7 Mathematical programming — A computational perspective

Abstract

Publisher Summary Mathematical programming studies the properties of optimization problems and techniques for computing their solution. A typical optimization problem has the form where the real-valued function f(x) is called either the “objective function” or the “cost function” and C is the constraint set. Because maximizing f(x) is equivalent to minimizing -f(x) , a maximization problem can always be posed as a minimization problem. The chapter focuses on the case where x is a vector with n components. Although in practice the constraint set C can be contained in an infinite dimensional space, the infinite dimensional problem must be discretized when a solution is computed numerically leading to a finite dimensional problem. Consequently the theory associated with finite dimensional optimization is relevant to infinite dimensional problems. Moreover, many finite dimensional algorithms extent directly to infinite dimensions. Mathematical programming has many applications in statistics. A linear program is a mathematical programming problem in which the cost function is linear and the constraint set is a polytope. The simplex method is one of the most popular methods for solving linear programming problems. A quadratic program is a mathematical programming problem in which the cost function is quadratic and the constraint set is a polytope. A special class of algorithms for the linear complementarity problem has been developed based on iterative matrix splitting techniques. Because of the connection between quadratic programming and the linear complementarity problem, these techniques are also applicable to the corresponding quadratic programs.