Strong Monotonicity Property Research Articles

Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations

The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an L^2-type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.

Covering a compact space by fixed-radius or growing random balls

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a weak concentration bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the former concerning coverage by i.i.d. random sets with arbitrary distribution, and the latter concerning hitting times for Markov chains with a strong monotonicity property. The growth model seems generally more tractable, and we record some basic results and open problems for that model.

Determining closest targets on the extended facet production possibility set in data envelopment analysis: Modeling and computational aspects

Within the framework of data envelopment analysis (DEA) methodology, the problem of determining the closest targets on the efficient frontier is receiving increased attention from both academics and practitioners. In the literature, the number of approaches to this problem are increasing, most of which are based on the computation of closest targets. Some of the existing approaches satisfy the important property of strong monotonicity. However, they tend to either propose a complex conceptual framework and multi-stage procedure or change the original definition of Hölder distance functions. Clearly, these approaches cannot be solved easily when there are many “extreme” efficient units with multiple inputs and multiple outputs. To solve this problem, we consider the notion of the extended facet production possibility set (EFPPS). In particular, we propose a Mixed Integer Linear Program (MILP) to find the closest efficient targets and that is related to a measure that satisfies the strong monotonicity property. Additionally, in this paper, the proposed approach is applied to real data from 38 universities involved in China's 985 university project.

Single-Timescale Distributed GNE Seeking for Aggregative Games Over Networks via Forward–Backward Operator Splitting

We consider aggregative games with affine coupling constraints, where agents have partial information on the aggregate value and can only communicate with neighbouring agents. We propose a single-layer distributed algorithm that reaches a variational generalized Nash equilibrium, under constant step sizes. The algorithm works on a single timescale, i.e., does not require multiple communication rounds between agents before updating their action. The convergence proof leverages an invariance property of the aggregate estimates and relies on a forward-backward splitting for two preconditioned operators and their restricted (strong) monotonicity properties on the consensus subspace.

General regularized nonconvex mixed equilibrium problems: iterative methods by auxiliary principle technique

This paper introduces a new class of equilibrium problems named general regularized nonconvex mixed equilibrium problem. By using the auxiliary principle technique, some predictor–corrector methods for solving such class of general regularized nonconvex mixed equilibrium problems are suggested and analyzed. The study of convergence analysis of the proposed iterative algorithms requires either jointly pseudomonotonicity or partially mixed relaxed and strong monotonicity property of the operator and bifunction involved in the considered problem. As a consequence of our main results, we provide the correct versions of the algorithms and results presented in the literature.

A multi-component enhanced Russell measure of efficiency: With application to water supply plans

Public water providers aim at developing a water supply plan (WSP) that not only provides a reliable and satisfactory level of service but also is efficient in terms of performance. This paper deals with evaluating the performance of WSPs within the framework of multi-component data envelopment analysis. Specifically, we consider the overall performance of each WSP as a decision making unit (DMU) so that economic, social, hygienic, technological, managerial and environmental performances of the WSP make up independent components of the defined DMU. To assess the performance of a set of WPSs, we propose a multi-component enhanced Russell measure of efficiency that takes all sources of inefficiency into account. We show that the proposed measure can be decomposed into individual efficiency measures at component level. This decomposition much enhances the efficiency of computing the proposed measure, noting the fact that this measure is obtained by solving a single linear program. It also guarantees the proposed measure to inherit two important—unit invariance and strong monotonicity—properties of the conventional enhanced Russell measure. In our empirical study, we apply our model to evaluate the efficiency of 10 urban WSPs in Qom city of Iran. In line with experts’ practical opinions, our findings reveal that the (relatively) most efficient WSP is to construct a potable water network and separation of non-drinking water network for urban usage.

On the Strong Monotonicity of the Two-Time-Level CABARET Scheme

The concept of the strong monotonicity of the two-time-level CABARET scheme is introduced. It suggests that the difference scheme does not increase the number of the generalized local extrema in the difference solution when passing from one time level to another. The special modification of the two-time-level CABARET scheme having a strong monotonicity property is proposed. The test calculations are presented that illustrate this property of the modified CABARET scheme.

The convexification effect of Minkowski summation

Let us define for a compact set A \subset \mathbb{R}^n the sequence A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big). It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k) approaches the convex hull of A in the Hausdorff distance induced by the Euclidean norm as k goes to \infty . We explore in this survey how exactly A(k) approaches the convex hull of A , and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on \mathbb R^n , the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of A(k) does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets A with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence A(k) , and both the Hausdorff distance and effective standard deviation are eventually monotone (once k exceeds n ). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets (showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.

The Incentive Core in Co-Investment Problems

We study resource-monotonicity properties of core allocations in co-investment problems: those where a set of agents pool their endowments of a certain resource or input in order to obtain a joint surplus or output that must be allocated among the agents. We analyze whether agents have incentives to raise their initial contribution (resource-monotonicity). We focus not only on looking for potential incentives to agents who raise their contributions, but also in not harming the payoffs to the rest of agents (strong monotonicity property). A necessary and sufficient condition to fulfill this property is stated and proved. We also provide a subclass of co-investment problems for which any core allocation satisfies the aforementioned strong resource-monotonicity property. Moreover, we introduce the subset of core allocations satisfying this condition, namely the incentive core.

Semiflows “Monotone with Respect to High-Rank Cones" on a Banach Space

We consider semiflows in general Banach spaces motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov functionals. These semiflows enjoy strong monotonicity properties with respect to “cones” of high ranks, which imply order-related structures on the $\omega$-limit sets of precompact semiorbits. We show that for a pseudoordered precompact semiorbit the $\omega$-limit set $\omega(x)$ either is ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. In particular, we show that if $\omega(x)$ contains no equilibrium, then $\omega(x)$ is ordered, and hence the dynamics of the semiflow on $\omega(x)$ is topologically conjugate to a compact flow on $\mathbb{R}^k$ with $k$ being the rank. We also establish a Poincare--Bendixson-type theorem in the case where $k=2$. All our results are established without the smoothness condition on the semiflow, allowing applications to many cellular or physiological feedback systems wit...

High-order filtered scheme for front propagation problems

In this work we develop a specific application of the scheme proposed and analyzed in [1] to front propagation problems. The approach is based on the level-set method which leads in the isotropic case to a classical evolutive first order Hamilton- Jacobi equation.We will apply to this equation high-order “filtered schemes”, for these schemes the strong monotonicity property will not be satisfied. However, a weak є-monotonicity property applies and this is enough to obtain a convergence result and a precise error estimate. In the last section we will present several examples where we solve front propagation problems by filtered scheme in two and three dimensions showing the accuracy of our method.

Weak Concentration for First Passage Percolation Times on Graphs and General Increasing Set-valued Processes

Author(s): Aldous, DJ | Abstract: A simple lemma bounds s.d.(T )/ET for hitting times T in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result concerns a model of first passage percolation on a finite graph, where the traversal times of edges are independent Exponentials with arbitrary rates. Consider the percolation time X between two arbitrary vertices. We prove that s.d.(X)/EX is small if and only if Ξ/EX is small, where Ξ is the maximal edge-traversal time in the percolation path attaining X.

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model.

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.

Lipschitz Continuity of the Solution Mapping of Symmetric ConeComplementarity Problems

This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform Cartesian P‐property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultra P‐property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultra P‐property, the Cartesian P‐property, and the Lipschitz continuity of the solutions are all equivalent to each other.

On the Lipschitz Continuity of the Solution Map in Semidefinite Linear Complementarity Problems

In this paper, we investigate the Lipschitz continuity of the solution map in semidefinite linear complementarity problems. For a monotone linear transformation defined on the space of real symmetric n × n matrices, we show that the Lipschitz continuity of the solution map implies the globally uniquely solvable (GUS)-property. For Lyapunov transformations with the Q-property, we prove that the Lipschitz continuity of the solution map is equivalent to the strong monotonicity property. For the double-sided multiplicative transformations, we show that the Lipschitz continuity of the solution map implies the GUS-property.

Relationship Between Strong Monotonicity Property, P2-Property, and the Gus-Property in Semidefinite Linear Complementarity Problems

In a recent paper on semidefinite linear complementarity problems, Gowda and Song (2000) introduced and studied the P-property, P2-property, GUS-property, and strong monotonicity property for linear transformation L: Sn → Sn, where Sn is the space of all symmetric and real n × n matrices. In an attempt to characterize the P2-property, they raised the following two questions: (i) Does the strong monotonicity imply the P2-property? (ii) Does the GUS-property imply the P2-property? In this paper, we show that the strong monotonicity property implies the P2-property for any linear transformation and describe an equivalence between these two properties for Lyapunov and other transformations. We show by means of an example that the GUS-property need not imply the P2-property, even for Lyapunov transformations.

Continuum Limit for Some Growth Models II

We continue our investigations on a class of growth models introduced in a previous paper. Given a nonnegative function $v: \mathbb{Z}^d \to \mathbb{Z}$ with $v(0) = 0$, we define the space of configurations $\Gamma$ to consist of functions $h: \mathbb{Z}^d \to \mathbb{Z}$ such that $h(i) - h(j) \leq v(i - j)$ for all i$i, j \epsilon \mathbb{Z}^d$. We then take two sequances of independent Poisson clocks $(p^{\pm}(i, t): i \epsilon \mathbb{Z}^d)$ of rates $\lambda^{\pm}$. We start with a possibly random configuration $h \epsilon \Gamma$. The function $h$ increases (respectively, decreases) by one unit at site $i$, when the clock $p^{+}(i, \cdot)$ [respectively, $p^{-}(i, \cdot)$] rings and the resulting configuration is still in $\Gamma$. Otherwise the change in h is suppressed. In this way we have a process $h(i, t)$ that after a rescaling $u^{\varepsilon}(x, t) = \varepsilonh([\frac{x}{\varepsilon}], \frac{t}{\varepsilon})$ is expected to converge to a function $u(x, t)$ that solves a Hamilton-Jacobi equation of the form $u_t + H(u_x) = 0$. We established this when $\lambda^{-}$ or $\lambda^{+} = 0$ in the previous paper, employing a strong monotonicity property of the process $h(i, t)$. Such property is no longer available when both $\lambda^{+}, \lambda^{-}$ are nonzero. In this paper we initiate a new approach to treat the problem when th dimension is 1 and the set $\Gamma$ can be described by local constraints on the configuration $h$. In higher dimensions, we can only show that any limit point of th processes $u^{varepsilon}$ is a process $u$ that satisfies a Hamilton-Jacobi equation for a suitable (possibly random) Hamiltonian $H$.

Monotonicity of the cd-index for polytopes

We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional polytope with n vertices is at most that of C(n,d), the d-dimensional cyclic polytope with n vertices.

On axiomatizations of the weighted Shapley values

The family of weighted Shapley values for cooperative n-person transferable utility games is studied. We assume first that the weights of the players are given exogenously and provide two axiomatic characterizations of the corresponding weighted Shapley value. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced by a new postulate called the ω-mutual dependence. In our second axiomatization we use among other things the strong monotonicity property of Young (1985, Int. J. Game Theory 14, 65–72). Finally, we give a new axiomatic characterization of the family of all weighted Shapley values. Journal of Economic Literature Classification Number: C71, D46.

Strongly order preserving semiflows generated by functional differential equations

The aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems, due to the authors [ 12, 13 J, to certain systems of functional differential equations (FDE’s) which do not enjoy the quasimonotone property considered by one of us in [9]. The present work extends our previous results in this direction for scalar equations in [ 111 to systems. In the fundamental paper [S], M. W. Hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space X tend to the set E of equilibria. Employing ideas of Hirsch and of H. Matano [7], the authors have extended the theory to obtain some improvements in the results of Hirsch and Matano under Matano’s weaker assumption that the semiflow is strongly order preserving. Recently, in [13], we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium. These conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium. P. PoliCik [S] has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties. In this paper, WC apply the convergence results developed in [ 131 to nonquasimonotone FDE’s. In this Introduction, WC briefly review the application of monotone dynamical systems theory to FDE’s enjoying the quasimonotone property. This research appeared in [9, 133. Then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property (M,) holds. Some simple examples are given which