A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity \(O(\varepsilon ^{-2})\) (first-order) and \(O(\varepsilon ^{-3/2})\) (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems.

数学成像、信号处理和计算统计中的一个关键问题是非凸目标函数的最小化，这些目标函数在可行集的相对边界处可能是不可微的。本文针对具有线性和圆锥约束的非凸优化问题，提出了一系列新的一阶和二阶内点方法，分别将对数齐次障碍与二次和三次正则化相结合。我们的方法基于势能减少机制，并且在相应导数相对于局部势垒诱导范数的 Lipschitz 连续性下，通过最坏情况迭代获得适当定义的近似一阶或二阶 KKT 点类复杂度分别为 \(O(\varepsilon ^{-2})\)（一阶）和\(O(\varepsilon ^{-3/2})\)（二阶）。基于这些发现，我们开发了新的路径跟踪方案，以获得相同的复杂性和模调整常数。已知这些复杂性界限在无约束的情况下是最优的，并且我们的工作表明它们在具有复杂约束的情况下也是上限。据我们所知，这项工作是第一个在一般圆锥约束非凸优化问题的弱条件下实现这些最坏情况复杂性界限的工作。