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

Optimizing Rayleigh quotient with symmetric constraints and their applications to perturbations of the structured polynomial eigenvalue problem

For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-an…

Fixed-rank Rayleigh Quotient Maximization by an $M$PSK Sequence

Certain optimization problems in communication systems, such as limited-feedback constant-envelope beamforming or noncoherent $M$-ary phase-shift keying ($M$PSK) sequence detection, result in the maximization of a fixed-rank positive semidefinite quadratic form over the $M$PSK alphabet. This form is a special case of the Rayleigh quotient of a matrix and, in general, its maximization by an $M$PSK sequence is $\mathcal{NP}$-hard. However, if the rank of the matrix is not a function of its size, then the optimal solution can be computed with polynomial complexity in the matrix size. In this work…

Linear Constrained Rayleigh Quotient Optimization: Theory and Algorithms

We consider the following constrained Rayleigh quotient optimization problem (CRQopt) $$ \min_{x\in \mathbb{R}^n} x^{T}Ax\,\,\mbox{subject to}\,\, x^{T}x=1\,\mbox{and}\,C^{T}x=b, $$ where $A$ is an $n\times n$ real symmetric matrix and $C$ is an $n\times m$ real matrix. Usually, $m\ll n$. The problem is also known as the constrained eigenvalue problem in the literature because it becomes an eigenvalue problem if the linear constraint $C^{T}x=b$ is removed. We start by equivalently transforming CRQopt into an optimization problem, called LGopt, of minimizing the Lagrangian multiplier of CRQopt,…

Eigenvalue Problems in $\mathrm{L}^\infty$: Optimality Conditions, Duality, and Relations with Optimal Transport

In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$ and relate it to a divergence-form PDE, similarly to what is known for $\mathrm{L}^p$ eigenvalue problems and the $p$-Laplacian for $p<\infty$. Contrary to existing methods, which study $\mathrm{L}^\infty$-problems as limits of $\mathrm{L}^p$-problems for $p\to\infty$, we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a …

二次函数商的零阶方法推广

Generalization of Zeroth-Order Method for Quotients of Quadratic Functions

本文提出了一种新的无约束采样方法，用于计算广义算子范数和极值广义Rayleigh商，通过与Riemannian优化方法的联系，改进了零阶方法的效率。

在有理函数和最小化中利用符号对称性

Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

本文研究基本半代数集上有理函数和的最小化问题，提出与广义矩方法对偶的SOS松弛层级，利用符号对称性与相关稀疏性构建高效稀疏半定松弛，通过数值实验验证有效性并给出广义Rayleigh商最大化的应用。

在流形上优化技术

Optimization Techniques on Riemannian Manifolds

本文提出在黎曼流形上求解优化问题的新方法，将欧几里得空间的经典优化技术推广到流形，并分析了两种新算法的收敛性。