The Analytic Center Cutting Plane Method with Semidefinite Cuts

The Analytic Center Cutting Plane Method with Semidefinite Cuts

We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semideenite cuts. At each iteration the oracle returns a p-dimensional semideenite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution set, is a compact set consists of piecewise algebraic surfaces. We prove that the analytic center is...

Multiple Cuts in the Analytic Center Cutting Plane Method

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 dir...

An Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems

Semidefinite feasibility problems arise in many areas of operations research. The abstract form of these problems can be described as finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m ×m symmetric positive semidefinite matrix Ŷ either confirms that Ŷ ∈ Γ or returns a cut, i.e....

Analytic Center Cutting-Plane Method

An Analytic Center Cutting Plane Method

Semideenite feasibility problems arise in many areas of operations research. The abstract form of these problems can be described as nding a point in a nonempty bounded convex body ? in the cone of symmetric positive semideenite matrices. Assume that ? is deened by an oracle, which, for any given m m symmetric matrix ^ Y , either connrms that ^ Y 2 ? or returns a cut, i.e., a symmetric matrix A...