Sparse SOS Relaxations for Minimizing Functions that are Summations of Small Polynomials

在球面积的多项式优化问题中的动量-SOS层次有限收敛性

Finite Convergence of the Moment-SOS Hierarchy on the Product of Spheres

该研究证明了在球面积上多齐次多项式优化问题中动量-SOS层次的有限收敛性，并通过微分几何和莫尔斯理论证明了局部最优条件的一般性成立。

Norm Bounds and Underestimators for Unconstrained Polynomial Integer Minimization

We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer minimizers as well. In case this condition holds, we use sos programming to compute the radius of a p-norm ball which contains all integer minimizers. We prove that this radius is smaller than the radius known from the literature. Furthermore, we derive a new class of underestimators of the polynomial function. Using a Stellensatz from real algebraic geometry and a…

收敛的提升Lasserre层级SDP用于最小化Wasserstein球上分段多项式损失的期望

Convergent Lifted Lasserre Hierarchy of SDPs for Minimizing Expectation of Piecewise Polynomial Loss over Wasserstein Balls

本文提出了一种收敛的提升Lasserre层级SDP方法，用于在Wasserstein球上最小化分段多项式损失的期望，通过结构化SOS表示和凸多项式条件实现有限收敛。

Minimizing Polynomials Over Semialgebraic Sets

This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in $\re^m$ defined by finitely many polynomial equations and inequalities, using the Karush-Kuhn-Tucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper \cite{njw_grad}, which considers minimizing polynomials on algebraic sets, i.e., sets in $\re^m$ defined by finitely many polynomial equations. Most of the theorems and conclusions in \cite{njw_grad} generalize to semialgebraic sets, even in the case where the semialgebraic set is not compact. …

具有动量和平方和的多凸性

Polyconvexity with Moments and Sums of Squares

本文研究多凸性问题，提出基于平方和的技术的可计算条件和基于动量-SOS层次的数值方法来计算多凸包。

通过1-范数中的最速下降分析并改进等式约束优化的贪婪2-坐标更新

Analyzing and Improving Greedy 2-Coordinate Updates for Equality-Constrained Optimization via Steepest Descent in the 1-Norm

本文通过建立贪婪2-坐标更新与1-范数最速下降的联系，在近端Polyak-Lojasiewicz假设下证明了快于随机选择且与维度无关的收敛速率，并针对含求和与边界约束的优化问题提出仅需$O(n \log n)$时间即可保证更大迭代进展的改进方法。

基于平方和泰勒模型的高阶正则化方法的全局收敛性

Global Convergence of High-Order Regularization Methods with Sums-of-Squares Taylor Models

本文针对非凸光滑优化问题，提出结合平方和（SoS）泰勒模型与自适应正则化的算法框架，证明其在一般非凸和强凸场景下的评估复杂度界，填补了相关算法全局收敛速率分析的空白。

最小化四次正则化三阶泰勒多项式的全局最优性刻画与算法

Global Optimality Characterizations and Algorithms for Minimizing Quartically-Regularized Third-Order Taylor Polynomials

研究最小化四次正则化三阶泰勒多项式问题，通过建立全局最优性条件，开发对角张量方法（DTM），证明其收敛性及复杂度，在特殊结构问题上性能优越，为优化问题提供了新算法和理论依据。

An Inexact Riemannian Proximal Gradient Method

This paper considers the problem of minimizing the summation of a differentiable function and a nonsmooth function on a Riemannian manifold. In recent years, proximal gradient method and its invariants have been generalized to the Riemannian setting for solving such problems. Different approaches to generalize the proximal mapping to the Riemannian setting lead versions of Riemannian proximal gradient methods. However, their convergence analyses all rely on solving their Riemannian proximal mapping exactly, which is either too expensive or impracticable. In this paper, we study the convergence…