ISSN:
1573-2878
Keywords:
Unconstrained optimization
;
variable-metric methods
;
quasi-Newton methods
;
numerical algorithms
;
nonlinear programming
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Quasi-Newton algorithms minimize a functionF(x),x ∈R n, searching at any iterationk along the directions k=−H kgk, whereg k=∇F(x k) andH k approximates in some sense the inverse Hessian ofF(x) atx k. When the matrixH is updated according to the formulas in Broyden's family and when an exact line search is performed at any iteration, a compact algorithm (free from the Broyden's family parameter) can be conceived in terms of the followingn ×n matrix: $$H{_R} = H - Hgg{^T} H/g{^T} Hg,$$ which can be viewed as an approximating reduced inverse Hessian. In this paper, a new algorithm is proposed which uses at any iteration an (n−1)×(n−1) matrixK related toH R by $$H_R = Q\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & K \\ \end{array} } \right]Q$$ whereQ is a suitable orthogonaln×n matrix. The updating formula in terms of the matrixK incorporated in this algorithm is only moderately more complicated than the standard updating formulas for variable-metric methods, but, at the same time, it updates at any iteration a positive definite matrixK, instead of a singular matrixH R. Other than the compactness with respect to the algorithms with updating formulas in Broyden's class, a further noticeable feature of the reduced Hessian algorithm is that the downhill condition can be stated in a simple way, and thus efficient line searches may be implemented.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00940543
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