Practical system problems are very often formulated with some constraints imposed on their variables. Optimization of such problems must be carried out within the limits of these constraints. The basic difference between the constrained and the unconstrained problems is the selection of admissible points that are eligible for optimization. Lagrange multipliers are known as an effective means of dealing with constraints. Two theorems for functions are given. The first theorem in each case gives a necessary condition and the second theorem gives a sufficient condition for minimization of constrained problems. The chapter introduces some necessary background in analytic computation that will be useful in optimization work. It includes matrix notation and some special properties that have significant applications in operation research and power systems in particular. Basic foundations in matrix algebra and functional calculus are required by the reader and the people has introduced standard background operations to test negative and positive semidefinite conditions on Jacobian and Hessian matrices.