AI 中文总结
针对QSOS优化问题中传统方法的计算瓶颈，提出无提升正则化方法，通过添加范数惩罚保留原始圆锥结构，有闭式原始更新和无约束凹对偶，加速一阶方法可最大化对偶，数值实验显示该方法比现有求解器更快且能处理更大问题。
AI 中文摘要
二次平方和（QSOS）优化问题出现在系统识别和机器学习中，但标准的舒尔补和二阶锥提升会扩大圆锥维度并给内点法带来计算瓶颈。本文引入一种无提升正则化方法，通过对SOS变量添加范数惩罚来保留原始圆锥结构，产生闭式原始更新和具有Lipschitz连续梯度的无约束凹对偶。加速一阶方法有效最大化此对偶，收敛分析表明能非渐近恢复解。数值实验表明该方法比现有求解器快40%，能处理更大问题且内存仅随等式约束数量缩放。
英文摘要
Quadratic Sum-Of-Squares (QSOS) optimization problems appear in system identification and machine learning, but standard Schur-complement and second-order cone liftings enlarge conic dimensions and create computational bottlenecks for interior-point methods. This paper introduces a lifting-free regularization that preserves the original conic structure by adding a norm penalty to SOS variables, yielding closed-form primal updates and an unconstrained, concave dual with Lipschitz-continuous gradient. Accelerated first-order methods efficiently maximize this dual, and convergence analysis shows non-asymptotic recovery of the solution. Numerical experiments on constrained regression problems show the proposed method can be 40\% faster than existing solvers such as SCS and handle larger problems than MOSEK, with memory scaling only in the number of equality constraints.
