Maximizing the sum of a generalized Rayleigh quotient and another Rayleigh quotient on the unit sphere via semidefinite programming

Abstract

The problem is a type of “sum-of-ratios” fractional programming and is known to be NP-hard. Due to many local maxima, finding the global maximizer is in general difficult. The best attempt so far is a critical point approach based on a necessary optimality condition. The problem therefore has not been completely solved. Our novel idea is to replace the generalized Rayleigh quotient by a parameter $\mu$ and generate a family of quadratic subproblems $(\hbox {P}_{\mu })'s$ subject to two quadratic constraints. Each $(\hbox {P}_{\mu })$, if the problem dimension $n\ge 3$, can be solved in polynomial time by incorporating a version of S-lemma; a tight SDP relaxation; and a matrix rank-one decomposition procedure. Then, the difficulty of the problem is largely reduced to become a one-dimensional maximization problem over an interval of parameters $[\underline{\mu },\bar{\mu }]$. We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find $\mu ^*$ numerically. Computational experiments show that our method solves the problem correctly and efficiently.