Some Properties of Interval Quadratic Programming Problem
International Journal of Systems Science and Applied Mathematics
Volume 2, Issue 5, September 2017, Pages: 105-109
Received: Jun. 5, 2017; Accepted: Jun. 27, 2017; Published: Oct. 24, 2017
Views 2509      Downloads 102
Authors
Qianqian Xu, School of Sciences, Hangzhou Dianzi University, Hangzhou, China
Shengnan Jia, School of Sciences, Hangzhou Dianzi University, Hangzhou, China
Haohao Li, School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou, China
Jinhua Huang, School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Article Tools
Follow on us
Abstract
For interval liner programming problems, Rohn proposed four equivalence relations regarding to the upper and lower bounds of the interval optimal value. In this paper, similar problems of interval quadratic programming problem have been discussed. Some interesting properties have been proved and an illustrative example and remarks are given to get an insight of the properties.
Keywords
Interval Quadratic Programming, Lower and Upper Bounds, Optimal Value
To cite this article
Qianqian Xu, Shengnan Jia, Haohao Li, Jinhua Huang, Some Properties of Interval Quadratic Programming Problem, International Journal of Systems Science and Applied Mathematics. Vol. 2, No. 5, 2017, pp. 105-109. doi: 10.11648/j.ijssam.20170205.15
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Worrawate Leela-apiradee, WeldonA. Lodwick , Phantipa Thipwiwatpotjana, An algorithm for solving two-sided interval system of max-plus linear equations, Information Sciences 399 (2017) 183–200.
[2]
R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to interval analysis, SIAM, Philadelphia, 2009.
[3]
A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
[4]
Fiedler M, Rohn J, Nedoma J. Linear optimization problems with inexact date [M]. New York: Springer, 2006:35-66.
[5]
Rohn J. A general method for enclosing solutions of interval linear equations [J]. Optimization Letters, 2012, 6(4): 709-717.
[6]
S. P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity, Reliab. Comput. 8(5) (2002), pp. 321-418.
[7]
Hladík M. Interval convex quadratic programming problems in a general form [J]. Central European Journal of Operations Research, 2016: 1-13.
[8]
Hladik M. Optimal value bounds in nonlinear programming with interval data[J]. Top, 2011, 19(1): 93-106.
[9]
Li W, Jin J, Xia M, et al. Some properties of the lower bound of optimal values in interval convex quadratic programming [J]. Optimization Letters, 1-16.
[10]
Li W, Xia M, Li H. Some results on the upper bound of optimal values in interval convex quadratic programming [J]. Journal of Computational and Applied Mathematics, 2016, 302: 38-49.
[11]
Li W, Xia M , Li H. New method for computing the upper bound of optimal value in interval quadratic program[J]. Journal of Computational and Applied Mathematics, 2015,(288):70–80.
[12]
Chinneck J W, Ramadan K. Linear programming with interval coefficients [J]. Journal of the operational research society, 2000: 209-220.
[13]
Hladık M. Interval linear programming: A survey [J]. Linear programming-new frontiers in theory and applications, 2012: 85-120.
[14]
Mráz F. Calculating the exact bounds of optimal valuesin LP with interval coefficients [J]. Annals of Operations Research, 1998, 81: 51-62.
[15]
Fiedler M, Nedoma J, Ramik J, et al. Linear optimization problems with inexact data [M]. Springer Science & Business Media, 2006.
[16]
Hladík M. Optimal value range in interval linear programming [J]. Fuzzy Optimization and Decision Making, 2009, 8(3): 283-294.
[17]
Hladík M. On approximation of the best case optimal value in interval linear programming [J]. Optimization Letters, 2014, 8(7): 1985-1997.
[18]
Liu S T, Wang R T. A numerical solution method to interval quadratic programming [J]. Applied mathematics and computation, 2007, 189(2): 1274-1281.
[19]
Li W, Tian X. Numerical solution method for general interval quadratic programming [J]. Applied mathematics and computation, 2008, 202(2): 589-595.
[20]
Wu X Y, Huang G H, Liu L, et al. An interval nonlinear program for the planning of waste management systems with economies-of-scale effects—a case study for the region of Hamilton, Ontario, Canada [J]. European Journal of Operational Research, 2006, 171(2): 349-372.
[21]
Li W, Liu X, Li H. Generalized solutions to interval linear programmes and related necessary and sufficient optimality conditions [J]. Optimization Methods and Software, 2015, 30(3): 516-530.
[22]
Prokopyev O A, Butenko S, Trapp A. Checking solvability of systems of interval linear equations and inequalities via mixed integer programming [J]. European Journal of Operational Research, 2009, 199(1): 117-121.
[23]
Inuiguchi M, Ramik J, Tanino T, et al. Satisficing solutions and duality in interval and fuzzy linear programming [J]. Fuzzy Sets and Systems, 2003, 135(1): 151-177.
[24]
Li W, Luo J, Wang Q, et al. Checking weak optimality of the solution to linear programming with interval right-hand side[J]. Optimization Letters, 2014, 8(4): 1287-1299.
[25]
Ishibuchi H, Tanaka H. Multiobjective programming in optimization of the interval objective function [J]. European journal of operational research, 1990, 48(2): 219-225.
[26]
Shary S P. A new technique in systems analysis under interval uncertainty and ambiguity [J]. Reliable computing, 2002, 8(5): 321-418.
[27]
Neumaier A. Interval methods for systems of equations [M]. Cambridge university press, 1990.
[28]
Steuer R E. Algorithms for linear programming problems with interval objective function coefficients [J]. Mathematics of Operations Research, 1981, 6(3): 333-348.
[29]
Gabrel V, Murat C, Remli N. Linear programming with interval right hand sides [J]. International Transactions in Operational Research, 2010, 17(3): 397-408.
[30]
Hladík M. How to determine basis stability in interval linear programming [J]. Optimization Letters, 2014, 8(1): 375-389.
[31]
Allahdadi M, Nehi H M. The optimal solution set of the interval linear programming problems [J]. Optimization Letters, 2013, 7(8): 1893-1911.
[32]
Hladík M. Weak and strong solvability of interval linear systems of equations and inequalities [J]. Linear Algebra and its Applications, 2013, 438(11): 4156-4165.
[33]
Li W, Wang H, Wang Q. Localized solutions to interval linear equations [J]. journal of computational and applied mathematics, 2013, 238: 29-38.
[34]
Luo J, Li W. Strong optimal solutions of interval linear programming [J]. Linear Algebra and its Applications, 2013, 439(8): 2479-2493.
[35]
Wang X, Huang G. Violation analysis on two-step method for interval linear programming [J]. Information Sciences, 2014, 281: 85-96.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186