Monotone Nondecreasing Functions Research Articles

Submodular batch scheduling on parallel machines

AbstractThis article studies a submodular batch scheduling problem motivated by the vacuum heat treatment. The batch processing time is formulated by a monotone nondecreasing submodular function characterized by decreasing marginal gain property. The objective is to minimize the makespan. We show the NP‐hardness of the problem on a single machine and of finding a polynomial‐time approximation algorithm with the worst‐case performance ratio strictly less than for the problem on parallel machines. We introduce a bounded interval to model the batch processing time using two parameters, that is, the total curvature and the quantization indicator. Based on the decreasing marginal gain property and the two parameters, we make a systematic analysis of the full batch longest processing time algorithm and the longest processing time greedy algorithm, and propose the instances with the bound of batch capacity for these two algorithms for the submodular batch scheduling problem. Moreover, we prove the submodularity of batch processing time function of the existing batch models including the parallel batch, serial batch, and mixed batch models. We compare the worst‐case performance ratios in the existing batch models with those deduced from our work in the submodular batch model. In most situations, the worst‐case performance ratios deduced from our work are comparable to the best‐known worst‐case performance ratios with tailored examinations.

Oscillation criteria for a second order half-linear differential equation with delay, with monotone nondecreasing delay function

Oscillation criteria for a second order half-linear differential equation with delay, with monotone nondecreasing delay function

Improved deep neural network performance under dynamic programming mode

For Deep Neural Networks (DNN), the standard gradient-based algorithms may not be efficient because of the raised computational expense resulting from the increase in the number of layers. This paper offers an alternative to the classic training solutions: an in-depth study to find conditions under which the underlying Artificial Neural Networks ANN minimisation problem can be addressed from a Dynamic Programming (DP) perspective. Specifically, we prove that any ANN with monotonic activation is separable when regarded as a parametric function. Particularly, when the ANN is viewed as a network representation of a dynamical system (as a coupled cell network), we also prove that the transmission-of-signal law is separable provided the activation function is a monotone non-decreasing function. This strategy may have a positive impact on the performance of ANNs by improving their learning accuracy, particularly for DNNs. For our purposes, ANNs are also viewed as universal approximators of continuous functions and as abstract compositions of an even number of functions. This broader representation makes it easier to analyse them from many other perspectives (universal approximation issues, inverse problem solving) leading to a general improvement in knowledge on NNs and their performance.

Maximization of nonsubmodular functions under multiple constraints with applications

We consider the problem of maximizing a monotone nondecreasing set function under multiple constraints, where the constraints are also characterized by monotone nondecreasing set functions. We propose two greedy algorithms to solve the problem with provable approximation guarantees. The first algorithm exploits the structure of a special class of the general problem instance to obtain a better time complexity. The second algorithm is suitable for the general problem. We characterize the approximation guarantees of the two algorithms, leveraging the notions of submodularity ratio and curvature introduced for set functions. We then discuss particular applications of the general problem formulation to problems that have been considered in the literature. We validate our theoretical results using numerical examples.

The span of singular tuples of a tensor beyond the boundary format

A singular k-tuple of a tensor T of format (n1,…,nk) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n1,…,nk) of rank at most one. A generic tensor has finitely many complex singular k-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k−1 dimensions ni, then the number of singular k-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable nk, that stabilizes when (n1,…,nk) reaches a boundary format.In this paper, we study the linear span of singular k-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-k tensors of format (2,…,2,n). As a consequence, if again we fix the first k−1 dimensions ni and let nk increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k>3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.

Узагальнення негативних результатів для інтерполяційного опуклого наближення функцій, що мають дробову похідну в просторі Соболєва

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function of Sobolev`s space by polynomials. The prob-lem of positive approximation is to estimate the pointwise degree of approxi-mation of a function of r times continuously differentiable and positive func-tions on [0, 1]. Estimates of the form (1) for positive approximation are known ([1, 2]). The problem of monotone approximation is that of estimating the de-gree of approximation of a monotone nondecreasing function bymonotone nondecreasing polynomials. Estimates of the form (1) for monotone approxi-mation were proved in [3, 4, 7]. In [3, 4] is consider r is natural and r not equal one. In [7] is consider r is real and rmore two. It was proved that for monotone approximation estimates of the form (1) are fails for r is real and r more two. The problem of convex approximation is that of estimating the degree of ap-proximation of a convex function by convex polynomials. The problem of con-vex approximation is consider in [8-10]. In [8] is consider r is natural and r not equal one. In [9] is consider r is real and r more two. It was proved that for con-vex approximation estimates of the form (1) are fails for r is real and r more two. In [10] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider, if the index of the Sobolev space is in the interval from three to four. It is proved that the estimate that gen-eralizes (1) is false This paper investigates the issue of approximation of con-vex functions from the Sobolev space by convex algebraic polynomials for a real index of the Sobolev space from the interval from two to three.Similarly to the paper [10], a counterexample is built, which shows that the estimate that generalizes the estimate (1) is false.This paper isthe generalization of results papers [9] and [11]. The main result is the analog of the theorem 2.3 in [11].

Iwueze’s Distribution and Its Application

Increased usage of single parameter life-time distributions for reference in development of other life-time distributions and data modeling has attracted the interest of researchers. Because performance ratings differ from one distribution to another and there are increased need for distributions that delivers improved fits, new distributions with a better performance rating capable of providing improved fits have evolved in the Literature. One of such distribution is the Iwueze’s distribution. Iwueze’s distribution is proposed as a new distribution with Gamma and Exponential baseline distributions. Iwueze’s distribution theoretical density, distribution functions and statistical features such as its moments, factors of variation, skewness, kurtorsis, reliability functions, stochastic ordering, absolute deviations from average, absolute deviations from mid-point, Bonferroni and Lorenz curves, Bonferroni and Gini indexes, entropy and the stress and strength reliability have been developed. Iwueze’s distribution curve is not bell-shaped, but rather skewed positively and leptokurtic. The risk measurement function is a monotone non-decreasing function, while the average residual measurement life-time function is a monotone non-increasing function. The parameter of the Iwueze’s distribution was estimated using the likelihood estimation approach. When used for a real-life data modeling, the new proposed Iwueze’s distribution delivers improved and superior fits better than the Akshya, Shambhu, Devya, Amarendra, Aradhana, Sujatha, Akash, Rama, Shanker, Suja, Lindley, Ishita, Prakaamy, Pranav, Exponential, Ram Awadh and Odoma distributions. Iwueze’s distribution is definitely tractable and offers a better distribution than a number of well-known distributions for modeling life-time data, with greater superiority of fit performance ratings.

Properties and Bounds of Jensen-Type Functionals via Harmonic Convex Functions

Dragomir introduced the Jensen-type inequality for harmonic convex functions (HCF) and Baloch et al. studied its different variants, such as Jensen-type inequality for harmonic h -convex functions. In this paper, we aim to establish the functional form of inequalities presented by Baloch et al. and prove the superadditivity and monotonicity properties of these functionals. Furthermore, we derive the bound for these functionals under certain conditions. Furthermore, we define more generalized functionals involving monotonic nondecreasing concave function as well as evince superadditivity and monotonicity properties of these generalized functionals.

Resource allocation in rooted trees subject to sum constraints and nonlinear cost functions

We study resource allocation problems in rooted trees in which demand values are given in the leaves. Single-type resources (weights) are to be assigned in the tree nodes such that the total weight in the rooted path from each leaf to the root equals its demand. The goal is to minimize the total costs of the allocated resources. It is known that the linear cost case, i.e., when the cost of a resource is proportional to its weight, is solvable in linear time. In this paper we show that when costs are monotone nondecreasing functions, which reflect, e.g., (dis)economies of scale, the problem becomes intractable, and design for it a fully polynomial time approximation scheme by formulating it as a dynamic program and using the technique of K-approximation sets and functions.

Opinion subset selection via submodular maximization

Current research on subset selection for opinion analysis assumes that their methods can retrieve the opinions expressed in documents from general text features. However, such relaxed conditions can hardly maintain the performance of the analysis in opinion mining, especially when given strict limitations on the subset size. In this paper, we propose a framework for opinion subset selection. This framework can select a small set of instances from original data to convey a subjective representation for opinion classification and regression. Compared with our framework, the conventional submodular based subset selection approach cannot capture the fine-grained opinion features expressed in the corpus. Specifically, we propose a monotone non-decreasing score function and a framework based on topic modeling and submodular maximization for filtering irrelevant information and selecting the subsets. Our work further introduces an opinion-sensitive algorithm for optimizing the proposed function for opinion subset construction. We perform extensive experiments and comparative analysis of different subset selection methods in this work. The experimental result shows that the proposed opinion subset selection framework can compress the original text training set and preserve the test set’s classification and regression metric performance at the same time.

A variation of DS decomposition in set function optimization

Any set function can be decomposed into the difference of two monotone nondecreasing submodular functions. This theorem plays an important role in the set function optimization theory. In this paper, we show a variation that any set function can be decomposed into the difference of two monotone nondecreasing supermodular functions. Meanwhile, we give an example in social network optimization and construct algorithmic solutions for the maximization problem of set functions with this variation of DS decomposition.

Деякі негативні результати для інтерполяційного монотонного наближення функцій, що мають дробову похідну

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

"Узагальнення поточкових інтерполяційних оцінок опуклого наближення функцій, що мають дробову похідну "

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function of Sobolev`s space by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function of r times continuously differentiable and positive functions on [0, 1]. Estimates of the form (1) for positive approximation are known ([1, 2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3, 4, 8]. In [3, 4] is consider r is natural and r not equal one. In [8] is consider r is real and r more two. It was proved that for monotone approximation estimates of the form (1) are fails for r is real and r more two. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5, 6]). In [5] is consider r is natural and r not equal one. In [6] is consider r is real and r more two. It was proved that for convex approximation estimates of the form (1) are fails for r is real and r more two. In [9] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider. It is proved, that for this function, estimate (1) is not true, if r is more three and less four generally speaking. In this paper the question of approximation of function Sobolev`s space and convex by algebraic convex polynomial is consider. This paper is the generalization of results papers [9] and [11]. It is proved, that for function of Sobolev`s space and convex, estimate of the type (1) is not true, generally speaking. The main result is the ana­log of the theorem 2.3 in [11].

Robust stability of mixed Cohen–Grossberg neural networks with discontinuous activation functions

PurposeThe purpose of this paper is to develop a method for the existence, uniqueness and globally robust stability of the equilibrium point for Cohen–Grossberg neural networks with time-varying delays, continuous distributed delays and a kind of discontinuous activation functions.Design/methodology/approachBased on the Leray–Schauder alternative theorem and chain rule, by using a novel integral inequality dealing with monotone non-decreasing function, the authors obtain a delay-dependent sufficient condition with less conservativeness for robust stability of considered neural networks.FindingsIt turns out that the authors’ delay-dependent sufficient condition can be formed in terms of linear matrix inequalities conditions. Two examples show the effectiveness of the obtained results.Originality/valueThe novelty of the proposed approach lies in dealing with a new kind of discontinuous activation functions by using the Leray–Schauder alternative theorem, chain rule and a novel integral inequality on monotone non-decreasing function.

An alternative distribution function estimation method using rational Bernstein polynomials

This paper gives a general method for nonparametric distribution function estimation using the rational Bernstein polynomials as an alternative to the current estimators. The proposed new method is compared with Bernstein polynomials and empirical distribution function methods by simulation studies. The new method guarantees monotone nondecreasing function by applying linear constraints on the coefficients of the rational Bernstein basis functions and smooth the empirical distribution function. Furthermore, as a special case, it reduces to Bernstein polynomial estimator method. Some theoretical properties of the new estimator are investigated. Simulation study shows that the proposed estimator is preferable to the Bernstein polynomials and empirical distribution function estimator methods.

Dynamics of a frictional system, accounting for hereditary-type friction and the mobility of the vibration limiter

Dynamics of a frictional system consisting of a rough body situated on a rough belt moving at a constant velocity is studied. The vibrations are limited by an elastic obstacle. The Coulomb-Hammonton dry friction characteristic is chosen, according to the hypothesis of Ishlinskiy and Kragelskiy, in the form of hereditary-type friction, where the coefficient of friction of relative rest (CFRR) is a monotone non-decreasing continuous function of the time of relative rest at the previous analogous time interval. A mathematical model and the structure of its phase space are presented, as well as the equations of point maps of the Poincare surface and the results of studying the dynamic characteristics of the parameters of the system (velocity of the belt, type of the functional relation of CFRR, rigidity of the elastic obstacle, etc.). A software product has been developed which makes it possible to find complex periodic motion regimes, as well as to calculate bifurcation diagrams used for determining the main variations of the motion regime from periodic to chaotic (the period doubling scenario).

Using a monotone single-index model to stabilize the propensity score in missing data problems and causal inference.

The augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems. An important component of this method is the propensity score. Popular parametric models for the propensity score include the logistic, probit, and complementary log-log models. A common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables. To avoid the need to choose a model, we model the propensity score via a semiparametric single-index model, in which the score is an unknown monotonic nondecreasing function of the given single index. Under this new model, the augmented inverse weighting estimator (AIWE) of the mean of the response is asymptotically linear, semiparametrically efficient, and more robust than existing estimators. Moreover, we have made a surprising observation. The inverse probability weighting and AIWEs based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model. A heuristic explanation of this phenomenon is provided. A real-data example is used to illustrate the proposed methods.

Network Deployment for Maximal Energy Efficiency in Uplink With Multislope Path Loss

This work aims to design the uplink (UL) of a cellular network for maximal energy efficiency (EE). Each base station (BS) is randomly deployed within a given area and is equipped with $M$ antennas to serve $K$ user equipments (UEs). A multislope (distance-dependent) path loss model is considered and linear processing is used, under the assumption that channel state information is acquired by using pilot sequences (reused across the network). Within this setting, a lower bound on the UL spectral efficiency and a realistic circuit power consumption model are used to evaluate the network EE. Numerical results are first used to compute the optimal BS density and pilot reuse factor for a Massive MIMO network with three different detection schemes, namely, maximum ratio combining, zero-forcing (ZF) and multicell minimum mean-squared error. The numerical analysis shows that the EE is a unimodal function of BS density and achieves its maximum for a relatively small density of BS, irrespective of the employed detection scheme. This is in contrast to the single-slope (distance-independent) path loss model, for which the EE is a monotonic non-decreasing function of BS density. Then, we concentrate on ZF and use stochastic geometry to compute a new lower bound on the spectral efficiency, which is then used to optimize, for a given BS density, the pilot reuse factor, number of BS antennas and UEs. Closed- form expressions are computed from which valuable insights into the interplay between optimization variables, hardware characteristics, and propagation environment are obtained.

Blocking Rumor by Cut

In this paper, a rumor blocking problem is studied with an objective function which is neither submodular or supermodular. We will prove that this problem is NP-hard and give a data-dependent approximation with sandwich method. In addition, we show that every set function has a pair of monotone nondecreasing modular functions as upper bound and lower bound, respectively.

Один контрприклад для опуклого наближення функцій, що мають дробову похідну, при r>4

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.