http://hdl.handle.net/1721.1/39115 http://hdl.handle.net/1721.1/41737 Computational issues and related mathematics of an exponential annealing homotropy for conic optimization Chen, Jeremy, S.M. Massachusetts Institute of Technology We present a further study and analysis of an exponential annealing based algorithm for convex optimization. We begin by developing a general framework for applying exponential annealing to conic optimization. We analyze the hit-and-run random walk from the perspective of convergence and develop (partially) an intuitive picture that views it as the limit of a sequence of finite state Markov chains. We then establish useful results that guide our sampling. Modifications are proposed that seek to raise the computational practicality of exponential annealing for convex optimization. In particular, inspired by interior-point methods, we propose modifying the hit-and-run random walk to bias iterates away from the boundary of the feasible region and show that this approach yields a substantial reduction in computational cost. We perform computational experiments for linear and semidefinite optimization problems. For linear optimization problems, we verify the correlation of phase count with the Renegar condition measure (described in [13]); for semidefinite optimization, we verify the correlation of phase count with a geometry measure (presented in [4]). Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007. Includes bibliographical references (p. 105-106). Sun, 29 Oct 2006 22:58:59 GMT http://hdl.handle.net/1721.1/41736 An analysis of the TR-BDF2 integration scheme Dharmaraja, Sohan We intend to try to better our understanding of how the combined L-stable 'Trapezoidal Rule with the second order Backward Difference Formula' (TR-BDF2) integrator and the standard A-stable Trapezoidal integrator perform on systems of coupled non-linear partial differential equations (PDEs). It was originally Professor KlausJiirgen Bathe who suggested that further analysis was needed in this area. We draw attention to numerical instabilities that arise due to insufficient numerical damping from the Crank-Nicolson method (which is based on the Trapezoidal rule) and demonstrate how these problems can be rectified with the TR-BDF2 scheme. Several examples are presented, including an advection-diffusion-reaction (ADR) problem and the (chaotic) damped driven pendulum. We also briefly introduce how the ideas of splitting methods can be coupled with the TR-BDF2 scheme and applied to the ADR equation to take advantage of the excellent modern day explicit techniques to solve hyperbolic equations. Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007. Includes bibliographical references (p. 75-76). Sun, 29 Oct 2006 22:58:59 GMT http://hdl.handle.net/1721.1/41735 A multiple secretary problem with switch costs Ding, Jiachuan In this thesis, we utilize probabilistic reasoning and simulation methods to determine the optimal selection rule for the secretary problem with switch costs, in which a known number of applicants appear sequentially in a random order, and the objective is to maximize the sum of the qualities of all hired secretaries over all time. It is assumed that the quality of each applicant is uniformly distributed and any hired secretary can be replaced by a better qualified one at a constant switch cost. A dynamic program is formulated and the optimal selection rule for the single secretary case is solved. An approximate solution is given for the multiple secretary case, in which we are allowed to have more than one secretary at a time. An experiment was designed to simulate the interview process, in which respondents were sequentially faced with random numbers that represent the qualities of different applicants. Finally, the experimental results are compared against the optimal selection strategy. Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007. Includes bibliographical references (p. 76). Sun, 29 Oct 2006 22:58:59 GMT http://hdl.handle.net/1721.1/41734 A diagnostic analysis of retail out-of-stocks Foo, Yong Ning In the highly competitive retail industry, merchandise out-of-stock (OOS) is a significant and pertinent problem. This thesis performs a diagnostic analysis on retail out-of-stocks using empirical data from a major retailer. In this thesis, we establish the empirical relationship of OOS rate with the amount of safety stock carried, the time between orders and the forecast error, providing insights to the effects of these three factors on the probability of OOS occurrences. The root causes of OOS are also examined in the thesis. We find that up to 34% of OOS can be attributed to forecast error while up to 22% can be attributed to delay in order replenishment. For the OOSs that were associated with order delay, we can trace 60% of these to out-of-stock at the store's distribution center (DC). The thesis also examines a peculiarity in the occurrence of OOSs. We found that the OOS rate of Class C items is significantly higher in stores with higher sales volume. We can attribute much of this phenomenon to three factors: stores with higher sales volume hold less safety stock for Class C items, have a shorter time between orders and have relatively larger forecast errors. Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007. Includes bibliographical references (p. 101). Sun, 29 Oct 2006 22:58:59 GMT