Linear Programming and the Simplex Method

Problems where the objective function to be optimized is linear in its design variables are of particular interest, because then a solution can be found very efficiently. Solutions of linear objective functions can only be unique if additional constraints exist. While the constraints bound the solution space, the linear nature of the objective ensures that the solution must be located at the border of this bounded solution space. In the case of linear constraints, the bounded solution space has the form of a polyhedron, and, hence, it suffices to seek for the solution at the vertices of the bounded space. As a consequence, the optimum can be found very fast. One method for solving linear programs is the simplex algorithm, which is one of the most famous optimization algorithms. It identifies the solution by moving from a vertex to one of its neighbors until the value of the objective cannot be reduced further for all neighboring vertices. In order to benefit from the speed advantage of linear programming, it sometimes is promising to approximate a given problem by linear relationships. Such an approximation is presented for the task of stereo disparity estimation at the end of this chapter.