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In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516). [ABSTRACT FROM AUTHOR]

In this paper, based on the previous work by Bai [On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J Matrix Anal Appl. 1999;21:67–78] and by Zhang et al. [Improved convergence theorems of multisplitting methods for the linear complementarity problem. Appl Math Comput. 2014;243:982–987], we further discuss the multisplitting AOR (MAOR) method for the linear complementarity problem. Some new convergence conditions of the MAOR method are obtained, which are different from those of the aforementioned papers. [ABSTRACT FROM AUTHOR]

In this paper, based on the previous work by Zhang and Li [The weaker convergence of modulus-based synchronous multisplitting multi-parameters methods for linear complementarity problems, Comput Math Appl. 2014;67:1954–1959], we further discuss the modulus-based synchronous multisplitting multi-parameters AOR (MSMMAOR) method for solving the linear complementarity problems. The new convergence conditions of the MSMMAOR method are obtained, which are weaker than those of the aforementioned paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the Continuous-Time Problem in an interval context and to obtain optimality conditions for this problem. In addition, we will find relationships between solutions of Interval Continuous-Time Problem and Interval Variational-like Inequality Problems, both Stampacchia and Minty type. Pseudo invex monotonicity condition ensures the existence of solutions of the problem. These results generalize similar conclusions obtained in Euclidean or Banach spaces inside classical mathematical programming problems or Continuous-Time Problems. We will finish generalizing the existence of Walrasarian equilibrium price model and the Wardrop's principle for traffic equilibrium problem to an environment of interval-valued functions. [ABSTRACT FROM AUTHOR]

In this paper, by transforming the vertical linear complementarity problem (VLCP) as a certain absolute value equation, we design a class of modulus-based matrix splitting iteration methods for solving the VLCP. The convergence properties of the proposed methods are discussed in depth. By making use of some numerical experiments, we confirm the efficiency of the proposed methods. Numerical results show that the proposed methods are superior to the classical modulus-based matrix splitting iteration methods. [ABSTRACT FROM AUTHOR]

This paper introduces interval union arithmetic, a new concept which extends the traditional interval arithmetic. Interval unions allow to manipulate sets of disjoint intervals and provide a natural way to represent the extended interval division. Considering interval unions lead to simplifications of the interval Newton method as well as of other algorithms for solving interval linear systems. This paper does not aim at describing the complete theory of interval union analysis, but rather at giving basic definitions and some fundamental properties, as well as showing theoretical and practical usefulness of interval unions in a few selected areas. [ABSTRACT FROM AUTHOR]

The Portuguese energy efficiency programs are usually run through grant awards that support energy efficient technologies, which require the establishment of rational investment plans considering distinct budget level inputs to each program. The current evaluation of energy efficiency programs mainly relies on the economic analysis of demand-side programs and projects. One of the weaknesses of this type of approach is that it cannot capture the complexities of decision-making processes, mainly relying on the device costs targeted for funding. This paper aims at developing a methodological framework to support public authorities in investment planning for energy efficiency programs based on portfolio theory explicitly considering the energy spent in the manufacturing and installation of each energy efficient technology. The applicability of the methodology herein proposed is illustrated by considering the potential investment in distinct portfolios of industrial lighting systems. Finally, a new solution methodology for computing possibly efficient solutions is also suggested which allows exploring distinct types of investment strategies, according to the public investor's preferences. Our findings suggest that LED lamps and T5 technologies should be considered as a valid option for replacing T8 and HPS technologies. Additionally, despite the investment cost involved in the installation of light control systems, they should be elected for funding. Finally, it is worth mentioning that the substitution of T8 and Halogen lamps with LED lamps has never been considered with the modelling framework used in spite of being effectively selected for public support, highlighting the need of adopting approaches that encompass a life-cycle perspective. [ABSTRACT FROM AUTHOR]

In this paper we present a Stone-Weierstrass type result in the context of continuous interval-valued functions defined on a compact Hausdorff space. We also provide a Jackson type approximation result involving the modulus of continuity of interval-valued functions based on the gH-difference of intervals. Finally we show how an interval-valued continuous function can be approximated using interval neural networks. All the proofs are constructive. [ABSTRACT FROM AUTHOR]

Solving large-scale l1 minimization problem is very important and has attracted much attentions in recent years. In this paper, we present an efficient method for solving this problem. To this end, we first reformulate this problem as a nonnegative constrained minimization problem. Then we propose a modulus-based iterative method to solve the nonnegative constrained minimization problem. Convergence analysis and the choice of near optimal parameter are presented. Experimental results are given to illustrate feasibility and effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, weak optimal inverse problems of interval linear programming (IvLP) are studied based on KKT conditions. Firstly, the problem is precisely defined. Specifically, by adjusting the minimum change of the current cost coefficient, a given weak solution can become optimal. Then, an equivalent characterization of weak optimal inverse IvLP problems is obtained. Finally, the problem is simplified without adjusting the cost coefficient of null variable. [ABSTRACT FROM AUTHOR]

A large number of free boundary problems can be formulated as linear-complementarity problems. In this paper, we propose an inexact alternating direction method of multipliers for solving linear complementarity problem arising from free boundary problems by using the special structure of these problems. The convergence of our proposed method is proved. Numerical results show that the proposed method is feasible and effective, and it is significantly faster than modified alternating direction implicit algorithm and many other methods, especially when dimension of the problem being solved is large. [ABSTRACT FROM AUTHOR]

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C2 continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves. [ABSTRACT FROM AUTHOR]

In this paper, we investigate three canonical forms of interval convex quadratic programming problems. Necessary and sufficient conditions for checking weak and strong optimality of given vector corresponding to various forms of feasible region, are established respectively. By using the concept of feasible direction, these conditions are formulated in the form of linear systems with both equations and inequalities. In addition, we provide two specific examples to illustrate the efficiency of the conditions. [ABSTRACT FROM AUTHOR]

In this paper, we prove some Milne type inequalities for interval-valued functions and, along with it, we explore some connections with other inequalities. More precisely, using the Aumann integral and the Kulisch–Miranker order and including-order on the space of real and compact intervals, we establish some Milne type inequalities for interval-valued functions. Also, using different orders, we obtain some connections with Chebyshev, Cauchy–Schwarz, and Hölder inequality. Finally, some new ideas and results based on submodular measures are explored as well as some examples and applications are presented for illustrating our results. [ABSTRACT FROM AUTHOR]

This paper addresses the common issue of locating an Underwater Vehicle. Usually, the positioning of a vehicle is based on the propagation of electromagnetic waves, using systems such as the Global Positioning System. However, water, and particularly salty water, makes the use of electromagnetic waves impractical due to the attenuation caused by the high conductivity of the medium. Hence, the most reliable way to transmit information underwater is by using sound waves and many technologies have emerged to solve the positioning problem in this way. Technologies such as Long BaseLine (LBL), Short BaseLine (SBL) and Ultra-Short BaseLine (USBL) are the most frequently used underwater. These technologies are based on the use of multiple emitters in the case of LBL and SBL, or multiple receivers in the case of USBL. This paper describes a way of finding a vehicle location, on-board, based on the measurement at the vehicle by a single receiver of the phase of an acoustic sine wave transmitted from a single emitter that is at a fixed and known location. The method also uses other proprioceptive measurements: vehicle's velocity and heading. An algorithm based on contractors and bisections scatters the solution space searching for all possible solutions (positions in this case) to the set of equations. Moreover, this paper introduces the Time Constraint Satisfaction Problem. Indeed, the proposed algorithm does not compute the solutions from measurements at a single point in time, rather it uses a set of measurements taken over a time window and stored in a buffer. As a result, the location is not only known at the latest instant but the past locations can be tracked back over the length of the chosen time window. [ABSTRACT FROM AUTHOR]

This paper deals with the fractional linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. A method is provided for the situation in which the feasible set is described by a linear interval system. Moreover, certain dependencies between the coefficients in the nominators and denominators can be involved. Also, we extend this approach for situations in which the same vector appears in different terms in nominators and denominators. The applicability of the approaches developed is illustrated in the context of the analysis of hospital performance. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of linear programming with inexact data represented by real intervals. We introduce the concept of duality gap to interval linear programming. We give characterizations of strongly and weakly zero duality gap in interval linear programming and its special case where the matrix of coefficients is real. We show computational complexity of testing weakly- and strongly zero duality gap for commonly used types of interval linear programming. [ABSTRACT FROM AUTHOR]

In this paper, a class of unconstrained discrete minimax problems is described, in which the objective functions are in C 1. The paper deals with this problem by means of taking the place of maximum-entropy function with adjustable entropy function. By constructing an interval extension of adjustable entropy function and some region deletion test rules, a new interval algorithm is presented. The relevant properties are proven. The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum-entropy algorithm. Both theoretical and numerical results show that the method is reliable and efficient. [ABSTRACT FROM AUTHOR]

Numerical solution of boundary value problems modelled by differential or integral equations is reduced to solving linear system of equations. Even in the systems of equations without intervals, the solution does not have to be unique, e.g., in the case of a singular (or at least ill-conditioned) matrix. Such problems are even more inconvenient in the interval linear systems of equations. In this paper, the various methods of solving such systems are tested. These systems have been generated during the numerical solution of boundary value problems modelled by Navier-Lamé equations. [ABSTRACT FROM AUTHOR]

Finding the solution of linear complementarity problem (LCP) is an important subject in scientific computing, engineering and economic applications. In the literature, many iterative methods have been proposed, especially, when the matrix of the problem is a real positive definite or an H− matrix. In this paper, we propose a new method using an alternating direction (AD) technique based on a modified symmetric successive overrelaxation (MSSOR) for LCP, assuming that the real matrix of the LCP is symmetric positive definite. This approach is an inner/outer iteration method, in which AD method is used as an outer iteration, and the MSSOR method as an inner one. We provide a detailed proof for the convergence of the proposed method. In addition, numerical results show that the proposed method is feasible and effective. [ABSTRACT FROM AUTHOR]

Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a nondifferentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke subdifferentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the G -- (F, ρ) convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond--Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs. [ABSTRACT FROM AUTHOR]

In this paper, we reformulate a class of nonlinear complementarity problems as the implicit fixed-point equations. We demonstrate accelerated modulus-based matrix splitting iteration method. We show their convergence by assuming that the system matrix is positive definite or the splitting of the system matrix are H+-compatible splitting and discuss the choice of the optimal parameter. Furthermore, we give two-step accelerated modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. Numerical experiments are presented to show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from the homogenous states is depicted, also showing coexistence of multiple solutions at the same parameter values. Combining analytical estimates and rigorous computations, the solutions are sought as fixed points of a operator in a suitable Banach space. To address the curse of dimensionality, a variation of the existing technique is presented, necessary to enable successful computations in reasonable time. [ABSTRACT FROM AUTHOR]

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to $$\varepsilon =0.9716$$ ), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map. [ABSTRACT FROM AUTHOR]

Verification methods compute intervals which contain the solution of a given problem with mathematical rigour. In order to compare the quality of intervals some measure is desirable. We identify some anticipated properties and propose a method avoiding drawbacks of previous definitions. [ABSTRACT FROM AUTHOR]

In this research, we describe the modulus-based synchronous multisplitting iteration method for solving a class of horizontal nonlinear complementarity problems by utilizing the matrix multisplitting technique. We derive several convergence criteria to guarantee the suggested method's convergence, and we provide numerical examples to demonstrate our method's efficacy. [ABSTRACT FROM AUTHOR]

This paper proposes a new set-membership state estimator for estimating the state vector of a nonlinear dynamic robot. The method combines a symbolic technique based on flatness concepts with rigorous numerical methods based on interval analysis. Two testcases related to the state estimation of a sailboat robot are proposed to illustrate the principle and the efficiency of the approach. [ABSTRACT FROM AUTHOR]

The approach to deriving interval-valued reliability measures described in this paper is distinctive from other imprecise reliability models in that it overcomes the issue of having to impose an upper bound on time to failure. It rests on the presupposition that a constant interval-valued failure rate is known possibly along with other reliability measures, precise or imprecise. The Lagrange method is used to solve the constrained optimization problem to derive new reliability measures of interest. The obtained results call for an exponential-wise approximation of failure probability density function if only partial failure information is available. An example is provided. [ABSTRACT FROM AUTHOR]

The interval Newton method can be used for computing an enclosure of a single simple zero of a smooth function in an interval domain. It can practically be extended to allow computing enclosures of all zeros in a given interval. This paper deals with the extended interval Newton method. An essential operation of the method is division by an interval that contains zero (extended interval division). This operation has been studied by many researchers in recent decades, but inconsistency in the research has occurred again and again. This paper adopts the definition of extended interval division redefined in recent documents (Kulisch in Arithmetic operations for floating-point intervals, ; Pryce in P1788: IEEE standard for interval arithmetic version 02.2, ). The result of the division is called the precise quotient set. Earlier definitions differ in the overestimation of the quotient set in particular cases, causing inefficiency in Newton's method and even leading to redundant enclosures of a zero. The paper reviews and compares some extended interval quotient sets defined during the last few decades. As a central theorem, we present the fundamental properties of the extended interval Newton method based on the precise quotient set. On this basis, we develop an algorithm and a convenient program package for the extended interval Newton method. Statements on its convergence are also given. We then demonstrate the performance of the algorithm through nine carefully selected very sensitive numerical examples and show that it can compute correct enclosures of all zeros of the functions with high efficiency, particularly in cases where earlier methods are less effective. [ABSTRACT FROM AUTHOR]

An automated general purpose method is introduced for computing a rigorous estimate of a bounded region in ℝ n whose points satisfy a given property. The method is based on calculations conducted in interval arithmetic and the constructed approximation is built of rectangular boxes of variable sizes. An efficient strategy is proposed, which makes use of parallel computations on multiple machines and refines the estimate gradually. It is proved that under certain assumptions the result of computations converges to the exact result as the precision of calculations increases. The time complexity of the algorithm is analyzed, and the effectiveness of this approach is illustrated by constructing a lower bound on the set of parameters for which an overcompensatory nonlinear Leslie population model exhibits more than one attractor, which is of interest from the biological point of view. This paper is accompanied by efficient and flexible software written in C++ whose source code is freely available at . [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the concept of inverse of a matrix with fuzzy numbers as its elements, which may be used to model uncertain and imprecise aspects of real-world problems. We pursue two main ideas based on employing real scenarios and arithmetic operators. In each case, exact and inexact strategies are provided. In the first idea, we give some necessary and sufficient conditions for invertibility of fuzzy matrices based on regularity of their scenarios. And then Zadeh's extension principle and interpolation on Rohn's approach for inverting interval matrices are followed to compute fuzzy inverse. In the second idea, Dubois and Prade's arithmetic operators will be employed for the same purpose. But with respect to the inherent difficulties which are derived from the positivity restriction on spreads of fuzzy numbers, the concept of ε-inverse of a fuzzy matrix and its relaxation are generalized and some useful theorems will be revealed. Finally fuzzifying the defuzzified version of the original problem for introducing fuzzy inverse, which can be followed by each idea, will be presented. [ABSTRACT FROM AUTHOR]

The application of interval-valued statistical models is often hindered by the rapid growth in imprecision that occurs when intervals are propagated through models. Is this deficiency inherent in the models? If so, what is the underlying cause of imprecision in mathematical terms? What kind of additional information can be incorporated to make the bounds tighter? The present paper gives an account of the source of this imprecision that prevents interval-valued statistical models from being widely applied. Firstly, the mathematical approach to building interval-valued models (discrete and continuous) is delineated. Secondly, a degree of imprecision is demonstrated on some simple reliability models. Thirdly, the root mathematical cause of sizeable imprecision is elucidated and, finally, a method of making the intervals tighter is described. A number of examples are given throughout the paper. [ABSTRACT FROM AUTHOR]

Interval analysis is a new uncertainty analysis method for engineering structures. In this paper, a new sensitivity analysis method is presented by introducing interval analysis which can expand applications of the interval analysis method. The interval analysis process of sensitivity factor matrix of soil parameters is given. A method of parameter intervals and decision-making target intervals is given according to the interval analysis method. With FEM, secondary developments are done for Marc and the Duncan-Chang nonlinear elastic model. Mutual transfer between FORTRAN and Marc is implemented. With practial examples, rationality and feasibility are validated. Comparison is made with some published results. [ABSTRACT FROM AUTHOR]