Topics in exact precision mathematical programming

The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.Ph.D.Committee Chair: Dr. William J. Cook; Committee Member: Dr. George Nemhauser; Committee Member: Dr. Robin Thomas; Committee Member: Dr. Santanu Dey; Committee Member: Dr. Shabbir Ahmed; Committee Member: Dr. Zonghao G

Scheduling a conference to minimize RSVP conflicts

Abstract This paper describes a conference scheduling problem where, at the time of registration, participants RSVP to events which they would like to attend. Based upon these preferences, an assignment of events to rooms and time slots should be determined that minimizes the number of RSVP conflicts and satisfies a number of hard constraints. We show that our problem, and several variants of it, are NP-hard. An integer programming model is developed to solve the problem and a computational study of this model is performed on instances generated from real data. Improvements to the model, including a symmetry breaking reformulation and a dualization of some hard constraints, are shown to significantly improve solution times, making the problem tractable for the desired real world application

Genetic architecture of voluntary exercise in an advanced intercross line of mice

Exercise is essential for health, yet the amount, duration, and intensity that individuals engage in is strikingly variable, even under prescription. Our focus was to identify the locations and effects of quantitative trait loci (QTL) controlling genetic predisposition for exercise-related traits utilizing a large advanced intercross line (AIL) of mice. This AIL (G4) population originated from a reciprocal cross between mice with genetic propensity for increased voluntary exercise (HR, selectively bred for increased wheel running) and the inbred strain C57BL/6J. After adjusting for family structure, we detected 32 significant and 13 suggestive QTL representing both daily running traits (distance, duration, average speed, and maximum speed) and the mean of these traits on days 5 and 6 (the selection criteria for HR) of a 6-day test conducted at 8 weeks of age, with many colocalizing to similar genomic regions. Additionally, 7 significant and 5 suggestive QTL were observed for the slope and intercept of a linear regression across all 6 days of running, some representing a combination of the daily traits. We also observed 2 significant and 2 suggestive QTL for body mass prior to exercise. These results, using a well-defined animal model, reinforce a genetic basis for the predisposition to engage in voluntary exercise, dissect this predisposition into daily segments across a continuous time period, and present unique QTL that may provide insight into the initiation, continuation, and temporal pattern of voluntary activity in mammals