Mathematical background and formulation of numerical minimization process are described in terms of gradient-based methods, whose ingredients include gradient, Hessian, directional derivatives, optimality conditions for minimization, Hessian eigensystem, conjugate number of Hessian, and conjugate vectors. Various minimization algorithms, such as the steepest descent method, Newton’s method, conjugate gradient method, and quasi-Newton’s method, are introduced along with practical examples.