Constrained Optimization - Independent Variables

Independent experimental variables, such as elemental components and process conditions, are often constrained by economic, safety, or physical limits. Incorporating these constraints into the library design can significantly reduce the number of samples to be screened, and in the case of safety are essential. Such constraints are ideally suited to be incorporated into software library design tools and data analysis tools so that sufficiently unattractive experiments can be given lower priority than more attractive experiments. 

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The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respec t to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form or the generahzed reduced gradient (GRG) approach to optimizing, one of the better ways to cany out optimization numerically. 

The constrained optimization procedure, originally developed from the simplex method and first described by Box, is ideally suited to model refinement (.8). It is a search method that searches for the minimum of a multidimensional function within given intervals. It possesses all the advantages of search methods, among them that calculation of derivatives is not necessary, a test to assure the independence of variables can be omitted, and diverse variables can be easily included. These are exactly the requirements of model refinement where bond lengths, bond angles, torsion angles, and other parameters are used within experimentally defined limits. 

The number of independent variables in a constrained optimization problem can be found by a procedure analogous to the degrees of freedom analysis in Chapter 2. For simplicity, suppose that there are no constraints. If there are Ny process variables (which includes process inputs and outputs) and the process model consists of Ne independent equations, then the number of independent variables is Np = Ny - Ne-This means Np set points can be specified independently to maximize (or minimize) the objective function. The corresponding values of the remaining (Ny - Np) variables can be calculated from the process model. However, the presence of inequality constraints that can become active changes the situation, because the Np set points cannot be selected arbitrarily. They must satisfy all of the equality and inequality constraints. 

The methods for solving an optimization task depend on the problem classification. Since the maximum of a function / is the minimum of the function —/, it suffices to deal with minimization. The optimization problem is classified according to the type of independent variables involved (real, integer, mixed), the number of variables (one, few, many), the functional characteristics (linear, least squares, nonlinear, nondifferentiable, separable, etc.), and the problem. statement (unconstrained, subject to equality constraints, subject to simple bounds, linearly constrained, nonlinearly constrained, etc.). For each category, suitable algorithms exist that exploit the problem s structure and formulation. 

The Excel Solver. Microsoft Excel, beginning with version 3.0 in 1991, incorporates an NLP solver that operates on the values and formulas of a spreadsheet model. Versions 4.0 and later include an LP solver and mixed-integer programming (MIP) capability for both linear and nonlinear problems. The user specifies a set of cell addresses to be independently adjusted (the decision variables), a set of formula cells whose values are to be constrained (the constraints), and a formula cell designated as the optimization objective. The solver uses the spreadsheet interpreter to evaluate the constraint and objective functions, and approximates derivatives, using finite differences. The NLP solution engine for the Excel Solver is GRG2 (see Section 8.7). 

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