About 4 years ago, I immigrated to the United States, where I found my home away from home in Los Angeles, California. Living in three different countries, I had to learn new languages and accustom to new traditions each time. However, one thing that never changed was Mathematics. While Mathematics has been my favorite subject since a very young age, my fascination grew even bigger when I saw the applications of the subject in various fields. I started my college journey at Los Angeles City College (LACC) as a President’s Scholar. I graduated from LACC with six associate’s degrees and had the opportunity to be involved in college activities, including student advocacy at a national level, being the President of the Mathematics Club that ranked #1 nationwide in the AMATYC competition during my presidency, and many more activities and student jobs on campus. After graduating from LACC, I transferred to Yale University as a Jack Kent Cooke Scholar, with a full-ride scholarship from Yale. Here, I am pursuing a degree in Applied Mathematics. So far, I have had the opportunity to apply my mathematical knowledge in research projects in the field of engineering, finance, data analysis, machine learning, political science, and etc, at institutions such as NASA, LACC, California State University, Fullerton, UCLA, Yale, and now MIT. My research here is in the field of Operations Research. I am planning on continuing my education beyond a bachelor’s degree, pursuing a Ph.D. doing research in the intersection of those fields. My dream is to establish a magnet school promoting the success of students who come from underrepresented communities. During my free time, I enjoy hiking while listening to an audiobook, assembling jigsaw puzzles, and watching detective TV-shows.
Cover vs Frank-Wolfe: Finding the Log-Optimal Portfolio
Mariam Alaverdian1, Robert M. Freund2
1Department of Applied Mathematics, Yale University
2Department of Operations Research, Massachusetts Institute of Technology
The goal of any financial portfolio is to maximize the returns on the assets it contains and minimize the risk. To achieve that goal, it is necessary to find the optimal proportion of each asset within the portfolio. We do so by optimizing the average of a log-linear objective function which represents our portfolio returns. Functions of this type can often take substantial time to find an optimal solution. Therefore, it is important to compare multiple algorithms and their variations with one another to determine which one is the most efficient. In this work, we compare the performance of Cover’s method to the Frank-Wolfe method, both with and without using a binary-search line-search technique for step-size selection. We used a data set that contained 100 weeks of data on 53 assets within the portfolio, from which, we calculated the fraction of return per asset. We evaluated the performance of the two algorithms by comparing the time required and the number of iterations by each method to find the optimal solution of the objective function. Our results show that Frank-Wolfe’s method with binary-search line-search is more efficient for the log-optimal portfolio problem. It takes less than a second to solve the problem and only three iterations, while both variations of Cover’s method take more than 10000 iterations and more than 350 seconds. This framework can also be extended to many other problems involving log-linear objective functions, including medical image reconstruction, and statistical experimental design.