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

The following theory is a systematic development of all functions covering properties of constrained and unconstrained functions that are continuous and differentiable to various specified degrees [

This approximate method utilizes the average of the

Let

For any

The above follows from the implicit function theorem (IFT) [

We seek the solutions for

Each

Then for an optimized

Define the Lagrangian to the problem as

Of interest is the theoretical relationship of the

The unconstrained LS solution to

The

The theorem, verification, and lemma above do not indicate topologically under what conditions a coincidence of solutions for the constrained and unconstrained models exists. Figure

Depiction of how the

The LS function metric such as (

A real function

For any region bounded by

Suppose in fact

The function that is optimized is

Any point

Hence we have demonstrated that it may be more realistic or accurate to fit parameters based on a function that represents different coupling sets such as

Whilst it is advantageous in science data analysis to optimize a particular multiparameter function by focusing on a few key variables (our

For the

As before

Methods (i)a and (i)b which are mutual variants of each other are applications of the implicit method to modeling problems to provide a best fit to a function. Here, another variant of the implicit methodology for optimization of a target or cost function

The stationary points

If

For nondegenerate coordinate choice, meaning that for a particular

We provide suggestions in pseudocode form for the above 3 proven methodologies. Real world applications of these methods are very involved undertakings that are separate research topics in their own right. Nevertheless, we provide a detailed and extended application of method (i)a suitable for a real world chemical kinetics problem where experimental data from the published literature are used for method (i)a in Section

Of the many variations possible, the following approach conforms to the theoretical development.

For any physical law, for a total of

Determine from the above set

Determine

Solve the 1D equation at

This is an “exact” method relative to LS variation of all parameters. A suitable algorithm based on the theory could be as follows.

Solve

Form the function

Solve

The solution set to the problem is

Here

For a particular

Form the function

Solve for

Solution to the optimization problem of

The utility of one of the above triad of methods is illustrated in the determination of two parameters in chemical reaction rate studies, of

Reaction (i) above corresponds to

For this reaction, the

The experimental estimates are

The experimental method involves adjusting the

Plot of the experimental and curve with optimized parameters showing the very close fit between the two. The slight difference between the two can probably be attributed to experimental errors.

To further test our method, we also analyze the second-order reaction

For Espenson, the above stoichiometry is kinetically equivalent to the reaction scheme [

The experimental estimates from the conventional methods are [

Again the two results are in close agreement. The graph of the experimental curve and the one that is derived from our optimization method are given in Figure

Graph of the experimental and calculated curve based on the current induced parameter-dependent optimization method.

The triad of associated implicit function optimization covers both the topics of modeling of data

the entire domain space of a function such as

refers to a function that is proposed to be a “law of nature” whose parameters

the theoretical law of nature if

an experimentally determined datapoint that ideally represents the range of

an averaged value for

least squares (LS) function to optimize

General cost or object function to be optimized, not necessarily in LS form (

Lagrange multipliers associated with the

Lagrangian to the optimization problem (

chemical kinetics rate constant for reaction

absorbance measurements for first-order chemical kinetics reactions at time

absorbance measurements for second-order chemical kinetics reactions at time

The author declares that there is no conflict of interests regarding the publication of this paper.

This work was supported by University of Malaya Grant UMRG(RG077/09AFR) and Malaysian Government grant FRGS(FP084/2010A). This work was initiated and completed during a Sabbatical research visit (2012-2013) to the Atomistic Simulation Centre (ASC), School of Mathematics and Physics, Queen’s University Belfast. I thank Ruth Lynden-Bell (Chemistry Department, Cambridge University) for facilitating this visit. Cordial discussions concerning real world applications with faculty at ASC are gratefully acknowledged. I thank my hosts, Jorge Kohanoff (ASC) and Christopher Hardacre (Chem. Dept., QUB) for congenial hospitality during this time.

_{∞}in reaction-rate studies