AI 中文总结
该研究提出基于无共轭缩放的原始-对偶内点法，将其嵌入齐次自对偶预测-校正框架，改进了迭代复杂度界，在特定锥优化问题上性能与专用求解器QICS相当。
AI 中文摘要
我们开发了一种基于无共轭缩放矩阵的非对称锥优化原始-对偶内点法，该缩放通过原始障碍海森矩阵的单割线BFGS更新得到。与多割线BFGS缩放不同，它不需要共轭障碍导数，这一特性对高维非对称锥至关重要——这类锥中共轭障碍导数可能无法以闭式形式表达或计算成本高昂。我们将无共轭缩放嵌入齐次自对偶预测-校正框架，采用分别控制锥变量与标量齐次变量的分裂中心路径邻域，证明了缩放矩阵与原始障碍海森矩阵始终保持一致可比。该比较界用于证明邻域保持性，并说明互补性测度与线性残差以均匀速率下降。因此，该方法达到$\tilde{O}(\nu\rho\boldsymbol{\text{log}}(1/\boldsymbol{\text{ε}}))$的迭代界，改进了Badenbroek和Dahl[《Optim. Methods Softw.》，37（2022），第1027-1064页]提出的$\boldsymbol{\text{O}}(\nu\boldsymbol{\text{log}}(1/\boldsymbol{\text{ε}}))$界，且匹配内点法的最佳已知复杂度阶。对包含算子视角上锥和量子相对熵锥的实例进行数值实验，结果表明该方法与QICS（一种针对量子信息领域锥模型的专用求解器）相比具有竞争力。
英文摘要
We develop a primal--dual interior-point method for nonsymmetric conic optimization based on a conjugate-free scaling matrix. The scaling is obtained from a single-secant BFGS update of the primal barrier Hessian. In contrast to multi-secant BFGS scalings, it does not require conjugate-barrier derivatives. This feature is important for high-dimensional nonsymmetric cones, where conjugate-barrier derivatives may be unavailable in closed form or expensive to compute. We embed the conjugate-free scaling in a homogeneous self-dual predictor--corrector framework. Using a split central-path neighborhood that separately controls the conic variables and the scalar homogeneous variables, we prove that the scaling matrix remains uniformly comparable to the primal barrier Hessian. This comparison bound is used to prove neighborhood preservation and to show that the complementarity measure and the linear residual decrease at a uniform rate. Consequently, the method attains an iteration bound of $\mathcal{O}(\sqrtν\log(1/\varepsilon))$, improving the $\mathcal{O}(ν\log(1/\varepsilon))$ bound of Badenbroek and Dahl [Optim. Methods Softw., 37 (2022), pp. 1027--1064] and matching the best-known complexity order for interior-point methods. Numerical experiments on instances involving the operator perspective epigraph cone and the quantum relative entropy cone show that the method is competitive with QICS, a specialized solver for conic models arising in quantum information.
Comments30 pages, 3 figures
