Showing papers on "Monotone polygon published in 2009"
TL;DR: It is shown that the PSPACE upper bounds cannot be substantially improved without a breakthrough on long standing open problems: the square-root sum problem and an arithmetic circuit decision problem that captures P-time on the unit-cost rational arithmetic RAM model.
Abstract: We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well-studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP).We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminatesq These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, and the quantitative problems of comparing the probabilities with a given bound, or approximating them to desired precision.We show that all these problems can be solved in PSPACE using a decision procedure for the Existential Theory of Reals. We provide a more practical algorithm, based on a decomposed version of multi-variate Newton's method, and prove that it always converges monotonically to the desired probabilities. We show this method applies more generally to any monotone polynomial system. We obtain polynomial-time algorithms for various special subclasses of RMCs. Among these: for SCFGs and MT-BPs (equivalently, for 1-exit RMCs) the qualitative problem can be solved in P-time; for linearly recursive RMCs the probabilities are rational and can be computed exactly in P-time.We show that our PSPACE upper bounds cannot be substantially improved without a breakthrough on long standing open problems: the square-root sum problem and an arithmetic circuit decision problem that captures P-time on the unit-cost rational arithmetic RAM model. We show that these problems reduce to the qualitative problem and to the approximation problem (to within any nontrivial error) for termination probabilities of general RMCs, and to the quantitative decision problem for termination (extinction) of SCFGs (MT-BPs).
632 citations
TL;DR: In this paper, a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem was introduced, and some strong convergence theorems for approximating the common elements of the above three sets are obtained.
Abstract: In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for α -inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings. The results presented in the paper extend and improve some recent results of Yao and Yao [Y.Y. Yao, J.C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Appl. Math. Comput. 186 (2) (2007) 1551–1558], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings, Appl. Math. Comput. (2007) doi:10.1016/j.amc.2007.07.075], S. Takahashi and W. Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for Equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006) 506–515], Su, Shang and Qin [Y.F. Su, M.J. Shang, X.L. Qin, An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. (2007) doi:10.1016/j.na.2007.08.045] and Chang, Cho and Kim [S.S. Chang, Y.J. Cho, J.K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dynam. Systems Appl. (in print)].
213 citations
04 Jan 2009
TL;DR: In this article, the problem of finding a monotone, submodular function f on a ground set of size n, after only poly(n) oracle queries was considered.
Abstract: Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization.In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function f such that, for every set S, f(S) approximates f(S) within a factor α(n), where α(n) = √n+1 for rank functions of matroids and α(n), = O(√n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids.Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω(√n/log n), even for rank functions of a matroid.
156 citations
TL;DR: The developed monotone method does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation and the second-order convergence rate is verified with numerical experiments.
152 citations
TL;DR: The existence of an entire solution which behaves as two monotone waves propagating from both sides of the x-axis is proved, where an entire solutions is meant by a classical solution defined for all space and time variables.
Abstract: We deal with a system of Lotka–Volterra competition-diffusion equations on $\mathbb{R}$, which is a competing two species model with diffusion. It is known that the equations allow traveling waves with monotone profile. In this article we prove the existence of an entire solution which behaves as two monotone waves propagating from both sides of the x-axis, where an entire solution is meant by a classical solution defined for all space and time variables. The global dynamics for this entire solution exhibits the extinction of the inferior species by the superior one invading from both sides. The proof is carried out by applying the comparison principle for the competition-diffusion equations, that is, using an appropriate pair of a subsolution and a supersolution.
136 citations
TL;DR: This article proposed a portfolio selection model based on a class of monotone preferences that coincide with mean-variance preferences on their domain of monotonicity, but differ where mean-variances preferences fail to be monotonous and are therefore not economically meaningful.
Abstract: We propose a portfolio selection model based on a class of monotone preferences that coincide with mean-variance preferences on their domain of monotonicity, but differ where mean-variance preferences fail to be monotone and are therefore not economically meaningful. The functional associated with this new class of preferences is the best approximation of the mean-variance functional among those which are monotonic. We solve the portfolio selection problem and we derive a monotone version of the capital asset pricing model (CAPM), which has two main features: (i) it is, unlike the standard CAPM model, arbitrage free, (ii) it has empirically testable CAPM-like relations. The monotone CAPM has thus a sounder theoretical foundation than the standard CAPM and a comparable empirical tractability.
134 citations
Journal Article•
TL;DR: The problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere is considered, after only poly(n) oracle queries, and it is shown that no algorithm can achieve a factor better than Ω(√n/log n), even for rank functions of a matroid.
Abstract: Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization.In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function f such that, for every set S, f(S) approximates f(S) within a factor α(n), where α(n) = √n+1 for rank functions of matroids and α(n), = O(√n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids.Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω(√n/log n), even for rank functions of a matroid.
128 citations
TL;DR: A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist.
Abstract: A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces, and its convergence is established under the assumption that solutions exist. Unlike existing alternating algorithms, which are limited to two variables and linear coupling, our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions, variational problems, best approximation, and network flows.
127 citations
TL;DR: In this paper, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.
Abstract: In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.
123 citations
TL;DR: In this paper, two Bayesian approaches to non-parametric monotone function estimation are proposed, one based on a hierarchical Bayes framework and a characterization of smooth functions given by Ramsay that allows unconstrained estimation, and the other using a Bayesian regression spline model of Smith and Kohn.
Abstract: The paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.
111 citations
TL;DR: A general projective framework for finding a zero of the sum of $n$ maximal monotone operators over a real Hilbert space is described, which gives rise to a family of splitting methods of unprecedented flexibility.
Abstract: We describe a general projective framework for finding a zero of the sum of $n$ maximal monotone operators over a real Hilbert space Unlike prior methods for this problem, we neither assume $n=2$ nor first reduce the problem to the case $n=2$ Our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator At the cost of a single, computationally simple projection step, this framework gives rise to a family of splitting methods of unprecedented flexibility: numerous parameters, including the proximal stepsize, may vary by iteration and by operator The order of operator evaluation may vary by iteration and may be either serial or parallel The analysis essentially generalizes our prior results for the case $n=2$ We also include a relative error criterion for approximately evaluating resolvents, which was not present in our earlier work
TL;DR: In this article, a spectral gradient projection algorithm for monotone nonlinear equations with convex constraints is proposed, which is obtained by combining a modified spectral gradient method and a projection method.
01 Jan 2009
TL;DR: In this paper, the authors consider convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property, and show that these maps have the same properties as risk measures.
Abstract: This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps defined on spaces of random variables, possibly with the so-called Fatou property.
Posted Content•
TL;DR: In this paper, a survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces, focusing on the comparison of the continuous time trajectories to sequences generated by implicit or explicit discrete time schemes.
Abstract: This survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces. The emphasis is in the comparison of the continuous time trajectories to sequences generated by implicit or explicit discrete time schemes. The analysis covers weak convergence for the average process, for the process itself and strong convergence and aims at highlighting the main ideas and unifying the proofs. We further make the connection with the analysis in terms of almost orbits that allows for a broader scope.
TL;DR: The Banach contraction mapping principle is used to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition recently introduced by Mastroeni.
Abstract: We make use of the Banach contraction mapping principle to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition recently introduced by Mastroeni. We then modify the proposed algorithm to obtain a line search-free algorithm which does not require the Lipschitz-type condition. We apply the proposed algorithms to implement inexact proximal methods for solving monotone (not necessarily strongly monotone) Ky Fan inequalities. Applications to variational inequality and complementarity problems are discussed. As a consequence, a linearly convergent derivative-free algorithm without line search for strongly monotone nonlinear complementarity problem is obtained. Application to a Nash-Cournot equilibrium model is discussed and some preliminary computational results are reported.
Journal Article•
TL;DR: In this paper, a survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces, focusing on the comparison of continuous time trajectories to sequences generated by implicit or explicit discrete time schemes.
Abstract: This survey is devoted to the asymptotic behavior of solutions of evolution equations generated by maximal monotone operators in Hilbert spaces. The emphasis is in the comparison of continuous time trajectories to sequences generated by implicit or explicit discrete time schemes. The analysis covers weak convergence for the average process, for the process itself and strong convergence. The aim is to highlight the main ideas and unifying the proofs. Furthermore the connection is made with the analysis in terms of almost orbits that allows for a broader scope.
TL;DR: It is proved that a monotone input-output behavior is always obtained for models with three input variables that apply TP and amonotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape.
Abstract: At first sight, it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model. However, this is not the case. In this study, we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone input-output behavior. In the Mamdani-Assilian models considered in this study, the linguistic values of the input variables, as well as the output variable, are described by trapezoidal membership functions that form a fuzzy partition, the rule base is monotone, and the crisp output is obtained by the center-of-gravity (COG) defuzzification method. It is verified that for each of the three basic t-norms, i.e., the minimum TM, the product TP, and the Lukasiewicz t-norm TL, a monotone input-output behavior is obtained for any monotone rule base, or at least for any monotone smooth rule base. The outcome of this study is a guideline for designers of monotone linguistic fuzzy models. For the t-norms TM and TL, models with a single input variable show a monotone input-output behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length. The latter restriction can easily be bypassed by an auxiliary interpolation procedure. For the t-norm TP, models with a single input variable show a monotone input-output behavior for any monotone rule base and any fuzzy output partition. When designing a monotone model with more than one input variable, one should opt for the t-norm TP and use a monotone smooth rule base. It is shown that monotonicity of models with two input variables that apply TP is guaranteed for any monotone smooth rule base and any fuzzy output partition. Finally, it is proved that a monotone input-output behavior is always obtained for models with three input variables that apply TP and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape.
TL;DR: Using the method of upper and lower solutions, an existence result for IVP of Riemann-Liouville fractional differential equation is studied in this paper, and the monotone iterative technique is developed and the existence results for maximal and minimal solutions are obtained.
Abstract: Using the method of upper and lower solutions, an existence result for IVP of Riemann–Liouville fractional differential equation is studied. Also, the monotone iterative technique is developed and the existence results for maximal and minimal solutions are obtained.
TL;DR: In this paper, a class of approximation schemes based on differencing and interpolation was introduced for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily degenerate.
Abstract: For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily degenerate. In general such schemes have to have a wide stencil. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes includes new first and higher order versions. The methods are easy to implement and more efficient than some other known schemes. We prove consistency and stability of the methods, and for the monotone first order methods, we prove convergence in the general case and robust error estimates in the convex case. The methods are extensively tested.
TL;DR: An H ∞ adaptive estimator that extends easily the obtained results to systems with unknown parameters in the presence of disturbances is proposed and is shown through a numerical example with a polynomial nonlinearity.
TL;DR: A new iterative scheme based on the extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem, theSet of fixed points of a family of finitely nonexpansive mappings and the setof solutions of the variational inequality for a monotone, Lipschitz continuous mapping is proposed.
TL;DR: It is proved that the proposed method is globally convergent if the equation is monotone and Lipschitz continuous without any differentiability requirement on the equation.
TL;DR: In this article, the dimension-free Harnack inequality was established for stochastic porous media equations and the existence of a spectral gap was shown for the associated transition semigroups.
Abstract: As a Generalization to Wang (Ann Probab 35:1333–1350, 2007) where the dimension-free Harnack inequality was established for stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space.
TL;DR: This work presents a parallel splitting augmented Lagrangian method (abbreviated to PSALM), which can be extended to solve the system of equilibrium problems with three separable operators and why it is explained why the same technique cannot be used to develop similar methods for problems with more than three separables operators.
Abstract: The typical structured variational inequalities can be interpreted as a system of equilibrium problems with a leader and two cooperative followers. Assume that, based on the instruction given by the leader, each follower can solve the individual equilibrium sub-problems in his own way. The responsibility of the leader is to give a more reasonable instruction for the next iteration loop based on the feedback information from the followers. This consideration leads us to present a parallel splitting augmented Lagrangian method (abbreviated to PSALM). The proposed method can be extended to solve the system of equilibrium problems with three separable operators. Finally, it is explained why we cannot use the same technique to develop similar methods for problems with more than three separable operators.
TL;DR: In this paper, the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions was investigated using iterative methods and numerical simulations were also presented.
Abstract: This work is concerned with the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions. The results are obtained by using the monotone iteration method and they extend early works on beams with null boundary conditions. Numerical simulations are also presented.
TL;DR: In this article, the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations is discussed and new comparison principles and existence results for extremal solutions are established using the monotone iterative technique.
Abstract: This paper discusses the periodic boundary value problem for first-order nonlinear impulsive integro-differential equations. We prove some new comparison principles, and then establish new existence results for extremal solutions by using these principles and the monotone iterative technique. These results improve and extend all known relevant conclusions from the literature. Some examples are also given to illustrate the advantage of our results.
TL;DR: A subset M of a topological vector space X is said to be dense-lineable in X if there exists an infinite dimensional linear manifold in M ∪ { 0 } and dense in X as mentioned in this paper.
TL;DR: In this paper, a cell-centered finite volume method for discretization of diffusion equations on conformal polyhedral meshes is proposed. But the method is not interpolation-free.
Abstract: Abstract We have developed a new monotone cell-centered finite volume method for the discretization of diffusion equations on conformal polyhedral meshes. The proposed method is based on a nonlinear two-point flux approximation. For problems with smooth diffusion tensors and Dirichlet boundary conditions the method is interpolation-free. An adaptive interpolation is applied on faces where diffusion tensor jumps or Neumann boundary conditions are imposed. The interpolation is based on physical relationships such as continuity of the diffusion flux. The second-order convergence rate is verified with numerical experiments.
TL;DR: This article considers monotone nonparametric regression in a Bayesian framework and finds that the two-sided power distribution function is well suited both from a computational and mathematical point of view.
Abstract: In this article, we consider monotone nonparametric regression in a Bayesian framework. The monotone function is modeled as a mixture of shifted and scaled parametric probability distribution functions, and a general random probability measure is assumed as the prior for the mixing distribution. We investigate the choice of the underlying parametric distribution function and find that the two-sided power distribution function is well suited both from a computational and mathematical point of view. The model is motivated by traditional nonlinear models for dose-response analysis, and provides possibilities to elicitate informative prior distributions on different aspects of the curve. The method is compared with other recent approaches to monotone nonparametric regression in a simulation study and is illustrated on a data set from dose-response analysis.
TL;DR: Tada and Takahashi as discussed by the authors introduced an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping.
Abstract: In this paper, we introduce an iterative scheme by a new hybrid method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings in a real Hilbert space. We show that the iterative sequence converges strongly to a common element of the above three sets under some parametric controlling conditions by the new hybrid method which is introduced by Takahashi et al. (J. Math. Anal. Appl., doi: 10.1016/j.jmaa.2007.09.062, 2007). The results are connected with Tada and Takahashi’s result [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mappings and an equilibrium problem, J. Optim. Theory Appl. 133, 359–370, 2007]. Moreover, our result is applicable to a wide class of mappings.