The Central Path
In this chapter, we begin our study of an alternative to the simplex method for solving linear programming problems. The algorithm we are going to introduce is called a path-following method. It belongs to a class of methods called interior-point methods. The path-following method seems to be the simplest and most natural of all the methods in this class, so in this book we focus primarily on it. Before we can introduce this method, we must define the path that appears in the name of the method. This path is called the central path and is the subject of this chapter. Before discussing the central path, we must lay some groundwork by analyzing a nonlinear problem, called the barrier problem, associated with the linear programming problem that we wish to solve.
KeywordsLagrange Multiplier Barrier Function Linear Programming Problem Central Path Negative Infinity
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- Research into interior-point methods has its roots in the work of Fiacco & McCormick (1968). Interest in these methods exploded after the appearance of the seminal paper Karmarkar (1984). Karmarkar’s paper uses clever ideas from projective geometry. It doesn’t mention anything about central paths, which have become fundamental to the theory of interior-point methods. The discovery that Karmarkar’s algorithm has connections with the primal-dual central path introduced in this chapter can be traced to Megiddo (1989). The notion of central points can be traced to preKarmarkar times with the work of Huard (1967). D.A. Bayer and J.C. Lagarias, in a pair of papers (Bayer & Lagarias 1989a,b), give an in-depth study of the central path.Google Scholar
- Deriving optimality conditions and giving conditions under which they are necessary and sufficient to guarantee optimality is one of the main goals of nonlinear programming. Standard texts on this subject include the books by Luenberger (1984), Bertsekas (1995), and Nash & Sofer (1996).Google Scholar