The Hakubi Project

   My study is on a field of applied mathematics known as mathematical optimization. In this talk, I will present some examples of optimization problems and demonstrate how a computer solves them to show the practical aspects of optimization. In order to solve a problem using a computer, we have to provide the computer with a procedure and to guarantee the validity of that procedure. Therefore, it is important to mathematically derive and analyze such procedures, that is, algorithms. After I explain the importance of deriving and analyzing such procedures, we will deal with constrained optimization problems. In particular, I will explain that if the set of variables that satisfy the constraints of a constrained optimization problem forms a Riemannian manifold, then the problem can be regarded as an unconstrained optimization problem on the manifold. Next, I will introduce my current and future research, such as geometric conjugate gradient and stochastic gradient descent methods, which are effective for large-scale problems. I will also explain that optimization, which is applied in various fields, has a solid mathematical foundation.