Showing papers on "Monotone polygon published in 2005"
TL;DR: In this article, the concept of isogeometric analysis is proposed and the basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model.
5,137 citations
TL;DR: It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset), which is surprising, as it is generally considered that convexity describes the local physical process of losing information.
Abstract: It is proven that logarithmic negativity does not increase on average under a general positive partial transpose preserving operation (a set of operations that incorporate local operations and classical communication as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is not a convex function this result is surprising, as it is generally considered that convexity describes the local physical process of losing information. The role of convexity and, in particular, its relation (or lack thereof) to physical processes is discussed and importance of continuity in this context is stressed.
1,087 citations
TL;DR: A prox-type method with efficiency estimate O(\epsilon^{-1}) for approximating saddle points of convex-concave C$^{1,1}$ functions and solutions of variational inequalities with monotone Lipschitz continuous operators is proposed.
Abstract: We propose a prox-type method with efficiency estimate $O(\epsilon^{-1})$ for approximating saddle points of convex-concave C$^{1,1}$ functions and solutions of variational inequalities with monotone Lipschitz continuous operators. Application examples include matrix games, eigenvalue minimization, and computing the Lovasz capacity number of a graph, and these are illustrated by numerical experiments with large-scale matrix games and Lovasz capacity problems.
980 citations
01 Jan 2005
TL;DR: In this article, a restricted but useful class of dynamical systems enjoying a comparison-principle with respect to a closed order relation on the state space is surveyed, variously called monotone, order-preserving or increasing.
Abstract: This chapter surveys a restricted but useful class of dynamical systems, namely, those enjoying a comparisonprinciple with respect to a closed order relation on the state space.Such systems, variously called monotone, order-preserving orincreasing, occur in many biological, chemical, physical and economicmodels.
763 citations
Book•
01 Jan 2005
TL;DR: This book discusses the formulation of a shape optimization problem, the problem of optimal partitions, and some open questions on Boundary variation for Neumann problems.
Abstract: * Preface * Introduction to Shape Optimization Theory and Some Classical Problems > General formulation of a shape optimization problem > The isoperimetric problem and some of its variants > The Newton problem of minimal aerodynamical resistance > Optimal interfaces between two media > The optimal shape of a thin insulating layer * Optimization Problems Over Classes of Convex Domains > A general existence result for variational integrals > Some necessary conditions of optimality > Optimization for boundary integrals > Problems governed by PDE of higher order * Optimal Control Problems: A General Scheme > A topological framework for general optimization problems > A quick survey on 'gamma'-convergence theory > The topology of 'gamma'-convergence for control variables > A general definition of relaxed controls > Optimal control problems governed by ODE > Examples of relaxed shape optimization problems * Shape Optimization Problems with Dirichlet Condition on the Free Boundary > A short survey on capacities > Nonexistence of optimal solutions > The relaxed form of a Dirichlet problem > Necessary conditions of optimality > Boundary variation > Continuity under geometric constraints > Continuity under topological constraints: Sverak's result > Nonlinear operators: necessary and sufficient conditions for the 'gamma'-convergence > Stability in the sense of Keldysh > Further remarks and generalizations * Existence of Classical Solutions > Existence of optimal domains under geometrical constraints > A general abstract result for monotone costs > The weak'gamma'-convergence for quasi-open domains > Examples of monotone costs > The problem of optimal partitions > Optimal obstacles * Optimization Problems for Functions of Eigenvalues > Stability of eigenvalues under geometric domain perturbation > Setting the optimization problem > A short survey on continuous Steiner symmetrization > The case of the first two eigenvalues of the Laplace operator > Unbounded design regions > Some open questions * Shape Optimization Problems with Neumann Condition on the Free Boundary > Some examples > Boundary variation for Neumann problems > General facts in RN > Topological constraints for shape stability > The optimal cutting problem > Eigenvalues of the Neumann Laplacian * Bibliography * Index
468 citations
Posted Content•
TL;DR: The authors proposed a portfolio selection model based on a class of preferences that coincide with mean-variance preferences on their domain of monotonicity, but differ where mean-variances preferences fail to be monotone.
Abstract: We propose a portfolio selection model based on a class of preferences that coincide with mean-variance preferences on their domain of monotonicity, but differ where mean-variance preferences fail to be monotone.
331 citations
TL;DR: In this article, an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space is introduced.
Abstract: In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.
293 citations
TL;DR: A general approach leading to a polynomial algorithm is presented for minimizing maximum power for a class of graph properties called monotone properties and a new approximation algorithm for the problem of minimizing the total power for obtaining a 2-node-connected graph is developed.
Abstract: Topology control problems are concerned with the assignment of power values to the nodes of an ad hoc network so that the power assignment leads to a graph topology satisfying some specified properties. This paper considers such problems under several optimization objectives, including minimizing the maximum power and minimizing the total power. A general approach leading to a polynomial algorithm is presented for minimizing maximum power for a class of graph properties called monotone properties. The difficulty of generalizing the approach to properties that are not monotone is discussed. Problems involving the minimization of total power are known to be NP-complete even for simple graph properties. A general approach that leads to an approximation algorithm for minimizing the total power for some monotone properties is presented. Using this approach, a new approximation algorithm for the problem of minimizing the total power for obtaining a 2-node-connected graph is developed. It is shown that this algorithm provides a constant performance guarantee. Experimental results from an implementation of the approximation algorithm are also presented.
244 citations
Book•
01 Jan 2005
TL;DR: In this article, the authors introduce the concepts of generalized convexity and quasiconvex analysis and define a set of criteria for generalized convexity in the differentiable case.
Abstract: The chapters are as follows: Introduction to Convex and Quasiconvex Analysis (J.B.G.Frenk, G. Kassay) Criteria for Generalized Convexity and Generalized Monotonicity in the Differentiable Case (J.-P. Crouzeix) Continuity and Differentiability of Quasiconvex Functions (J.-P. Crouzeix) Generalized Convexity and Optimality Conditions in Scalar and Vector Optimization (A. Cambini, L. Martein) Generalized Convexity in Vector Optimization (D. T. Luc) Generalized Convex Duality and Its Economic Applications (J.-E. Martinez-Legaz) Abstract Convexity (A.Rubinov, J.Dutta) Fractional programming (J.B.G. Frenk, S.Schaible) Generalized Monotone Maps (N.Hadjisavvas, S.Schaible) Generalized Convexity and Generalized Derivatives (S.Komlosi) Generalized Convexity, Generalized Monotonicity and Nonsmooth Analysis (N.Hadjisavvas) Pseudomonotone Complementarity Problems and Variational Inequalities (J.-C. Yao, O. Chadly) Generalized Monotone Equilibrium Problems and Variational Inequalities (I. Konnov) Uses of Generalized Convexity and Monotonicity in Economics (R. John)
229 citations
22 May 2005
TL;DR: This paper presents a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent, and shows that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding.
Abstract: This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle.
214 citations
TL;DR: This paper proposes a globally and quadratically convergent algorithm, which is based on smoothing and regularization methods, for solving monotone SOCCP, and studies strong semismoothness and Jacobian consistency, which play an important role in establishing quadratic convergence of the algorithm.
Abstract: The second-order cone complementarity problem (SOCCP) is a wide class of problems containing the nonlinear complementarity problem (NCP) and the second-order cone programming problem (SOCP). Recently, Fukushima, Luo, and Tseng [SIAM J. Optim., 12 (2001), pp. 436--460] extended some merit functions and their smoothing functions for NCP to SOCCP. Moreover, they derived computable formulas for the Jacobians of the smoothing functions and gave conditions for the Jacobians to be invertible. In this paper, we propose a globally and quadratically convergent algorithm, which is based on smoothing and regularization methods, for solving monotone SOCCP. In particular, we study strong semismoothness and Jacobian consistency, which play an important role in establishing quadratic convergence of the algorithm. Furthermore, we examine the effectiveness of the algorithm by means of numerical experiments.
TL;DR: A brief review of the main results in the theory of discrete-time monotone dynamics can be found in this paper, where the authors provide a brief summary of the results.
Abstract: The aim of this paper is to provide a brief review of the main results in the theory of discrete-time monotone dynamics.
TL;DR: This work presents near-linear time approximation algorithms that, given a parameter ε > 0, compute a simplified polygonal curve P’ whose error is less than ε and size at most the size of an optimal simplified Polygonal Curve with error ε/2.
Abstract: We consider the problem of approximating a polygonal curve P under a given error criterion by another polygonal curve P' whose vertices are a subset of the vertices of P. The goal is to minimize the number of vertices of P' while ensuring that the error between P' and P is below a certain threshold. We consider two different error measures: Hausdorff and Frechet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter ź > 0, compute a simplified polygonal curve P' whose error is less than ź and size at most the size of an optimal simplified polygonal curve with error ź/2. We consider monotone curves in ź2 in the case of the Hausdorff error measure under the uniform distance metric and arbitrary curves in any dimension for the Frechet error measure under Lp metrics. We present experimental results demonstrating that our algorithms are simple and fast, and produce close to optimal simplifications in practice.
TL;DR: The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton--Jacobi--Bellman equation.
Abstract: We obtain error bounds for monotone approximation schemes of Hamilton--Jacobi--Bellman equations. These bounds improve previous results of Krylov and the authors. The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton--Jacobi--Bellman equation.
TL;DR: The convergence of the approximations to the solution of the equations is proved and the Stochastic evolutional equations with monotone operators are considered in Banach spaces.
Abstract: Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.
TL;DR: In this article, it was shown that the shrinker entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders, and that W+ remains bounded for all time.
Abstract: Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij/∂t = − 2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ+ and v+. The forward reduced volume θ+ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/tn/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like tn/2(maximal volume growth) then W+, θ+ and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.
06 Jul 2005
TL;DR: A policy iteration algorithm for monotone self-maps of complete lattices for lattices arising in the interval abstraction of values of variables is introduced and analyzed.
Abstract: We present a new method for solving the fixed point equations that appear in the static analysis of programs by abstract interpretation. We introduce and analyze a policy iteration algorithm for monotone self-maps of complete lattices. We apply this algorithm to the particular case of lattices arising in the interval abstraction of values of variables. We demonstrate the improvements in terms of speed and precision over existing techniques based on Kleene iteration, including traditional widening/narrowing acceleration mecanisms.
TL;DR: In this paper, the authors explore representability of monotone operators by convex functions and characterize the class of operators that admit a convex representation as the one consisting of intersections of maximal monotones.
Abstract: A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.
TL;DR: The asymptotic behavior of the composition of two resolvents in a Hilbert space is investigated in this paper, where connections are made between the solutions of associated monotone inclusion problems and their dual versions.
Abstract: The asymptotic behavior of the composition of two resolvents in a Hilbert space is investigated. Connections are made between the solutions of associated monotone inclusion problems and their dual versions. The applications provided include a study of an alternating minimization procedure and a new proof of von Neumann's classical result on the method of alternating projections.
TL;DR: In this article, the existence of monotone positive solutions and a corresponding iterative scheme for the following multipoint boundary value problem with p-Laplacian operator was established. But the problem was not studied.
Abstract: In the paper, we obtain the existence of monotone positive solutions and establish acorresponding iterative scheme for the following multipoint boundary value problem with p-Laplacian operator, (@f"p(u^'))^'(t)+q(t)f(t,u(t))=0,0
TL;DR: This paper extends the Shu--Osher representations for any Runge--Kutta method giving sufficient conditions for monotonicity and shows how optimal Shu-- Osher representations can be constructed from the Butcher tableau of a Runge-KutTA method.
Abstract: Over the last few years a great effort has been made to develop monotone high order explicit Runge--Kutta methods by means of their Shu--Osher representations. In this context, the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation. In this paper we extend the Shu--Osher representations for any Runge--Kutta method giving sufficient conditions for monotonicity. We show how optimal Shu--Osher representations can be constructed from the Butcher tableau of a Runge--Kutta method.
The optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the Runge--Kutta method [L. Ferracina and M. N. Spijker, SIAM J. Numer. Anal., 42 (2004), pp. 1073--1093], and hence if this radius is zero, the method is not monotone. In the Shu--Osher representation, methods with zero radius require negative coefficients, and to deal with them, an extra associate problem is considered. In this paper we interpret these schemes as representations of perturbed Runge--Kutta methods. We extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity. Optimal representations can be constructed from the Butcher tableau of a perturbed Runge--Kutta method.
TL;DR: It is demonstrated that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem.
Abstract: We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.
TL;DR: In this paper, the authors consider some category-theoretic properties of Pos-S of all S-posets (posets equipped with a compatible right action of a pomonoid S), with monotone action-preserving maps between them.
Abstract: In this paper, we consider some category-theoretic properties of the
category Pos-S of all S-posets (posets equipped with a compatible right action of a pomonoid S), with monotone action-preserving maps between them. We first discuss some general category-theoretic ingredients of Pos-S; specifically, we characterize several kinds of epimorphisms and monomorphisms. Then, we present some adjoint relations of Pos-S with Pos, Set, and Act-S. In particular, we discuss free and cofree objects. We also examine other category-theoretic properties, such as cartesian closedness and monadicity. Finally, we consider projectivity in Pos-S with respect to regular epimorphisms and show that it is the same asprojectivity, although projectives are not generally retracts of free
objects over posets.
TL;DR: A new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems, and it is proved that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of $1-1/O(\sqrt{n})$.
Abstract: In this paper we propose a new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems. At each iteration, the method would select a target on the central path with a large update from the current iterate, and then the Newton method is used to get the search directions, followed by adaptively choosing the step sizes, which are, e.g., the largest possible steps before leaving a neighborhood that is as wide as the given ${\cal N}^-_{\infty}$ neighborhood. The only deviation from the classical approach is that we treat the classical Newton direction as the sum of two other directions, corresponding to, respectively, the negative part and the positive part of the right-hand side. We show that if these two directions are equipped with different and appropriate step sizes, then the method enjoys the low iteration bound of $O(\sqrt{n}\log L)$, where $n$ is the dimension of the problem and $L=\frac{(x^0)^Ts^0}{\ep}$ with $\ep$ the required precision and $(x^0,s^0)$ the initial interior solution. For a predictor-corrector variant of the method, we further prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of $1-1/O(\sqrt{n})$. Additionally, if the problem has a strict complementary solution, then the predictor steps converge Q-quadratically.
TL;DR: In this article, the authors provide necessary and sufficient behavioral conditions ensuring the countable additivity and non-atomicity of all priors in a multiple priors model, a la Gilboa and Schmeidler (1989).
Abstract: In a multiple priors model a la Gilboa and Schmeidler (1989), we provide necessary and sufficient behavioral conditions ensuring the countable additivity and non-atomicity of all priors.
TL;DR: In this paper, the concept of strictly monotone and absolutely continuous functions on was established and a criterion for absolute continuity of the inverse function of any strictly non-monotone function on is also obtained.
Abstract: In this paper, we establish the concept of absolutely continuous function on and we prove a characterization of such functions, which generalizes the one given for the real case in the classical Banach-Zarecki Theorem. Moreover, we prove that this kind of functions satisfy the Fundamental Theorem of Calculus. A criterion for absolute continuity of the inverse function of any strictly monotone and absolutely continuous function on is also obtained. †Research partially supported by D. G. I. and F.E.D.E.R. project BFM2001-3884-C02-01, and by Xunta of Galicia and F.E.D.E.R. project PGIDIT020XIC20703PN, Spain.
TL;DR: In this article, the authors considered the problem of finding solutions of systems of monotone equations and showed that the Newton-type algorithm still holds under a local error-bound assumption that is weaker than the standard nonsingularity condition.
Abstract: We consider the problem of finding solutions of systems of monotone equations. The Newton-type algorithm proposed in Ref. 1 has a very nice global convergence property in that the whole sequence of iterates generated by this algorithm converges to a solution, if it exists. Superlinear convergence of this algorithm is obtained under a standard nonsingularity assumption. The nonsingularity condition implies that the problem has a unique solution; thus, for a problem with more than one solution, such a nonsingularity condition cannot hold. In this paper, we show that the superlinear convergence of this algorithm still holds under a local error-bound assumption that is weaker than the standard nonsingularity condition. The local error-bound condition may hold even for problems with nonunique solutions. As an application, we obtain a Newton algorithm with very nice global and superlinear convergence for the minimum norm solution of linear programs.
TL;DR: In this article, the authors further developed a method for proving the global attractivity of steady states in certain classes of dynamical systems, where one views the given system as a negative feedback loop of a monotone controlled system, and the question of expressing a given system in the required negative feedback form is addressed.
Abstract: This paper further develops a method,
originally introduced by Angeli and the second author, for proving
global attractivity of steady states in certain classes of
dynamical systems. In this approach, one views the given system as
a negative feedback loop of a monotone controlled system. An
auxiliary discrete system, whose global attractivity implies that
of the original system, plays a key role in the theory, which is
presented in a general Banach space setting. Applications are
given to delay systems, as well as to systems with multiple inputs
and outputs, and the question of expressing a given system in the
required negative feedback form is addressed.
TL;DR: In this paper, several methods to construct triangular norms on partially ordered sets are proposed and several methods via Galois connections between bounded posets based on a categorical analysis of the construction of triangularnorm on closed intervals via monotone functions and their pseudo-inverses are proposed.
28 Oct 2005
TL;DR: The radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality was established in this paper.
Abstract: In this paper, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated with the classical and weighted Hardy-Littlewood-Sobolev inequality.