Constrained optimization techniques

Sharma and Prasad (2006b) optimized the process parameters for the microwave-convective drying of garlic cloves using a constrained optimization technique (Tukey s multicomparison pair-wise test), in an effort to identify the best compromise between quality and drying time. The results revealed that a... 

Most practical multivariable problems include constraints, which must be treated using enhancements of unconstrained optimization algorithms. The next two sections describe two classes of constrained optimization techniques that are used extensively in the process industries. When constraints are an important part of an optimization problem, constrained techniques must be employed, because an unconstrained method might produce an optimum that violates the constraints, leading to unrealistic values of the process variables. The general form of an optimization problem includes a nonlinear objective function (profit) and nonlinear constraints and is called a nonlinear programming problem. 

The steepest descent method is quite old and utilizes the intuitive concept of moving in the direction where the objective function changes the most. However, it is clearly not as efficient as the other three. Conjugate gradient utilizes only first-derivative information, as does steepest descent, but generates improved search directions. Newton s method requires second derivative information but is veiy efficient, while quasi-Newton retains most of the benefits of Newton s method but utilizes only first derivative information. All of these techniques are also used with constrained optimization. 

A significant advantage is that the constrained optimization can usually be carried out using only the first derivative of tlie energy. This avoids an explicit, and computationally expensive, calculation of the second derivative matrix, as is nomially required by Newton-Raphson techniques. 

Slightly different constraints are used to illustrate the mathematical technique. In this example, the constrained optimization problem is to locate levels of stearic acid (X ) and starch (X2) that minimize the time of in vitro release (y2) such that the average tablet volume (jy) did not exceed 9.422 cm2 and the average friability (y3) did not exceed 2.72%. 

Techniques for unconstrained and constrained optimization problems generally involve repeated use of a one-dimensional search as described in Chapters 6 and 8. 

The application of the standard nonlinear programming techniques of constrained optimization on analyzing the mean and variance response surfaces has been investigated by Del Castillo and Montgomery [34]. These techniques are appropriate since both the primary and secondary responses are usually quadratic functions. 

Another useful program (E04HAA) provides constrained optimization with bounds for each parameter using a sequential penalty function technique, which effectively operates around unconstrained minimization cycles. 

A variety of rules have been developed to control the movement and adaptation of the simplex, of which the most famous set is due to Nelder and Mead (Olsson and Nelson, 1975). The Nelder-Mead simplex procedure has been successfully used for a wide range of optimization problems and, due to its simple implementation, is amongst the most widely used of all optimization techniques. Importantly for the current application, simplex optimization is a black-box technique since it uses only the comparative values of the function at the vertices of the simplex to advance the position of the simplex, and it therefore requires no knowledge of the underlying mathematical function. It is also well suited to the optimization of expensive functions since as few as one new measurement is needed to advance the simplex one step. In its usual form, simplex optimization is suitable only for unconstrained optimization, but effective constrained versions have also been developed (Parsons et al., 2007 ... 

Closed-loop multivariable boiler control has to be planned and performed carefully because plant operators are not traditionally willing to reduce air-fuel ratios due to concerns about CO and other symptoms associated with Oz-deficient combustion. Model predictive control (MPC) is by far the most widely used technique for conducting multivariable boiler optimization and control. Forms of MPC that are inherently multivariable and that include real-time constrained optimization in the design are best suited for boiler application. 

Chamberlain, R. M., C. Lemarechal, H. C. Pedersen, and M. J. D. Powell. The Watchdog Technique for Forcing Convergence in Algorithms for Constrained Optimization, Math. Prog. Study 16 (1982). 

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