In mathematical optimization, a trust region is the subset of the region of the objective function that is approximated using a model function (often a quadratic). If an adequate model of the objective function is found within the trust region, then the region is expanded; conversely, if the approximation is poor, then the region is contracted.
The fit is evaluated by comparing the ratio of expected improvement from the model approximation with the actual improvement observed in the objective function. Simple thresholding of the ratio is used as the criterion for expansion and contraction—a model function is "trusted" only in the region where it provides a reasonable approximation.
Trust-region methods are in some sense dual to line-search methods: trust-region methods first choose a step size (the size of the trust region) and then a step direction, while line-search methods first choose a step direction and then a step size.
The general idea behind trust region methods is known by many names; the earliest use of the term seems to be by Sorensen (1982).[1] A popular textbook by Fletcher (1980) calls these algorithms restricted-step methods.[2] Additionally, in an early foundational work on the method, Goldfeld, Quandt, and Trotter (1966) refer to it as quadratic hill-climbing.[3]
^Sorensen, D. C. (1982). "Newton's Method with a Model Trust Region Modification". SIAM J. Numer. Anal. 19 (2): 409–426. doi:10.1137/0719026.
^Fletcher, Roger (1987) [1980]. "Restricted Step Methods". Practical Methods of Optimization (Second ed.). Wiley. ISBN 0-471-91547-5.
^Goldfeld, Stephen M.; Quandt, Richard E.; Trotter, Hale F. (1966). "Maximization by Quadratic Hill-Climbing". Econometrica. 34 (3): 541–551. doi:10.2307/1909768. JSTOR 1909768.
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