1 Abstract An interior point of a nite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k +1 interior points. We proved that h(3) = 3 in an earlier pa...

We consider simplices in [w” with lattice point vertices, no other boundary lattice points and n interior lattice points, with an emphasis on the barycentric coordinates of the interior points. We completely classify such triangles under unimodular equivalence and enumerate. For example, in a lattice point triangle with exactly one interior point, that point must be the centroid. We discuss the...

A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short step pathfollowing method is established.

The Semidefinite Program (SDP) is a fundamental problem in mathematical programming. It covers a wide range of applications, such as combinatorial optimization, control theory, polynomial optimization, and quantum chemistry. Solving extremely large-scale SDPs which could not be solved before is a significant work to open up a new vista of future applications of SDPs. Our two software packages S...

Interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In [31], these off-central paths are shown to be well-defined analytic curves ...

An interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In [C.-K. Sim and G. Zhao, Math. Program. Ser. A, 110 (2007), pp. 475–499],...

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 ...