schemes, we perform only two projections onto the feasible set in each iteration, rather than one projection for each tentative step during the search, which represents a considerable saving when the projection is computationally expensive. A full convergence analysis is given, without any Lipschitz continuity assumption

Vector Equilibrium Problems with Generalized Monotone Bifunctions

Abstract: A vector equilibrium problem is defined as follows: given a closed convex subset K of a real topological Hausdorff vector space and a bifunction F(x, y) valued in a real ordered locally convex vect...

Enlargement of Monotone Operators with Applications to Variational Inequalities

Abstract: Given a point-to-set operator T, we introduce the operator Te defined as Te(x)= {u: 〈 u − v, x − y 〉 ≥ −e for all y ɛ Rn, v ɛ T(y)}. When T is maximal monotone Te inherits most properties of the e-subdifferential, e.g. it is bounded on bounded sets, Te(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of Te, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 ɛ He(x), while the subproblems of the exact method are of the form 0 ɛ H(x). If ek is the coefficient used in the kth iteration and the ek's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.

Alternating Projection-Proximal Methods for Convex Programming and Variational Inequalities

Abstract: We consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem. We propose a solution method for this problem that alternates between a proximal step (for the maximal monotone operator part) and a projection-type step (for the monotone variational inequality part) and analyze its convergence and rate of convergence. This method extends a decomposition method of Chen and Teboulle [Math. Programming, 64 (1994), pp. 81--101] for convex programming and yields, as a by-product, new decomposition methods.

Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators

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Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

Abstract: In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented.

Monotonicity Considerations for Saturated--Unsaturated Subsurface Flow

Abstract: It is demonstrated that monotonicity is a sufficient condition to ensure that no new nonphysical local maxima and minima can be produced in the discrete nonlinear unsaturated flow equation. Monotonicity conditions are derived for various types of weighting for the mobility term. The basic discretization is of finite element type, but the results can be extended to finite volume discretizations. Central weightings are only conditionally monotone, while upstream weightings are unconditionally monotone. Sample computations are given for highly heterogeneous problems.

Finding a zero of the sum of two maximal monotone operators

Abstract: In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brezis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.

Lectures on maximal monotone operators

Abstract: These lectures will focus on those properties of maximal monotone operators which are valid in arbitrary real Banach spaces.

Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method

Abstract: The monotone iteration method is employed to establish the existence of traveling wave fronts in delayed reaction-diffusion systems with monostable nonlinearities.

The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws

Abstract: We are interested in the rate of convergence in L1 of the approximate solution of a conservation law generated by a monotone finite difference scheme. Kuznetsov has proved that this rate is 1/2 [USSR Comput. Math. Math. Phys., 16 (1976), pp. 105--119 and Topics Numer. Anal. III, in Proc. Roy. Irish Acad. Conf., Dublin, 1976, pp. 183--197], and recently Teng and Zhang have proved this estimate to be sharp for a linear flux [SIAM J. Numer. Anal., 34 (1997), pp. 959--978]. We prove, by constructing appropriate initial data for the Cauchy problem, that Kuznetsov's estimates are sharp for a nonlinear flux as well.

Monotone quantities and unique limits for evolving convex hypersurfaces

Abstract: The aim of this paper is to introduce a new family of monotone integral quantities associated with certain parabolic evolution equations for hypersurfaces; and to deduce from these some results about the limiting behaviour of the evolving hypersurfaces. A variety of parabolic equations for hypersurfaces have been considered. One of the earliest was the Gauss curvature flow; introduced in [Fi] as a model for the changing shape of a stone wearing on a beach. The stone is represented by a bounded convex region; and each point on its surface moves in the inward normal direction with speed equal to the Gauss curvature: If the surface at timet is given by an embeddingxt; then @x @t DiK n; whereK is the Gauss curvature; and n the outward unit normal. Firey showed that stones which are symmetric about the origin shrink to points in finite time; and are asymptotically spherical in shape. Other evolution equations have been considered since then; of the form

Definition and existence of renormalized solutions of elliptic equations with general measure data

Abstract: We introduce a new definition of solution for the nonlinear monotone elliptic problem-div(a(a;, ∇u)) = μ in Ω u = 0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω. We prove the existence of such a solution, a stability result, and partial uniqueness results.

Nonovershooting and monotone nondecreasing step responses of a third-order SISO linear system

Abstract: This paper presents the necessary and sufficient conditions for a third-order single-input/single-output (SISO) linear system to have a nonovershooting (or monotone nondecreasing) step response. If the transfer function of an overall system has real poles, a necessary and sufficient condition is found for the nonovershooting (or monotone nondecreasing) step response. In the case of complex poles, one sufficient condition and two necessary conditions are obtained. The resulting conditions are all in terms of the coefficients of the numerator of the transfer function. Simple calculations can be used to check a system for the nonovershooting (or monotone nondecreasing) step response. Another feature is that the conditions in terms of pole-zero configurations can be easily derived from the present results.

Which Fixed Point Does the Iteration Method Select

Abstract: An approximate diagonal version of the iteration method for a nonexpansive self mapping of a Banach space (hence with possible non unique fixed-point if any) is considered. Two kinds of properties satisfied by this method that cover widely recent results on the prox or gradient-prox method for convex optimization and monotone inclusions are presented: localization and selection.

Pythagorean-hodograph quintic transition curves of monotone curvature

Abstract: Walton and Meek [Walton, D.J. and Meek, D.S., A Pythagorean-hodograph quintic spiral. Computer Aided Design , 1996, 28 , 943–950] have recently advocated the use of Pythagorean-hodograph quintics of monotone curvature, or “PH spirals” for short, as transitional elements that give G 2 connections of linear and circular arcs in applications such as layout of highways and railways—in which context PH curves provide the important advantage of rational offsets and exact rectifications . They construct a PH quintic, interpolating an initial point and tangent and a final tangent, with monotone curvature variation from zero to a given (extremum) final value. We show that using the complex representation for PH curves greatly simplifies this problem and also reveals that the method of Walton and Meek yields a special instance among a one-parameter family of solutions. The additional degree of freedom of PH spirals identified herein relaxes constraints otherwise required to guarantee their existence, and offers the designer precise control over their total length and/or the ability to fine-tune their curvature distributions.

Matrix Searching with the Shortest-Path Metric

Abstract: We present an O(n) time algorithm for computing row-wise maxima or minima of an implicit, totally monotone $n \times n$ matrix whose entries represent shortest-path distances between pairs of vertices in a simple polygon. We apply this result to derive improved algorithms for several well-known problems in computational geometry. Most prominently, we obtain linear-time algorithms for computing the geodesic diameter, all farthest neighbors, and external farthest neighbors of a simple polygon, improving the previous best result by a factor of O(log n) in each case.

An Infeasible Path-Following Method for Monotone Complementarity Problems

Abstract: We propose an infeasible path-following method for solving the monotone complementarity problem. This method maintains positivity of the iterates and uses two Newton steps per iteration---one with a centering term for global convergence and one without the centering term for local superlinear convergence. We show that every cluster point of the iterates is a solution, and if the underlying function is affine or is sufficiently smooth and a uniform nondegenerate function on $\Re_{++}^n$, then the convergence is globally Q-linear. Moreover, if every solution is strongly nondegenerate, the method has local quadratic convergence. The iterates are guaranteed to be bounded when either a Slater-type feasible solution exists or when the underlying function is an R0-function.

A priori error estimates for numerical methods for scalar conservation laws part II: flux-splitting monotone schemes on irregular Cartesian grids

Abstract: This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (Δx) 1/2 in L∞(L 1 ) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p-consistent schemes, and prove that they converge to the entropy solution with the rate of (Δx) min{1/2,P} in L∞(L 1 ); again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (Δx) 1/2 in L∞(L 1 ). We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.

Simultaneous vector variational inequalities and vector implicit complementarity problem

Abstract: The concepts of monotone pair of mappings and vector implicit complementarity problem (VICP) in ordered Banach spaces are introduced. Existence theorems for simultaneous vector variational inequalities (VVI) and VICP are proved.

Fitting Monotone Surfaces to Scattered Data Using C 1 Piecewise Cubics

Abstract: We derive sufficient conditions on the Bezier net of a Bernstein--Bezier polynomial defined on a triangle in the plane to insure that the corresponding surface is monotone. We then apply these conditions to construct a new algorithm for fitting a monotone surface to gridded data. The method uses C1 cubic splines defined on the triangulation obtained by drawing both diagonals of each subrectangle. In addition, we present an algorithm for the monotone scattered data interpolation problem which is based on a method for creating gridded data from the scattered data. Numerical results for several test examples are presented.

The computational intractability of training sigmoidal neural networks

Abstract: We demonstrate that the problem of approximately interpolating a target function by a neural network is computationally intractable. In particular the interpolation training problem for a neural network with two monotone Lipschitzian sigmoidal internal activation functions and one linear output node is shown to be NP-hard and NP-complete if the internal nodes are in addition piecewise ratios of polynomials. This partially answers a question of Blum and Rivest (1992) concerning the NP-completeness of training a logistic sigmoidal 3-node network. An extension of the result is then given for networks with n monotone sigmoidal internal nodes and one convex output node. This indicates that many multivariate nonlinear regression problems may be computationally infeasible.

Malicious Omissions and Errors in Answers to Membership Queries

Abstract: We consider two issues in polynomial-time exact learning of concepts using membership and equivalence queries: (1) errors or omissions in answers to membership queries, and (2) learning finite variants of concepts drawn from a learnable class. To study (1), we introduce two new kinds of membership queries: limited membership queries and malicious membership queries. Each is allowed to give incorrect responses on a maliciously chosen set of strings in the domain. Instead of answering correctly about a string, a limited membership query may give a special “I don‘t know” answer, while a malicious membership query may give the wrong answer. A new parameter L is used to bound the length of an encoding of the set of strings that receive such incorrect answers. Equivalence queries are answered correctly, and learning algorithms are allowed time polynomial in the usual parameters and L. Any class of concepts learnable in polynomial time using equivalence and malicious membership queries is learnable in polynomial time using equivalence and limited membership queries; the converse is an open problem. For the classes of monotone monomials and monotone k -term DNF formulas, we present polynomial-time learning algorithms using limited membership queries alone. We present polynomial-time learning algorithms for the class of monotone DNF formulas using equivalence and limited membership queries, and using equivalence and malicious membership queries. To study (2), we consider classes of concepts that are polynomially closed under finite exceptions and a natural operation to add exception tables to a class of concepts. Applying this operation, we obtain the class of monotone DNF formulas with finite exceptions. We give a polynomial-time algorithm to learn the class of monotone DNF formulas with finite exceptions using equivalence and membership queries. We also give a general transformation showing that any class of concepts that is polynomially closed under finite exceptions and is learnable in polynomial time using standard membership and equivalence queries is also polynomial-time learnable using malicious membership and equivalence queries. Corollaries include the polynomial-time learnability of the following classes using malicious membership and equivalence queries: deterministic finite acceptors, boolean decision trees, and monotone DNF formulas with finite exceptions.