威尼斯人

 

- Abstract:

Optimization problems involving objective functions composed of a smooth function and multiple Rayleigh Quotients are commonly encountered in practice. Traditional optimization techniques for these problems can result in local optima and substantial computational costs. In this talk, we introduce a novel approach to address these challenges by reformulating the problem as an optimization over the joint numerical range. This reformulation can help eliminate certain local optima present in the original problem. Additionally, by exploiting potential convexity in the reformulation, we can express the first-order optimality conditions as eigenvector-dependent nonlinear eigenvalue problems. The latter can be efficiently solved by exploiting state-of-the-art eigensolvers. In addition, we will provide a geometric interpretation of the Self-Consistent Field (SCF) iteration for solving the NEPv and develop a geometric scheme to identify the global optimizer in the original problem. Numerical experiments will illustrate the effectiveness of our new approaches.