Univariate optimization strategies

Despite the multivariate nature of experimental designs, univariate optimization strategies can be applied when there is no interaction between factors. An interaction can be better understood with the help of Figure 2. 

The complexity of the response surface is what makes the optimization of chromatographic selectivity stand out as a particular optimization problem rather than as an example to which known optimization strategies from other fields can be readily applied. This is illustrated by the application of univariate optimization. In univariate optimization (or univariate search) methods the parameters of interest are optimized sequentially. An optimum is located by varying a given parameter and keeping all other parameters constant. In this way, an optimum value is obtained for that particular parameter. From this moment on the optimum value is assigned to the parameter and another parameter is varied in order to establish its optimum value . 

Whenever a new CE method is being developed, optimization strategies are usually applied to improve analysis speed, sensitivity, and resolution, using these three parameters or a combination of them as the monitored output (also called response or performance criteria). Very frequently, a step-by-step approach in which each factor is varied sequentially is followed. In this case, all parameters are kept constant, while the parameter of interest is varied and the response is measured. Depending on the problem (especially when the number of factors to optimize is very low) and on the performance criteria, univariate optimization can be useful, that is, the analysis of a single compound with only one component of the BGE. However, in most cases, a step-by-step optimization is laborious and tedious because it typically requires a high number of experiments. Furthermore, and more important, it does not consider possible interactions between factors. 

The optimization of the variables is a critical step in the design of new analytical methods. Optimization involves the selection of the chemical and instrumental factors which may affect the analytical signal, and the choice of the values of the variables to obtain the best response from the chemical system. For this purpose, two different strategies can be used. In the traditional univariate optimization, all values of the different factors except one are constant, and this one is the object of the examination. The alternative to this strategy is the use of chemometric techniques based mainly on the use of experimental designs (Tarley, et al. 2009). 

The line search procedure in step 3 of Algorithm [Al] is an approximate univariate minimization problem. It is typically performed via quadratic or cubic polynomial interpolation of the one-dimensional function X 0(X) = f k + Xpt). For the polynomial interpolant of ensure that the minimum of is located within the feasible region [xt, x -l- Xpt]. Typically, the initial trial value for X is one. Thus, the line search can be considered as a backtracking algorithm for X in the interval (0, 1]. 

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