Globally Convergent Interior Path Following Methods for Nonlinear Programming and Brouwer Fixed Point Problems

Abstract

Since the remarkable paper of Karmarkar [3] was published, interior point methods for liner and quadratic programming, linear and nonlinear complementarity problems and for convex nonlinear programming have been well studied. However, to our knowledge, no such an algorithm for nonconvex programming is published.

In this paper, we present two kinds of globally convergent interior path following methods for nonlinear programming problems, which are not necessarily convex. One is the combined homotopy interior point method, and the other is called aggregate constraint homotopy method. Paralleled to the combined homotopy interior point method for nonlinear programming problems, a homotopy method for solving a class of nonconvex Brouwer fixed point problem is also given.

In 1984, Karmarkar presented a polynomial-time interior point method for linear programming in [14]. Its practical efficiency as well as its theoretical efficiency have motivated that interior point methods (or, equivalently, interior path following methods) for linear programming and related problems have been intensively studied (we will not cite the rich amount of literature on interior point methods for linear programming, see [3], [8] and [9] for surveys) and have made the classical barrier methods (see [5]) relive. Deep investigations into interior point methods for nonlinear programming, mainly on convex pro¬gramming and a few on nonconvex programming, were made in [10-13], [17], [24], [26], [27], [29], [30], [32], [33], [38] and etc.. To develop an interior path following method, an interior path from an interior point to a solution of the problem to be solved must be proven to exist. For a linear programming problem or a convex nonlinear programming problem, with strict convexity of the logarithmic barrier function and boundedness of the solution set, the existence of the central path is trivial. For nonconvex problem, however, the thing becomes more complex. It is needed to study that: how to construct the homotopy (or barrier function) and under what conditions that a smooth curve from an interior point to a solution or a first order necessary condition solution can be proven to exist. To our knowledge, no globally convergent interior point method, which converges from any interior point, for nonconvex programming is published.

Recently (see [4], [19] and [37]), in cooperation with Lin, Z.H. and Wang, Y., we have made some efforts to develop globally convergent methods. Utilizing differential topology techniques and ideals in interior point methods for convex programming, we presented two kinds of globally convergent interior path following methods for nonlinear programming problems, which are not necessarily convex. One is the combined homotopy interior point method, and the other is called the aggregate constraint homotopy method. For nonconvex problems, under a so-called normal cone condition, existence of paths from almost all interior point to a solution of the Karush-Kuhn-Tucker system of the nonlinear programming problem were proven. For convex problems, without strict convexity of the logarithmic barrier function and boundedness of the solution set, the convergence was also proven. Inspired by the combined homotopy interior point method for nonlinear programming problems, a homotopy method for solving a class of nonconvex Brouwer fixed point problem is also given (see [34]. It essentially carried the milestone work [2] and [15] a step forward.

In this paper, a summary of our main results in [4], [19], [34] and [37] is made and some new results are included.