Primal–Dual–Infeasible Newton Approach for the Analytic Center Deep-Cutting Plane Method

The convergence and complexity of a primal–dual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a deep cut separation oracle is studied. The primal–dual–infeasible Newton method is used to generate a primal–dual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is tangent to the primal–dual variant of Dikin's ellipsoid.     

Long-Step Interior-Point Algorithms for a Class of Variational Inequalities with Monotone Operators

This paper describes two interior-point algorithms for solving a class of monotone variational inequalities defined over the intersection of an affine set and a closed convex set. The first algorithm is a long-step path-following method, and the second is an extension of the first, incorporating weights in the gradient of the barrier function. Global convergence of the algorithms is proven under the assumptions of monotonicity and differentiability of the operator.