Nonnegative Interval Research Articles

Kernel-Free Quadratic Surface Support Vector Regression with Non-Negative Constraints.

In this paper, a kernel-free quadratic surface support vector regression with non-negative constraints (NQSSVR) is proposed for the regression problem. The task of the NQSSVR is to find a quadratic function as a regression function. By utilizing the quadratic surface kernel-free technique, the model avoids the difficulty of choosing the kernel function and corresponding parameters, and has interpretability to a certain extent. In fact, data may have a priori information that the value of the response variable will increase as the explanatory variable grows in a non-negative interval. Moreover, in order to ensure that the regression function is monotonically increasing on the non-negative interval, the non-negative constraints with respect to the regression coefficients are introduced to construct the optimization problem of NQSSVR. And the regression function obtained by NQSSVR matches this a priori information, which has been proven in the theoretical analysis. In addition, the existence and uniqueness of the solution to the primal problem and dual problem of NQSSVR, and the relationship between them are addressed. Experimental results on two artificial datasets and seven benchmark datasets validate the feasibility and effectiveness of our approach. Finally, the effectiveness of our method is verified by real examples in air quality.

Solving Multi-Objective Linear Programming Problems with Fuzzy Interval Based on Decomposition Method

Most of the problems in real-world situations have a multi-objective, in these situations, available information in the system is not exact or imprecise. In this paper, solving multi-objective linear programming problems with fuzzy non-negative intervals such as objective function coefficients, technical coefficients, and fuzzy variables by using an approximation but a convenient method called decomposition method has been proposed. In the composition method, ranking functions are not used. With the help of numerical examples, the method is illustrated.

Weyl Integrals on Weighted Spaces

Abstract Weighted spaces of continuous functions are introduced such that Weyl fractional integrals with orders from any finite nonnegative interval define equicontinuous sets of continuous linear endomorphisms for which the semigroup law of fractional orders is valid. The result is obtained from studying continuity and boundedness of convolution as a bilinear operation on general weighted spaces of continuous functions and measures.

Double resonance for Robin problems with indefinite and unbounded potential

We study a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential term. The nonlinearity \begin{document}$f(x, s)$\end{document} is a Caratheodory function which is asymptotically linear as \begin{document}$ s\to ± ∞$\end{document} and resonant. In fact we assume double resonance with respect to any nonprincipal, nonnegative spectral interval \begin{document}$ \left[ \hat{λ}_k, \hat{λ}_{k+1}\right]$\end{document} . Applying variational tools along with suitable truncation and perturbation techniques as well as Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of constant sign.

New Error Measures and Methods for Realizing Protein Graphs from Distance Data

The interval distance geometry problem consists in finding a realization in $$\mathbb {R}^K$$RK of a simple undirected graph $$G=(V,E)$$G=(V,E) with non-negative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. In this paper, we focus on the application to the conformation of proteins in space, which is a basic step in determining protein function: given interval estimations of some of the inter-atomic distances, find their shape. Among different families of methods for accomplishing this task, we look at mathematical programming based methods, which are well suited for dealing with intervals. The basic question we want to answer is: what is the best such method for the problem? The most meaningful error measure for evaluating solution quality is the coordinate root mean square deviation. We first introduce a new error measure which addresses a particular feature of protein backbones, i.e. many partial reflections also yield acceptable backbones. We then present a set of new and existing quadratic and semidefinite programming formulations of this problem, and a set of new and existing methods for solving these formulations. Finally, we perform a computational evaluation of all the feasible solver $$+$$+ formulation combinations according to new and existing error measures, finding that the best methodology is a new heuristic method based on multiplicative weights updates.

Multiplication of convex sets in $C(K)$ spaces

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. We study this for a special class of closed convex sets, namely closed intervals. Our main interest is in correlations between properties of the product of order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a first-countable space K is finite whenever it has the property that the product of two nonnegative intervals is closed, or the property that the product of an interval with itself is convex. That some assumption on K is needed, can be seen from the fact that, if K is the Stone–Cech compactification of N, then the product of two intervals in C(K) with continuous boundary functions is always an interval. For any K, it is proved that the product of two positive intervals is always an interval, and that the product of two nonnegative intervals is always convex. Finally, square roots of intervals are investigated, with results of similar type. Mathematics Subject Classification: primary 52A05; secondary 46J10, 46E15.

The interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem with inexact distances

The Distance Geometry Problem in three dimensions consists in finding an embedding in $${\mathbb{R}^3}$$ of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where we consider the full-atom representation of the protein backbone and some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances.

Reduction approaches for robust shortest path problems

We investigate the uncertain versions of two classical combinatorial optimization problems, namely the Single-Pair Shortest Path Problem (SP-SPP) and the Single-Source Shortest Path Problem (SS-SPP). The former consists of finding a path of minimum length connecting two specific nodes in a finite directed graph G; the latter consists of finding the shortest paths from a fixed node to the remaining nodes of G. When considering the uncertain versions of both problems we assume that cycles may occur in G and that arc lengths are (possibly degenerating) nonnegative intervals. We provide sufficient conditions for a node and an arc to be always or never in an optimal solution of the Minimax regret Single-Pair Shortest Path Problem (MSP-SPP). Similarly, we provide sufficient conditions for an arc to be always or never in an optimal solution of the Minimax regret Single-Source Shortest Path Problem (MSS-SPP). We exploit such results to develop pegging tests useful to reduce the overall running time necessary to exactly solve both problems.

Perron vectors of an irreducible nonnegative interval matrix

As is well known, an irreducible nonnegative matrix possesses a uniquely determined Perron vector. As the main result of this article we give a description of the set of Perron vectors of all the matrices contained in an irreducible nonnegative interval matrix A. This result is then applied to show that there exists a subset A * of A parameterized by n parameters (instead of n 2 ones in the description of A) such that for each A∈ A there exists a matrix A′∈ A * having the same spectral radius and the same Perron vector as A.

Oscillatory Störmer–Cowell methods

We consider explicit methods for initial-value problems for special second-order ordinary differential equations where the right-hand side does not contain the derivative of y and where the solution components are known to be periodic with frequencies ω j lying in a given nonnegative interval [ ω ̄ , ω ̄ ] . The aim of the paper is to exploit this extra information and to modify a given integration method in such a way that the method parameters are “tuned” to the interval [ ω ̄ , ω ̄ ] . Such an approach has already been proposed by Gautschi in 1961 for linear multistep methods for first-order differential equations in which the dominant frequencies ω j are a priori known. In this paper, we only assume that the interval [ ω ̄ , ω ̄ ] is known. Two “tuning” techniques, respectively based on a least squares and a minimax approximation, are considered and applied to the classical explicit Störmer–Cowell methods and the recently developed parallel explicit Störmer–Cowell methods.

E‐optimal designs for regression models with quantitative factors— a reasonable choice?

AbstractFor regression models with quantitative factors it is illustrated that the E‐optimal design can be extremely inefficient in the sense that it degenerates to a design which takes all observations at only one point. This phenomenon is caused by the different size of the elements in the covariance matrix of the least‐squares estimator for the unknown parameters. For these reasons we propose to replace the E‐criterion by a corresponding standardized version. The advantage of this approach is demonstrated for the polynomial regression on a nonnegative interval, where the classical and standardized E‐optimal designs can be found explicitly. The described phenomena are not restricted to the E‐criterion but appear for nearly all optimality criteria proposed in the literature. Therefore standardization is recommended for optimal experimental design in regression models with quantitative factors. The optimal designs with respect to the new standardized criteria satisfy a similar invariance property as the famous D‐optimal designs, which allows an easy calculation of standardized optimal designs on many linearly transformed design spaces.

Duration Modeling of Spatial Point Patterns

This paper attempts to further the research by Odland and Ellis (1992) in applying event history methodology to the analysis of spatial point patterns (that is, event patterns). Its empirical focus is the event pattern derived from the adoption of an agricultural innovation, the Harvestore, in southern Ontario, Canada, from 1963 to 1986. Event history analysis involves the use of discrete‐state, continuous‐time stochastic models to investigate a temporal longitudinal record on discrete variables. Event history models are usually concerned with durations of time between events and the effects of intertemporal time dependencies on future event occurrences. As such, they are often referred to as duration models. Many of the methods used in event history analysis allow the use of other nonnegative interval measurements in place of standard temporal intervals to investigate a series of events. In particular, spatial intervals (or durations) of distances between events may also be accommodated by event history models. Our analysis extends the previous research of Odland and Ellis (1992) by using a wider range of parametric models to explore duration dependence, investigating the role of spatial censoring, and using a more extensive set of explanatory variables. In addition, simulation experiments and graphical tests are used to evaluate the empirical event pattern against one generated from Complete Spatial Randomness. Results indicate that the event pattern formed by the Harvestore adopter farms is clustered (that is, is described by positive duration dependency), the sales agent is a significant factor in the distribution of adopters, and that contrasting results are obtained from the analysis using censored data versus uncensored data.

Comments on "A necessary and sufficient condition for the stability of nonnegative interval discrete systems" by B. Shafai et al

In the above-titled paper (ibid., vol.36, no.6, p.742-6, June 1991), the authors gave necessary and sufficient conditions for the Schur stability of nonnegative interval matrices. The commenter suggests that these conditions may be more conveniently stated in terms of characteristics of M-matrices and are easy consequences of some well-known facts in nonnegative matrix theory. The M-matrix characterization further reveals that the conditions in the above-titled paper fall into a special case of a known result for the general interval matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comments on "A necessary and sufficient condition for the stability of nonnegative interval discrete systems" by B. Shafai et al

It is shown that a necessary and sufficient condition for the stability of nonnegative interval discrete systems reported in the above-named work (ibid., vol.36, no.6, p.742-6, June 1991) can be alternatively be established by using a previous work of the present authors (Memoirs, Fac. Eng., Kyoto Univ. vol.49, no.4, p.273-9, 1987) and properties of M-matrices, and furthermore it ensures the stability of an even wider class of systems. >

A necessary and sufficient condition for the stability of nonnegative interval discrete systems

The stability of nonnegative discrete systems with interval uncertainty is discussed. For this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals. An illustrative example is used as a guideline to demonstrate the effectiveness of the result. Simple computational tests are suggested, and their usefulness in this framework is discussed. The possible extension of this result to more general classes of systems is addressed. >

Non-connectedness of the set of equilibrium money prices: The overlapping-generations economy

Consider the overlapping-generations economy with nominal taxes and transfers. Under some conditions, the set of equilibrium money prices is a non-negative interval. It has not been known whether this set can consist of two or more disjoint intervals. Three examples are provided here in which the set of equilibrium money prices is a non-connected set. The examples are for a finite-horizon balanced economy, an infinite-horizon balanced economy, and an infinite-horizon non-balanced economy.

On the analysis of joint production

AbstractIt is well known that n-process, n-commodity (or square), models of productive single-product industries have positive solutions to their price and quantity systems if the rates of profit and growth lie in appropriate non-negative intervals. On the other hand, negative prices and quantities can occur in formal solutions of models of square, productive, multiple-product industries even when the rates of profit and growth are less than their respective maximum positive values. It is shown in this paper that these differences can be attributed to the presence in joint production of dominance, in either row or column versions. Results on positive solutions to the price (respectively, quantity) system are derived in terms of the absence of column (respectively, row) dominance of the net output matrix. As the concepts of row and column dominance are defined in terms of linear inequalities, the basic mathematical results to be applied are theorems of the alternative.

Concerning Nonnegative Valued Interval Functions

Concerning Nonnegative Valued Interval Functions