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 present a dual-scaling interior-point algorithm and show how it exploits the structure and sparsity of some large scale problems. We solve the positive semideenite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints. We report the rst computational results of interior-point algorithms for approximating the maximum cut semideenite programs with dimen...

We present a full-Newton step infeasible interior-point algorithm. It is shown that at most O(n) (inner) iterations suffice to reduce the duality gap and the residuals by the factor 1 e . The bound coincides with the best known bound for infeasible interior-point algorithms. It is conjectured that further investigation will improve the above bound to O( √ n).

Algorithms are presented for the determination of whether a given point in is interior to, exterior to or on an arbitrary polygonal boundary and for the determination of whether a point in is interior to, exterior to or on a simple polyhedral boundary. The algorithms are based on the principle of using binary coded coordinate systems and parity counting of the number of intersections of the pol...

in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

We develop and compare multilevel algorithms for solving constrained nonlinear variational problems via interior point methods. Several equivalent formulations of the linear systems arising at each iteration of the interior point method are compared from the point of view of conditioning and iterative solution. Furthermore, we show how a multilevel continuation strategy can be used to obtain go...

The central path is the most important concept in the theory of interior point methods. It is an analytic curve in the interior of the feasible set which tends to an optimal point at the boundary. The analyticity properties of the paths are connected with the analysis of the superlinear convergence of the interior point algorithms for semidefinite programming. In this paper we study the analyti...

“Interior point algorithms are a good choice for solving pure LPs or QPs, but when you solve MIPs, all you need is a dual simplex.” This is the common conception which disregards that an interior point solution provides some unique structural insight into the problem at hand. In this paper, we will discuss some of the benefits that an interior point solver brings to the solution of difficult MI...