A pr 2 01 3 Intrinsic complexity estimates in polynomial optimization 1

Operator Scaling via Geodesically Convex Optimization, Invariant Theory and Polynomial Identity Testing

We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive de nite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, commutative metric (for which the abo...

Intrinsic complexity estimates in polynomial optimization

It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using (s d)O(n) arithmetic operations, where n and s are the numbers of variables and constraints and d is the maximal degree of the polynomials involved. We associate to each of these problems an intrinsic system degree which becomes in wor...

New complexity analysis of a full Nesterov-Todd steps IIPM for semidefinite optimization

The power of local search for clustering

We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) k-median and k-means in edge-weighted planar graphs; (3) k-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the p-th po...

The Complexity of DC-Switching Problems

This report provides a comprehensive complexity study of line switching in the Linear DC model for the feasibility problem and the optimization problems of maximizing the load that can served (maximum switching flow, MSF) and minimizing generation cost (optimal transmission switching, OTS). Our results show that these problems are NP-complete and that there is no fully polynomial-time approxima...