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

• Published in: Journal of global optimization Vol. 64; no. 2; pp. 399 - 416
• Main Authors: Nguyen, Van-Bong, Sheu, Ruey-Lin, Xia, Yong
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2016; Springer; Springer Nature B.V
• Subjects: Algorithms; Computer Science; Discriminant analysis; Eigenvalues; Intervals; Investment analysis; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Maximization; Operations Research/Decision Theory; Optimization; Polynomials; Programming; Quadratic programming; Quotients; Rankings; Ratios; Rayleigh number; Real Functions; Semidefinite programming; Studies; Texts; Semidefinite programming; Quadratic fit line search; S-Lemma; 90C25; 90C22; Fractional programming; Quadratically constrained quadratic programming; (Generalized) Rayleigh quotient
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-015-0315-2

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 μ and generate a family of quadratic subproblems ( P μ ) ′ s subject to two quadratic constraints. Each ( P μ ) , if the problem dimension n ≥ 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 [ μ ̲ , μ ¯ ] . We propose a two-stage scheme incorporating the quadratic fit line search algorithm to find μ ∗ numerically. Computational experiments show that our method solves the problem correctly and efficiently.