Lecture notes: Semidefinite programs and harmonic analysis

谐波载荷下的桁架拓扑设计：基于半定规划的峰值功率最小化

Truss topology design under harmonic loads: Peak power minimization with semidefinite programming

针对无阻尼谐波振动下桁架拓扑优化问题，提出基于半定可表示函数的新型框架，通过最小化谐波载荷下结构峰值功率克服传统方法假设局限，引入凸松弛并获有前景计算结果。

具有受限调和支撑的平方和多项式 Gram 矩阵的结构

The structure of Gram matrices of sum of squares polynomials with restricted harmonic support

本文利用 Gram 矩阵结构将 $SL(2)$ 与 $SO(2)$ 的表示论联系起来，证明存在于 $SO(2)$ 不变子空间的平方和多项式具有存在于 $SL(2)$ 特定不变子空间的半正定 Gram 矩阵。

优化器的量子算法

Quantum algorithms for optimizers

本讲义面向应用数学家和工程师，无需量子力学背景，系统介绍量子优化算法相关内容，涵盖计算模型、量子搜索等主题，为研究生课程提供学习资料。

How Well Do Local Algorithms Solve Semidefinite Programs?

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erdős-Renyi random graphs with bounded average degree $d>1$, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking…

对单相电力转换器最优调制中最小电流谐波失真的界值分析

Bounding the Minimal Current Harmonic Distortion in Optimal Modulation of Single-Phase Power Converters

本文提出了一种将最优脉冲模式设计转化为混合系统周期性模式选择最优控制问题的方法，通过凸松弛技术生成半正定规划层次结构，有效界定了单相电力转换器最优调制中最小电流谐波失真。