Cubic overestimation and secant updating for unconstrained optimization of C functions.
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| Title: | Cubic overestimation and secant updating for unconstrained optimization of C functions. |
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| Authors: | Griewank, Andreas1 (AUTHOR) griewank@math.hu-berlin.de, Fischer, Jonathan2 (AUTHOR), Bosse, Torsten1 (AUTHOR) |
| Source: | Optimization Methods & Software. Aug2014, Vol. 29 Issue 5, p1075-1089. 15p. |
| Subjects: | Estimation theory, Secant function, Mathematical optimization, Approximation theory, Mathematical models, Iterative methods (Mathematics) |
| Abstract: | The discrepancy between an objective functionfand its local quadratic modelf(x)+∇f(x)⊤s+s⊤H(x)s/2 ≈f(x+s) at the current iteratexis estimated using a cubic termq|s|3/3. Potential steps are chosen such that they minimize (or at least significantly reduce) the overestimating function ∇f(x)⊤s+s⊤B s/2+q|s|3/3 withB≈H(x). This ensuresf(x+s) |
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| Database: | Engineering Source |
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| Abstract: | The discrepancy between an objective functionfand its local quadratic modelf(x)+∇f(x)⊤s+s⊤H(x)s/2 ≈f(x+s) at the current iteratexis estimated using a cubic termq|s|3/3. Potential steps are chosen such that they minimize (or at least significantly reduce) the overestimating function ∇f(x)⊤s+s⊤B s/2+q|s|3/3 withB≈H(x). This ensuresf(x+s)<f(x) unless the approximating HessianB=B⊤differs significantly fromH(x) or the scalarq>0 is too small. Either one or both quantities may be updated after unsuccessful and successful steps alike. For an algorithm employing both the symmetric rank one update and a shifted version of the BFGS formula we show that either∈f|∇f|=0 or sup |B|=∞, provided the HessianH(x) is Lipschitz on some neighbourhood of a bounded level set. Superlinear convergence is theoretically expected and numerically observed but not yet proven. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 10556788 |
| DOI: | 10.1080/10556788.2013.863308 |
