A Polynomial Cutting Surfaces Algorithm for the ConvexFeasibility Problem Defined by Self-Concordant Inequalities

  • Authors:

  • Zhi-Quan Luo;Jie Sun

  • Affiliations:

  • Communication Research Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, L8S 4L7, Canada. luozq@mcmail.cis.mcmaster.ca;Department of Decision Sciences, National University of Singapore, Singapore 119260. fbasunj@leonis.nus.sg

  • Venue:
  • Computational Optimization and Applications
  • Year:
  • 2000

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Abstract

Consider a nonempty convex set in R^m whichis definedby a finite number of smooth convex inequalities and which admits aself-concordant logarithmicbarrier. We study the analytic center basedcolumn generation algorithm forthe problem of finding a feasible point in this set. At eachiteration the algorithm computes an approximate analytic center ofthe set defined by the inequalities generated in the previousiterations. If this approximate analytic center is a solution, thenthe algorithm terminates; otherwise either an existing inequality isshifted or a new inequality is added into the system. As the numberof iterations increases, the set defined by the generatedinequalities shrinks and the algorithm eventually finds a solution ofthe problem. The algorithm can be thought of as an extension of theclassical cutting plane method. The difference is that we useanalytic centers and “convex cuts” instead of arbitrary infeasiblepoints and linear cuts. In contrast to the cutting plane method, thealgorithm has a polynomial worst case complexity ofO(N\log\frac{1}{\varepsilon}) on the total number of cuts to beused, where N is the number of convex inequalities in the originalproblem and &epsis; is the maximum common slack of the originalinequality system.