Interval Analysis on Directed Acyclic Graphs for Global Optimization

• Published in: Journal of global optimization Vol. 33; no. 4; pp. 541 - 562
• Main Authors: Schichl, Hermann, Neumaier, Arnold
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Nature B.V 01.12.2005
• Subjects: Graph representations; Mathematical functions; Mathematical problems; Operations research; Optimization algorithms
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-005-0937-x

A directed acyclic graph (DAG) representation of optimization problems represents each variable, each operation, and each constraint in the problem formulation by a node of the DAG, with edges representing the flow of the computation. Using bounds on ranges of intermediate results, represented as weights on the nodes and a suitable mix of forward and backward evaluation, it is possible to give efficient implementations of interval evaluation and automatic differentiation. It is shown how to combine this with constraint propagation techniques to produce narrower interval derivatives and slopes than those provided by using only interval automatic differentiation preceded by constraint propagation. The implementation is based on earlier work by L.V. Kolev, (1997), Reliable Comput., 3, 83-93 on optimal slopes and by C. Bliek, (1992), Computer Methods for Design Automation, PhD Thesis, Department of Ocean Engineering, Massachusetts Institute of Technology on backward slope evaluation. Care is taken to ensure that rounding errors are treated correctly. Interval techniques are presented for computing from the DAG useful redundant constraints, in particular linear underestimators for the objective function, a constraint, or a Lagrangian. The linear underestimators can be found either by slope computations, or by recursive backward underestimation. For sufficiently sparse problems the work is proportional to the number of operations in the calculation of the objective function (resp. the Lagrangian). [PUBLICATION ABSTRACT]