A piecewise‐linear homotopy method for nonlinear programming

Abstract

AbstractIn nonlinear programming problems, an objective function f(x) is optimized (maximized or minimized) subject to some constraints. Such problems are also called constrained optimization problems. Most of the algorithms in nonlinear programming are classified into two categories: 1) transformation methods; and 2) projection methods.The homotopy methods, which are the subject of this paper, belong to the category of projection methods. The main feature of the homotopy methods compared with other projection methods is that they are good at global convergence (which is lacking in most of the projection methods) but are not good at convergence speed (which is the strong point of most of the projection methods).This paper discusses the homotopy methods in nonlinear programming and show that the piecewise‐linear homotopy method using the Newton homotopy and polyhedral subdivision is very effective for solving nonlinear optimization problems. A new algorithm is proposed that exploits the partial separability and linearity of the Kuhn‐Tucker equations (which appear in the nonlinear programming problems). By this exploitation, the computation efficiency is improved markedly compared with the conventional homotopy methods using simplicial subdivision. Moreover, the proposed algorithm converges quadratically, thus accurate solutions can be obtained rapidly. It is proved also that the proposed algorithm is globally convergent for the constrained convex optimization problems. Except for the shortcoming that the programming is complicated, the proposed algorithm has wellbalanced effectiveness.