工程及科技实践中，多种不确定系统的数学模型可以用区间系统以及区间优化模型来刻画.文章主要探讨混合区间系统的区间解问题.我们定义了混合区间线性系统几种新的区间解，包括弱区间解、强区间解、容许区间解和控制区间解，并研究了它们的相关性质.特别地，我们还刻画了区间方程和区间不等式之各类区间解的特征，这些特征与区间分析中著名的经典不等式Oettli-Prager不等式和Gerlach不等式在形式上完全类似.同时，文章给出了一个区间解的应用背景实例和一个区间解计算实例.
Abstract
The mathematical models of various uncertain systems in engineering and scientific practice can be described by interval systems and interval optimization models. In this paper, we discuss the problem of interval solutions for interval systems. We define several new interval solutions for mixed interval linear systems, including weak interval solutions, strong interval solutions, tolerance interval solution and control interval solution and study their related properties. In particular, the characterizations of weak interval solution, strong interval solution, tolerance interval solution and control interval solution of the interval equations and the interval inequalities are established. These characterizations are similar to the classical Oettli-Prager inequality and Gerlach inequality. At the same time, we give some examples to illustrate the application background and calculation method of interval solutions.
关键词
区间系统 /
区间方程 /
区间不等式 /
区间解
{{custom_keyword}} /
Key words
Interval system /
interval equation /
interval inequality /
interval solution
{{custom_keyword}} /
中图分类号：
15A06
65G40
{{custom_clc.code}}
({{custom_clc.text}})
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Fiedler M, Nedoma J, Ramik J, et al. Linear Optimization Problems with Inexact Data. New York:Springer-Verlag, 2006.
[2] Li W, Tian X. Fault detection in discrete dynamic systems with uncertainty based on interval optimization, Math. Model. Anal., 2011, 16(4):549-557.
[3] Efimov D, Raissi T. Design of interval observers for uncertain dynamical systems. Automat. Rem. Contr., 2016, 77(2):191-225.
[4] Oliver B, Bernard P. Linear interval tolerance problem and linear programming techniques. Reliab. Comput., 201, 7(6):433-447.
[5] Rohn J. A general method for enclosing solutions of interval linear equations. Optim. Lett., 2012, 6:709-717.
[6] Hladík M. Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl., 2013, 438:4156-4165.
[7] Li W, Wang H, Wang Q. Localized solutions to interval linear equations. J. Comput. Appl. Math., 2013, 238(15):29-38.
[8] Li H. Necessary and sufficient conditions for unified optimality of interval linear program in the general form. Linear Algebra Appl., 2015, 484:154-174.
[9] 李炜, 夏梦雪, 李好好, 一般区间线性系统的Farkas型定理, 系统科学与数学, 2016, 36(11):1959-1971. (Li W, Xia M X, Li H H. Farkas-type theorems for general interval linear systems. Journal of Systems Science and Mathematical Sciences, 2016, 36(11):1959-1971.)
[10] Huang J, Wang C, Li H. EA solutions and EA solvability to general interval linear systems. Linear Multilinear A., 2021, 69(15):2865-2881.
[11] Thipwiwatpotjana P, Gorka A, Leela-apirade W. Solution types of two-sided interval linear system and their application on interval linear programming problems. J. Comput. Appl. Math., 2021, 388:113-294.
[12] Polyak B T, Nazin S A. Interval solutions for interval algebraic equations. Math. Comput. Simulat., 2004, 66:207-217.
[13] Allahviranloo T, Ghanbari M. A new approach to obtain algebraic solution of interval linear systems. Soft Comput., 2012, 16(1):121-133.
[14] Ghanbari M. An estimation of algebraic solution for a complex interval linear system. Soft Comput., 2018, 22:2881-2890.
[15] Oettli W, Prager W. Compatibility of approximate solution of linear equations with given error bounds for coeficients and right-hand sides. Numer. Math., 1964, 6:405-409.
[16] Gerlach W. Zur Losung hearer Ungleichungssysteme bei Storung der rechten Seite und der Koefizientenmatrix. Mathematische Operationsforschung und Statistik. Ser. Optim., 1981, 12:41-43.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
基金
浙江省自然科学基金（LY21A010021），国家自然科学基金（11701506）资助课题.
{{custom_fund}}
PDF(354 KB)
