On Murty\u27s gravitational interior point method for quadratic programming

This thesis presents a modification of the gravitational interior point method for quadratic programming [7]. Murty presented the algorithm as a generalization of his gravitational method for linear programming [8]. Murty claims that this method is matrix inverse free unlike other interior point methods, however convergence of his algorithm is not guaranteed. This thesis introduces modifications in the centering step of the algorithm and, using a MatlabR2009a implementation, demonstrates the centering step

p-Cone Optimization with Application to Radiotherapy Planning

The work presented in this thesis is dedicated to the study of two facets of the so-called p-cone optimization: one methodological and mostly theoretical, while the other applied and computational. Motivated by optimal radiotherapy design for cancer treatment, we investigate the computational hardness of p-cone optimization, namely the complexity of forming a tractable approximation to such a problem, as well as the suitability of p-cone optimization to optimal radiotherapy design. It is demonstrated by Ben-Tal and Nemirovski that an ε-approximation to a Second Order Conic Optimization (SOCO) problem is polynomial reducible to Linear Optimization (LO). In this thesis, we consider a generalization of SOCO to p-Order Conic Optimization (pOCO) where the second order cone is replaced with the p-norm cone. In the first part of the thesis, we develop three conic approximation schemes for p-norm cone and estimate their complexities. We proposed a second order conic approximation with complexity of O(4√1/ε) on the number of affine inequalities giving ε cone approximation. In the second part of the thesis, a novel unconstrained convex moment based model is proposed to enable embedding of so-called dose volume histogram in the context of radiation therapy planning optimization. We investigate the suitability of the newly proposed method and compare its performance to other three known methods to replicate a set of dose volume histograms. We provide a benchmark of performance of each method in terms of dosimetery and run time