In this thesis, we investigate the cost of finding collisions in a black-box function, a problem that is of fundamental importance in cryptanalysis. Inspired by the excellent performance of the heuristic rho method of collision finding, we define several new models of complexity that take into account the cost of moving information across a large space, and lay the groundwork for studying the performance of classical and quantum algorithms in these models.

The Trust Region Subproblem (TRS) is the problem of minimizing a quadratic (possibly non-convex) function over a sphere. It is the main step of the trust region method for unconstrained optimization problems. Two cases may cause numerical difficulties in solving the TRS, i.e., (i) the so-called hard case and (ii) having a large trust region radius. In this thesis we give the optimality characteristics of the TRS and review the major current algorithms. Then we introduce some techniques to solve the TRS efficiently for the two difficult cases. A shift and deflation technique avoids the hard case; and a scaling can adjust the value of the trust region radius. In addition, we illustrate other improvements for the TRS algorithm, including: rotation, approximate eigenvalue calculations, and inverse polynomial interpolation. We also introduce a warm start approach and include a new treatment for the hard case for the trust region method. Sensitivity analysis is provided to show that the optimal objective value for the TRS is stable with respect to the trust region radius in both the easy and hard cases. Finally, numerical experiments are provided to show the performance of all the improvements.

Thesis (M.A. Sc.)--University of Toronto, 2005. / Source: Masters Abstracts International, Volume: 44-02, page: 0952.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 100-104). Also available via the Internet.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 125-133).

Dissertation (Ph.D.)--Auburn University, 2007. / Abstract. Vita. Includes bibliographic references.

Thesis (Ph. D.)--Lehigh University, 1997. / Includes vita. Bibliography: leaves 119-120.

Thesis (Ph.D.) -- University of Glasgow, 1981. / Includes bibliographical references (p. 329-336). Print version also available.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 74-80). Also available via the Internet.

Thesis (M.S. in Operations Research) Naval Postgraduate School, December 2001. / Thesis Advisor (s): Dell, Robert F. "December 2001." Includes bibliographical references (p. 35). Also available online.