In the last lecture, we were introduced to the Interior Point Method. The simplex method solves linear programming problems (LP) by visiting extreme points (vertices) on the boundary of the feasible set. In contrast, the interior point method is based on algorithms that find an optimal solution while moving in the interior of the feasible set. Intuitively, in each iteration of the interior poin...

We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the norm of the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained usi...

Linear programming is now included in algorithm undergraduate and postgraduate courses for computer science majors. We give a self-contained treatment of an interior-point method which is particularly tailored to the typical mathematical background of CS students. In particular, only limited knowledge of linear algebra and calculus is assumed.

We show that for every convex body K in R, there is a convex body H such that H ⊂ K ⊂ c ·H with c = 2.24 and H has at most O(n) contact points with the minimal volume ellipsoid that contains it. When K is symmetric, we can obtain the same conclusion for every constant c > 1. We build on work of Rudelson [5], who showed the existence of H with O(n log n) contact points. The approximating body H ...

We present a new strategy for choosing primal and dual steplengths in a primal-dual interior-point algorithm for convex quadratic programming. Current implementations often scale steps equally to avoid increases in dual infeasibility between iterations. We propose that this method can be too conservative, while safeguarding an unequally-scaled steplength approach will often require fewer steps ...

There is a large number of implementational choices to be made for the primal-dual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide an elaborate discussion of the implementation ...

In [H. Mansouri and C. Roos, Numer. Algorithms 52 (2009) 225-255.], Mansouri and Ross presented a primal-dual infeasible interior-point algorithm with full-Newton steps whose iteration bound coincides with the best known bound for infeasible interior-point methods. Here, we introduce a slightly different algorithm with a different search direction and show that the same complexity result is obt...

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic program, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained pro...