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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 108184, 13 pages
http://dx.doi.org/10.1155/2014/108184
Research Article

Numerical Reduced Variable Optimization Methods via Implicit Functional Dependence with Applications

Department of Chemistry and Center for Theoretical and Computational Physics, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 14 March 2014; Accepted 24 May 2014; Published 27 August 2014

Academic Editor: Mariano Torrisi

Copyright © 2014 Christopher Gunaseelan Jesudason. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A systematic theoretical basis is developed that optimizes an arbitrary number of variables for (i) modeling data and (ii) the determination of stationary points of a function of several variables by the optimization of an auxiliary function of a single variable deemed the most significant on physical, experimental or mathematical grounds from which all the other optimized variables may be derived. Algorithms that focus on a reduced variable set avoid problems associated with multiple minima and maxima that arise because of the large numbers of parameters. For (i), both approximate and exact methods are presented, where the single controlling variable k of all the other variables passes through the local stationary point of the least squares metric. For (ii), an exact theory is developed whereby the solution of the optimized function of an independent variation of all parameters coincides with that due to single parameter optimization of an auxiliary function. The implicit function theorem has to be further qualified to arrive at this result. A nontrivial real world application of the above implicit methodology to rate constant and final concentration parameter determination is made to illustrate its utility. This work is more general than the reduction schemes for conditional linear parameters since it covers the nonconditional case as well and has potentially wide applicability.