An Interval Partitioning Approach for Continuous Constrained Optimization
Constrained Optimization Problems (COP’s) are encountered in many scientific fields concerned with industrial applications such as kinematics, chemical process optimization, molecular design, etc. When non-linear relationships among variables are defined by problem constraints resulting in non-convex feasible sets, the problem of identifying feasible solutions may become very hard. Consequently, finding the location of the global optimum in the COP is more difficult as compared to bound-constrained global optimization problems.
This chapter proposes a new interval partitioning method for solving the COP. The proposed approach involves a new subdivision direction selection method as well as an adaptive search tree framework where nodes (boxes defining different variable domains) are explored using a restricted hybrid depth-first and best-first branching strategy. This hybrid approach is also used for activating local search in boxes with the aim of identifying different feasible stationary points. The proposed search tree management approach improves the convergence speed of the interval partitioning method that is also supported by the new parallel subdivision direction selection rule (used in selecting the variables to be partitioned in a given box). This rule targets directly the uncertainty degrees of constraints (with respect to feasibility) and the uncertainty degree of the objective function (with respect to optimality). Reducing these uncertainties as such results in the early and reliable detection of infeasible and sub-optimal boxes, thereby diminishing the number of boxes to be assessed. Consequently, chances of identifying local stationary points during the early stages of the search increase.
The effectiveness of the proposed interval partitioning algorithm is illustrated on several practical application problems and compared with professional commercial local and global solvers. Empirical results show that the presented new approach is as good as available COP solvers.
Key wordscontinuous constrained optimization interval partitioning approach practical applications
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- [Aem]www.aemdesign.com/FSQPmanyobj.htmGoogle Scholar
- [BFG87]Benhabib, B., Fenton, R.G., Goldberg, A.A.: Analytical trajectory optimization of seven degrees of freedom redundant robot. Transactions of the Canadian Society for Mechanical Engineering, 11, 197–200 (1987)Google Scholar
- [BLW80]Berna, T., Locke, M., Westerberg, A.W.: A new approach to optimization of chemical processes. American Institute of Chemical Engineers Journal, 26, 37–43 (1980)Google Scholar
- [BMV94]Benhamou, F., McAllester, D., Van Hentenryck, P.: CLP (Intervals) revisited. Proc. of ILPS’94, International Logic Programming Symposium, 124–138 (1994)Google Scholar
- [DNS97]Dallwig, S., Neumaier, A., Schichl, H.: GLOPT-a program for constrained global optimization. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P.M. (eds) Developments in Global Optimization. Kluwer, Dordrecht, 19–36 (1997)Google Scholar
- [Dru96]Drud, A.S.: CONOPT: A System for Large Scale Nonlinear Optimization. Reference Manual for CONOPT Subroutine Library, ARKI Consulting and Development A/S, Bagsvaerd, Denmark (1996)Google Scholar
- [GMS97]Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: An SQP Algorithm for Large-scale Constrained Optimization. Numerical Analysis Report 97-2, Department of Mathematics, University of California, San Diego, La Jolla, CA (1997)Google Scholar
- [HS80]Hansen, E., Sengupta, S.: Global constrained optimization using interval analysis. In: Nickel, K.L. (ed) Interval Mathematics. Academic Press, New York (1980)Google Scholar
- [Kea94]Kearfott, R.B.: On Verifying Feasibility in Equality Constrained Optimization Problems. preprint (1994)Google Scholar
- [Kea96a]Kearfott, R.B.: A review of techniques in the verified solution of constrained global optimization problems. In: Kearfott, R.B., Kreinovich, V. (eds) Applications of Interval Computations. Kluwer, Dordrecht, Netherlands, pp. 23–60 (1996)Google Scholar
- [Kea96b]Kearfott, R.B.: Test results for an interval branch and bound algorithm for equality-constrained optimization. In: Floudas, C, Pardalos, P. (eds) State of the Art in Global Optimization: Computational Methods and Applications. Kluwer, Dordrecht, Netherlands, pp. 181–200 (1996)Google Scholar
- [Kea03]Kearfott, R.B.: An Overview of the GlobSol Package for Verified Global Optimization. talk given for the Department of Computing and Software, McMaster University, Ontario, Canada (2003)Google Scholar
- [Kea04]Kearfott, R.B.: Empirical comparisons of linear relaxations and alternate techniques in validated deterministic global optimization. Optimization Methods and Software, accepted (2004)Google Scholar
- [Kea05]Kearfott, R.B.: Improved and simplified validation of feasible points: inequality and equality constrained problems. Mathematical Programming, submitted (2005)Google Scholar
- [LZT97]Lawrence, C.T, Zhou, J.L., Tits, A.L.: User’s Guide for CFSQP Version 2.5: A Code for Solving (Large Scale) Constrained Nonlinear (minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints. Institute for Systems Research, University of Maryland, College Park, MD (1997)Google Scholar
- [Mar03]Markót, M.C.: Reliable Global Optimization Methods for Constrained Problems and Their Application for Solving Circle Packing Problems. PhD dissertation, University of Szeged, Hungary (2003)Google Scholar
- [MFCC05]Markót, M.C., Fernandez, J., Casado, L.G., Csendes, T.: New interval methods for constrained global optimization. Mathematical Programming, accepted (2005)Google Scholar
- [MNW]Morales, J.L., Nocedal, J., Waltz, R., Liu, G., Goux, J.P.: Assessing the Potential of Interior Methods for Nonlinear Optimization. Optimization Technology Center, Northwestern University, USA (2001)Google Scholar
- [Moo66]Moore, R.E.: Interval Anlaysis. Prentice-Hall, Englewood Cliffs, New Jersey (1966)Google Scholar
- [MS87]Murtagh, B.A., Saunders, M.A.: MINOS 5.0 User’s Guide. Report SOL 83-20, Department of Operations Research, Stanford University, USA (1987)Google Scholar
- [Pin97]Pintér, J.D.: LGO — a program system for continuous and Lipschitz global optimization. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P.M. (eds) Developments in Global Optimization. Kluwer Academic Publishers, Boston/Dordrecht/London, pp. 183–197 (1997)Google Scholar
- [PR02]Pardalos, P.M., Romeijn, H.E.: Handbook of Global Optimization Volume 2. Nonconvex Optimization and Its Applications. Springer, Boston/Dordrecht/London (2002)Google Scholar
- [Pri]PrincetonLib.: Princeton Library of Nonlinear Programming Models. www.gamsworld.org/performance/princetonlib/princetonlib.htmGoogle Scholar
- [Sah03]Sahinidis, N.V.: Global optimization and constraint satisfaction: the branch-and-reduce approach. In: Bliek, C, Jermann, C, Neumaier, A. (eds) COCOS 2002. LNCS, 2861, 1–16 (2003)Google Scholar
- [San88]Sandgren, E.: Nonlinear integer and discrete programming in mechanical design. Proceeding of the ASME Design Technology Conference, Kissimmee, FL, 95–105 (1988)Google Scholar
- [SNS]Scherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.-H., Nguyen, T.-V.: Benchmarking global optimization and constraint satisfaction codes. Global Optimization and Constraint Satisfaction: First International Workshop on Global Constraint Optimization and Constraint Satisfaction, COCOS 2002, Valbonne-Sophia Antipolis, France (2002)Google Scholar
- [TZ89]Torn, A., Žilinskas, A.: Global Optimization. Lecture Notes in Computer Science, 350, Springer, Berlin (1989)Google Scholar