Reduced variable optimization methods via implicit functional dependence with applications

TL;DR

This paper introduces a theoretical framework for optimizing multiple variables by reducing the problem to a single variable, with applications in physical sciences and chemical reaction analysis.

Contribution

It develops new exact and approximate methods for variable reduction in optimization, extending the implicit function theorem for broader applications.

Findings

Methods effectively reduce multi-variable optimization to single-variable problems.

Application to chemical reaction data demonstrates practical utility.

The approach surpasses existing reduction schemes and does not require boundary conditions.

Abstract

Optimization methods have been broadly applied to two classes of objects viz. (i) modeling and description of data and (ii) the determination of the stationary points of functions. Here, a theoretical basis is developed that optimizes an arbitrary number of variables for classes (i) and (ii) by the minimization of a function of a single variable. Algorithms that focus on a reduced variable set also avoid problems associated with multiple minima and maxima that arise because of the large numbers of parameters. The methods described could have applications in the physical sciences where the optimization of one physically significant variable has priority over the other variables. For (i), we develop both an approximate but computationally more tractable method and an exact method where the single controlling variable k of all the other variables (P,k) passes through the local stationary point of the least squares (LS) metric. For (ii), an exact theory is developed whereby the optimized function of an independent variation of all parameters coincides with that due to single parameter optimization. The implicit function theorem has to be further qualified to arrive at this result. The topology of the surfaces of constant value of the target or cost function are considered for all the methods. A real world application of the above implicit methodology to rate constant and final concentration parameter determination for first and second order chemical reactions from published data. This work is different from and more general than all the reduction schemes for conditional linear parameters nor is it a subset of the Adomian decomposition method (ADM) used for estimating solutions of differential equations, which still require boundary conditions that do not feature in topics (i) and (ii).