A Center Cutting Plane Algorithm for a Likelihood Estimate Problem
Computational Optimization and Applications
Multiple Cuts with a Homogeneous Analytic Center Cutting Plane Method
Computational Optimization and Applications
An Analytic Center Cutting Plane Method for Solving Semi-Infinite Variational Inequality Problems
Journal of Global Optimization
A cutting plane method for solving KYP-SDPs
Automatica (Journal of IFAC)
Analytic center of spherical shells and its application to analytic center machine
Computational Optimization and Applications
A second-order cone cutting surface method: complexity and application
Computational Optimization and Applications
On Interior-Point Warmstarts for Linear and Combinatorial Optimization
SIAM Journal on Optimization
Specialized fast algorithms for IQC feasibility and optimization problems
Automatica (Journal of IFAC)
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We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variance-covariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid.We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(plog (p+1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix---primal, dual, or primal-dual---that is used in the computations.The computation of the optimal direction uses Newton's method applied to a self-concordant function of p variables.The convergence result of [Ye, Math. Programming, 78 (1997), pp. 85--104] holds here also: the algorithm stops after $O^*(\frac{\bar p^2n^2}{\varepsilon^2})$ cutting planes have been generated, where $\bar p$ is the maximum number of cuts generated at any given iteration.