Research On Theory And Algorithm Of Rayleigh Quotient Problem

In order to find the appropriate generalized coordinates,Rayleigh put forward a special form of quotient in the 1930 s when studying the small oscillations of the vibration system.This kind of quotient is called Rayleigh quotient by later generations.It can be expressed as the ratio of two quadratic functions.In recent years,Rayleigh quotient has been widely used in optimization theory,signal processing,pattern recognition and communication technology,rather than being limited to its original scope of application.Recently,the sum of Rayleigh quotients has attracted more and more scholars' attention and research.Maximization the sum of generalized Rayleigh quotient problem is a kind of fractional programming problem of "sum-of-ratio",which is known as NP hard problem,and it is very difficult to find the optimality condition.At present,the research on the sum of generalized Rayleigh quotients problem mainly focuses on the sum of two generalized Rayleigh quotients problem.On the basis of previous work,the local and global optimality conditions for maximizing the sum of two or more generalized Rayleigh quotients are studied extensively.The generalized Rayleigh quotient studied in this paper mainly focuses on the case where matrices can be diagonalized simultaneously.Firstly,some useful results are obtained by using the hidden convexity of quadratic transformation.Hidden convexity is different from ordinary convexity,but it inherits some properties of convex functions.In this paper,we use hidden convexity instead of convexity to connect the problem with semidefinite programming problem.Combined with the extended S lemma,by using the separation theorem of convex sets and Lagrange duality,the corresponding relaxation model is established,and then the necessary and sufficient conditions for the optimal solution of the sum of generalized Rayleigh quotients problem are obtained.Finally,an improved Nelder-Mead algorithm is proposed in this paper.Numerical experiments show that the algorithm achieves good results in finding local and global optimal solutions.