TITLE:  Extending and Simplifying Existing Piecewise-linear Homotopy Methods for Solving Nonlinear Systems of Equations

ABSTRACT:

This research extends and simplifies existing piecewise-linear homotopy (PL) methods to solve G(x) = 0, with G : Rn → Rm. Existing PL methods are designed to solve F (x) = 0, with F : Rn → Rn and some related point-to-set mappings. PL methods are a component of what is also known as numerical continuation methods, and they are known for being globally convergent methods. First, we present a new PL method for computing zeros of functions of the form f : Rn → R by mimicking classical PL methods for computing zeros of functions of the form f : R → R. Our PL method avoids traversing subdivisions of Rn × [0, 1] and instead uses an object that we refer to as triangulation-graph, which is essentially a triangulation of R × [0, 1] with hypercubes of Rn as its vertices. The hypercubes are generated randomly, and a sojourn time of an associated discrete-time Markov chain is used to show that not too many cubes are generated. Thereafter, our PL method is applied to solving G(x) = 0 for G : Rn → Rm under inequality constraints. The resultant method for solving G(x) = 0 translates into a new type of iterative method for solving systems of linear equations. Some computational illustrations are reported. A possible application to optimization problems is also indicated as a direction for further work.

Bio:  Ira Monroe Wheaton Jr. was born and raised in Chicago, IL to parents Ira Sr. and Laura. From an early age, Ira excelled in school and discovered a great affinity for learning. After graduating as the valedictorian of his high school class, Ira decided to pursue a Bachelor of Science degree in Mathematics. In 2007, Florida A&M University (FAMU) provided Ira with a full academic scholarship to accomplish this goal. During his time at FAMU, Ira was active on campus as a tutor in the FAMU Math Lab and a member of many organizations including Florida-Georgia Louis Stokes Alliance for Minority Participation (FGLSAMP), Kemetic Mathematical Society, and Honors Student Association. Furthermore, he participated in summer internships at the NASA Johnson Space Center in Houston, TX and The Northern Trust Company in Chicago, IL.

After obtaining his B.S. degree with Summa Cum Laude honors in April 2011, Ira enrolled in the Florida State University (FSU) Department of Mathematics and obtained an M.S. in Financial Mathematics in May 2013. Thereafter, Ira enrolled in the FSU Department of Industrial and Manufacturing Engineering (IME) to pursue a PhD. During that time, he served as a teaching assistant, working closely with professors to assist them with Operations Research I & II courses. With the support of the Florida Education Fund’s McKnight Doctoral Fellowship, Ira has been involved in research on homotopy methods, which has resulted in a publication in the Journal of Computational and Applied Mathematics, as well as three more articles in progress. Ira graduated with a PhD in Industrial Engineering on May 6, 2017.

Ira is currently serving as an Instructor in the Morehouse College Department of Mathematics. In his spare time, Ira enjoys spending time with his wife and daughter, playing the piano, and playing basketball. He hopes to continue to contribute to the field and to inspire new researchers in the same way that his advisor and others have done for him.