# An efficient global optimization algorithm for maximizing the sum of two generalized Rayleigh quotients

## TL;DR

This paper introduces a branch-and-bound algorithm for globally maximizing the sum of two generalized Rayleigh quotients, reformulating the problem into a one-dimensional optimization with SDP subproblems, and demonstrating superior efficiency over existing heuristics.

## Contribution

The paper presents a novel branch-and-bound method that explicitly overestimates the objective using dual SDP subproblems for efficient global optimization.

## Key findings

- The proposed algorithm outperforms recent SDP-based heuristics in efficiency.
- Reformulation reduces the problem to a one-dimensional optimization with SDP subproblems.
- Numerical results confirm the effectiveness of the method.

## Abstract

Maximizing the sum of two generalized Rayleigh quotients (SRQ) can be reformulated as a one-dimensional optimization problem, where the function value evaluations are reduced to solving semi-definite programming (SDP) subproblems. In this paper, we first use the dual SDP subproblem to construct an explicit overestimation and then propose a branch-and-bound algorithm to globally solve (SRQ). Numerical results demonstrate that it is even more efficient than the recent SDP-based heuristic algorithm.