Abstract
We consider nonlinear programming problems the input data of which are not fixed, but vary in some real compact intervals. The aim of this paper is to determine bounds of the optimal values. We propose a general framework for solving such problems. Under some assumption, the exact lower and upper bounds are computable by using two non-interval optimization problems. While these two optimization problems are hard to solve in general, we show that for some particular subclasses they can be reduced to easy problems. Subclasses that are considered are convex quadratic programming and posynomial geometric programming.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press, London
Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming. Theory and algorithms, 2nd edn. Wiley, New York
Boţ RI, Grad SM, Wanka G (2006) Fenchel-Lagrange duality versus geometric duality in convex optimization. J Optim Theory Appl 129(1):33–54
Chinneck JW, Ramadan K (2000) Linear programming with interval coefficients. J Oper Res Soc 51(2):209–220
Dorn WS (1960) Duality in quadratic programming. Q Appl Math 18(2):155–162
Duffin RJ, Peterson EL (1966) Duality theory for geometric programming. SIAM J Appl Math 14(6):1307–1349
Fiedler M, Nedoma J, Ramik J, Rohn J, Zimmermann K (2006) Linear optimization problems with inexact data. Springer, New York
Gerlach W (1981) Zur Lösung linearer Ungleichungssysteme bei Störung der rechten Seite und der Koeffizientenmatrix. Math Oper Stat Ser Optim 12:41–43
Hansen E, Walster GW (2004) Global optimization using interval analysis, 2nd edn. Dekker, New York
Hladík M (2007) Optimal value range in interval linear programming. Fuzzy Optim Decis Mak (Submitted). Available as a technical report KAM-DIMATIA Series 2007-824, Dep of Appl Math, Prague
Hu B, Wang S (2006a) A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers. J Ind Manage Optim 2(4):351–371
Hu B, Wang S (2006b) A novel approach in uncertain programming. II: a class of constrained nonlinear programming problems with interval objective functions. J Ind Manage Optim 2(4):373–385
Huang GH (1998) A hybrid inexact-stochastic water management model. Eur J Oper Res 107(1):137–158
Huang GH, Baetz BW, Patry GG, Terluk V (1997) Capacity planning for an integrated waste management system under uncertainty: a North American case study. Waste Manage Res 15(5):523–546
Levin VI (1999) Nonlinear optimization under interval uncertainty. Cybern Syst Anal 35(2):297–306
Liu ST (2006a) Computational method for the profit bounds of inventory model with interval demand and unit cost. Appl Math Comput 183(1):499–507
Liu ST (2006b) Posynomial geometric programming with parametric uncertainty. Eur J Oper Res 168(2):345–353
Liu ST (2008) Posynomial geometric programming with interval exponents and coefficients. Eur J Oper Res 186(1):17–27
Liu ST, Wang RT (2007) A numerical solution method to interval quadratic programming. Appl Math Comput 189(2):1274–1281
Martos B (1975) Nonlinear programming. Theory and methods. Akadémiai Kiadó, Budapest
Mráz F (1998) Calculating the exact bounds of optimal values in LP with interval coefficients. Ann Oper Res 81:51–62
Neumaier A (2004) Complete search in continuous global optimization and constraint satisfaction. Acta Numer 13:271–369
Rohn J, Kreslová J (1994) Linear interval inequalities. Linear Multilinear Algebra 38(1/2):79–82
Wu XY, Huang GH, Liu L, Li JB (2006) 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. Eur J Oper Res 171(2):349–372
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hladík, M. Optimal value bounds in nonlinear programming with interval data. TOP 19, 93–106 (2011). https://doi.org/10.1007/s11750-009-0099-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-009-0099-y