AI 中文总结
该研究针对凸约束优化问题，开发并分析了一种应用于对数障碍函数的带线搜索的正则化牛顿法，在特定条件下，从严格可行点出发，能在$\tilde{O}(\epsilon^{-2/3})$次迭代中找到$\epsilon$-近似最优解，弥补了无自和谐假设的约束凸优化方法分析的不足。
AI 中文摘要
内点法因其在求解线性、凸和非凸优化问题时具有较高的实际效率而被广泛采用。对于凸优化，其性能在理论上有很好的支持：自和谐障碍设置有很强的复杂度保证。无约束凸优化已得到充分研究，但对无自和谐假设的约束凸优化方法分析有限。我们开发并分析了一种应用于对数障碍函数的带线搜索的正则化牛顿法，在目标函数和约束条件三次可微且其一阶和二阶导数具有Lipschitz连续性的情况下，从严格可行点开始，该方法能在$\tilde{O}(\epsilon^{-2/3})$次迭代中找到$\epsilon$-近似最优解。
英文摘要
Interior-point methods (IPMs) are widely adopted due to their high practical efficiency in solving linear, convex, and nonconvex optimization problems. For convex optimization, this performance is theoretically well-supported: there are strong complexity guarantees for self-concordant barrier setups \cite{nesterov1994interior}, which cover linear and conic optimization. On the other hand, unconstrained convex optimization is well-studied. However, there is limited analysis of constrained convex optimization methods without the self-concordance assumption. We develop and analyze a regularized Newton method with line search applied to the log barrier function in the setting that the objective and constraints are thrice differentiable and have Lipschitz continuous first and second derivatives. Starting from a strictly feasible point, our method finds an $ε$-approximately optimal solution in $\tilde{O}(ε^{-2/3})$ iterations.
