# Solving Nonconvex Process Optimisation Problems Using Interval Subdivision Algorithms

• R. P. Byrne
• I. D. L. Bogle
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 9)

## Abstract

Many Engineering Design problems are nonconvex. A particular approach to global optimisation, the class of ‘Covering Methods’, is reviewed in a general framework. The method can be used to solve general nonconvex problems and provides guarantees that solutions are globally optimal. Aspects of the Interval Subdivision method are presented with the results of their application to some illustrative test problems. The results show the care that must be taken in constructing inclusion functions and demonstrate the effects of some different implementation decisions. Some particular difficulties of applying the method to constrained problems are brought to light by the results.

## Keywords

Feasible Point Interval Analysis Inclusion Function Tight Bound Interval Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Rinnoy Kan, A.H.G, Timmer, G.T (1987). “Stochastic Global Optimization Methods Part II: Multi—Level Methods.” Math. Prog. 39 (1) 57–78.
2. [2]
Rinnoy Kan, A.H.G, Timmer, G.T (1987). “Stochastic Global Optimization Methods Part I: Clustering Methods.” Math. Prog. 39 (1) 27–56.
3. [3]
Androulakis, I.P and Venkatasubramanian, V. (1991) “A Genetic Algorithmic Framework for Process Design and Optimization.” Computer Chem. Engng. 15 (4) 217–228.
4. [4]
Floudas, C.A., Visweswaran, V. (1990) “A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs. 1 Theory.” Comp. Chem. Eng., 14 (12), 1397–1417.
5. [5]
Floudas, C.A., Aggarwal, A., Ciric, A.R. (1989) “Global Optimum Search for Nonconvex NLP and MINLP Problems.” Computer. Chem. Eng 13 (10) 1117–1132.
6. [6]
Quesada, I., Grossmann, I.E (1993) “Global Optimization Algorithm for Heat-Exchanger Networks.” Ind. Eng. Chem. Res. 32 (3) 487–499.
7. [7]
Hansen, E. (1992) “Global Optimization Using Interval Analysis.” Marcel Dekker, New York.Google Scholar
8. [8]
Kocis, G.R, Grossmann, I.E (1991). “Global Optimization of Nonconvex Mixed—Integer Nonlinear—Programming (MINLP) Problems in Process Synthesis.”Ind. Eng. Chem. Res. 27 (8) 1407–1421.Google Scholar
9. [9]
Piyayskii, S.A (1972). “An Algorithm for Finding the Absolute Extremum of a Function.” USSR Comp. Mat. Si Mat. Phys. 12 57–67.Google Scholar
10. [10]
Meewela, C.C, Mayne, D.Q (1988) “An Algorithm for Global Optimization of Lipschitz Continuous Functions.” J.Optim. Theory. Appl 57 (2) 307–322.
11. [11]
Törn, A., 2ilinskas, A. (1989), “Global Optimization. Lecture Notes in Computer Science.” Springer—Verlag, Berlin.
12. [12]
Hansen, P., Jaumard, B., Lu, S.H (1992). “On Using Estimates of Lipschitz-Constants in Global Optimization.” J. Optim. Theory. Appl. 75 (1) 195200.Google Scholar
13. [13]
Hansen, P., Jaumard, B., Lu, S.H (1992). “Global Optimization of Uni-variate Lipschitz Functions.1. Survey and Properties.” Math. Prog. 55 (3) 251–272.
14. [14]
Moore, R.E. (1966), “Interval Analysis.”, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
15. [15]
Ratschek, H., Rockne, J. (1988), “New Computer Methods for Global Optimization”, Ellis Horwood Ltd., Chichester, West Sussex, England.Google Scholar
16. [16]
Neumaier, A. “Interval Methods for Systems of Equations.” Cambridge University Press, London.Google Scholar
17. [17]
Csendes, T., Pintér, J. (1993) “The Impact of Accelerating Tools on the Subdivision Algorithm for Global Optimization.” European J. of Ops. Res. 65 314–320.
18. [18]
Schnepper, C.A, Stadtherr, M.A (1993) “Application of a Parallel Interval Newton/Generalized Bisection Algorithm to Equation-based Chemical Process Flowsheeting.”, in Proc. International Conference on Numerical Analysis with Automatic Result Verification, Lafayette, LA.Google Scholar