Title: The Joint Numerical Range and Its Uses in Optimization

Abstract: In this talk, we consider the benefits of the Joint Numerical Range(JNR) as it relates to optimization problems. After reviewing the basics of the JNR, we show how to compute/visualize it. We then reformulate optimization problems of the Robust Rayleigh Quotient and the Sum of 2 Rayleigh Quotients(SRQ2) into optimizations over the JNR. Applications of these problems include Fair PCA and Eigenvalue Backward Errors respectively. These reformulations provide better solutions, assuming that the JNR is convex. However,  JNR is only generally proven to be convex for a small number of matrices. Thus, future work is to be done to expand what is known about the convexity of JNR. We shall also consider the use of the approximate JNR to solve more difficult optimizations.