Approximating Data in $$\mathfrak{R}^{n}$$ by a Quadratic Underestimator with Specified Hessian Minimum and Maximum Eigenvalues

Autor: John Glick, J. B. Rosen
Rok vydání: 2006
Předmět:
Zdroj: Journal of Global Optimization. 36:461-469
ISSN: 1573-2916
0925-5001
DOI: 10.1007/s10898-006-9021-4
Popis: The problem of approximating m data points (x i , y i ) in $$\mathfrak{R}^{n+1}$$ , with a quadratic function q(x, p) with s parameters, m ? s, is considered. The parameter vector $$p\in \mathfrak{R}^s$$ is to be determined so as to satisfy three conditions: (1) q(x, p) must underestimate all m data points, i.e. q(x i , p) ? y i , i=1,...,m. (2) The error of the approximation is to be minimized in the L1 norm. (3) The eigenvalues of H are to satisfy specified lower and upper bounds, where H is the Hessian of q(x, p) with respect to x. This is called the Quadratic Underestimator with Bounds on Eigenvalues (QUBE) problem. An algorithm for its solution (QUBE algorithm) is given and justified, and computational results presented. The QUBE algorithm has application to finding the global minimum of a basin (or funnel) shaped function with a large number of local minima. Such problems arise in computational biology where it is desired to find the global minimum of an energy surface, in order to predict native protein-ligand docking geometry (drug design) or protein structure. Computational results for a simulated docking energy surface, with n=15, are presented. It is shown that specifying a small condition number for H improves the ability of the underestimator to correctly predict the global minimum point.
Databáze: OpenAIRE
Externí odkaz: