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

  title={An efficient global optimization algorithm for maximizing the sum of two generalized Rayleigh quotients},
  author={Xiaohui Wang and Longfei Wang and Yong Xia},
  journal={Computational and Applied Mathematics},
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… 

A Linear-Time Algorithm for Globally Maximizing the Sum of a Generalized Rayleigh Quotient and a Quadratic Form on the Unit Sphere

The problem, which is referred to as problem (P), of maximizing the sum of a generalized Rayleigh quotient and a quadratic form on the unit sphere is studied.

Nonlinear dimension reduction for surrogate modeling using gradient information

It is shown that building a nonlinear feature map g can permit more accurate approximation of u than a linear g, for the same input data set.



Solving the Sum-of-Ratios Problem by an Interior-Point Method

An algorithm for computing the global minimum of the problem by means of an interior-point method for convex programs is proposed.

On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients

Fractional programming: The sum-of-ratios case

A recent survey of applications, theoretical results and various algorithmic approaches for the sum-of-ratios problem is provided.

Range division and compression algorithm for quadratically constrained sum of quadratic ratios

A novel linear relaxation approach is proposed for deriving linear relaxation programming, which is used to obtain a lower bound of the optimal value of this problem and a range compression technique is presented to contract the investigated range.

Towards a joint optimization of scheduling and beamforning for MIMO downlink

This paper addresses the downlink of a multi-user MIMO system by investigating a novel greedy solution that approximately maximizes the sum-rate and compared with known solutions based on combinatorial search.

New Results on Quadratic Minimization

This paper proposes a polynomial-time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint, and introduces a parameterized problem and proves the existence of a trajectory that will lead to an optimal solution.

On Cones of Nonnegative Quadratic Functions

It is shown that optimizing a general quadratic function over the intersection of an ellipsoid and a half-plane can be formulated as semidefinite programming (SDP), thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming.

A Survey of the S-Lemma

A unified analysis of the S-lemma is given by providing three different proofs for the theory and revealing hidden connections with various areas of mathematics, to prove some new duality results and present applications from control theory, error estimation, and computational geometry.

Nonlinear Programming: Theory and Algorithms

The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.