Linearly Constrained Optimization Problems

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The problems we consider in this chapter have general objective functions but the constraints are linear. Section 4.1 gives a short introduction to linear optimization (LO) — also referred to as linear programming, which is the historically entrenched term. LO is the simplest type of constrained optimization: the objective function and all constraints are linear. The classical, and still well usable algorithm to solve linear programs is the Simplex Method. Quadratic problems which we treat in section 4.2 are linearly constrained optimization problems with a quadratic objective function. Quadratic optimization is often considered to be an essential field in its own right. More important, however, it forms the basis of several algorithms for general nonlinearly constrained problems. In section 4.3 we give a concise outline of projection methods, in particular the feasible direction methods of Zoutendijk, Rosen and Wolfe. They are extensions of the steepest descent method and are closely related to the simplex algorithm and the active set method. Then we will discuss some basic ideas of SQP methods — more generally treated in section 5.2 — which have proven to be very efficient for wide classes of problems.