Cutting Surfaces And Analytic Center: A Polynomial Algorithm For The Convex Feasibility Problem Defined By Self-Concordant Inequalities

Z.-Q. Luo and J. Sun

Consider a nonempty convex set in $R^m$ which is defined by a finite number of convex differentiable inequalities and which admits a self-concordant logarithmic barrier. We study the analytic center based column generation algorithm for the problem of finding a feasible point in this set. At each iteration, the algorithm computes an approximate analytic center of the set defined by the inequalities generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise either an existing inequality is shifted or a new inequality is added into the system. As the number of iterations increases, the set defined by the generated inequalities shrinks and the algorithm eventually finds a solution of the problem. The algorithm can be thought of as an extension of the classical cutting plane method. The difference is that we use analytic centers and ``convex cuts" instead of linear cuts. Our method has a polynomial worst case complexity of $O(n\ln \frac{1}{\varepsilon})$ on the total number of cuts to be used, where $n$ is the number of convex inequalities in the original problem and $\varepsilon$ is the maximum common slack of the original inequality system. In contrast, the early column generation methods using linear cuts can only solve the convex feasibility problem in pseudo-polynomial time.