ABSTRACT

This chapter focuses on calculus-based methods for optimization. These methods use the derivatives of the objective function U and of the constraints to determine the location where the objective function is a minimum or a maximum. The method of Lagrange multipliers is introduced and the system of equations, whose solution yields the optimum, is derived. Derivatives are needed, making it a requirement for applying calculus methods that the objective function and the constraints must be continuous and differentiable. In addition, only equality constraints can be treated by this approach, though inequality constraints may be converted to equality constraints in some cases. The importance of this method lies not only in solving relatively simple problems, but also in providing basic concepts and strategies that can be used for other optimization methods. The physical interpretation and proof of the Lagrange multiplier method is discussed, using a single constraint and only two independent variables. The characteristics and solutions of more complicated problems are discussed. The method is used for both unconstrained and constrained problems, including cases where a constrained problem may be converted into an unconstrained one by substitution. The Lagrange multipliers are shown to be related to the sensitivity of the objective function to changes in the constraints. The application of these methods to thermal systems is considered. A few examples of thermal systems and processes are given. A computational approach for solving relatively complicated optimization problems using these methods is also presented.