Yuriy Zinchenko

Polytopes and arrangements: Diameter and curvature

We introduce a continuous analogue of the Hirsch conjecture and a discrete analogue of the result of Dedieu, Malajovich and Shub. We prove a continuous analogue of the result of Holt and Klee, namely, we construct a family of polytopes which attain the conjectured order of the largest total curvature.

Central Path Curvature and Iteration-Complexity for Redundant Klee—Minty Cubes

We consider a family of linear optimization problems over the n-dimensional Klee—Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2 n –2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2 n ) iterations to solve this problem, whereas in practice typically only a few iterations (e.g., 50) suffices to obtain a high-quality solution. Thus, the construction potentially exhibits the worst-case iteration-complexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n).

A moment-based approach for DVH-guided radiotherapy treatment plan optimization

The dose-volume histogram (DVH) is a clinically relevant criterion to evaluate the quality of a treatment plan. It is hence desirable to incorporate DVH constraints into treatment plan optimization for intensity modulated radiation therapy. Yet, the direct inclusion of the DVH constraints into a treatment plan optimization model typically leads to great computational difficulties due to the non-convex nature of these constraints. To overcome this critical limitation, we propose a new convex-moment-based optimization approach. Our main idea is to replace the non-convex DVH constraints by a set of convex moment constraints. In turn, the proposed approach is able to generate a Pareto-optimal plan whose DVHs are close to, or if possible even outperform, the desired DVHs. In particular, our experiment on a prostate cancer patient case demonstrates the effectiveness of this approach by employing two and three moment formulations to approximate the desired DVHs.

A Continuous d -Step Conjecture for Polytopes

The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as a continuous analogue of its diameter. We prove an analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytopes defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.

On hyperbolicity cones associated with elementary symmetric polynomials

Elementary symmetric polynomials can be thought of as derivative polynomials of

Optimal Transmission Network Topology for Resilient Power Supply

E_n(x)=\prod_{i=1,\ldots,n} x_i

SU-F-T-333: Deliverability Considerations in Modulated Photon Radiotherapy (XMRT)

. Their associated hyperbolicity cones give a natural sequence of relaxations for