The complexity of solving feasibility problems is considered in this work. It is assumed that the constraints that define the problem can be divided into expensive and cheap constraints. At each iteration, the introduced method minimizes a regularized pth-order model of the sum of squares of the expensive constraints subject to the cheap constraints. Under a Hölder continuity property with constant $\beta \in (0,1]$ on the pth derivatives of the expensive constraints, it is shown that finding a feasible point with precision $\epsilon >0$ or an infeasible point that is stationary with tolerance $\gamma >0$ of minimizing the sum of squares of the expensive constraints subject to the cheap constraints has iteration complexity $O(|\log (\epsilon )|{\gamma }^{\zeta (p,\beta )}{\omega }_p^{1+(1/2)\zeta (p,\beta )})$ and evaluation complexity (of the expensive constraints) $O(|\log (\epsilon )|[{\gamma }^{\zeta (p,\beta )}{\omega }_p^{1+(1/2)\zeta (p,\beta )}+(1-\beta )/(p+\beta -1)|\log (\gamma \sqrt{\epsilon })|])$, where $\zeta (p,\beta )=-(p+\beta )/(p+\beta -1)$ and ${\omega }_p=\epsilon$ if p = 1, while ${\omega }_p=\mathrm{\Phi }(x^0)$ if p>1. Moreover, if the derivatives satisfy a Lipschitz condition and a uniform regularity assumption holds, both complexities reduce to $O(|\log (\epsilon )|)$, independently of p.

在这项工作中考虑了解决可行性问题的复杂性。假设定义问题的约束可以分为昂贵的和廉价的约束。在每次迭代中，引入的方法最小化了受廉价约束约束的昂贵约束的平方和的正则化p阶模型。在具有常数的 Hölder 连续性下$\beta \in (0,1]$在代价高昂的约束的p次导数上，表明找到一个精确的可行点 $\epsilon >0$或具有容差的静止不可行点 $\gamma >0$使受廉价约束的昂贵约束的平方和最小化具有迭代复杂性$○(|\mathrm{日志}(\epsilon )|{\gamma }^{\zeta (p,\beta )}{\omega }_p^{1+(1/2)\zeta (p,\beta )})$和评估复杂性（昂贵的约束）$○(|\mathrm{日志}(\epsilon )|[{\gamma }^{\zeta (p,\beta )}{\omega }_p^{1+(1/2)\zeta (p,\beta )}+(1-\beta )/(p+\beta -1)|\mathrm{日志}(\gamma \sqrt{\epsilon })|])$， 在哪里$\zeta (p,\beta )=-(p+\beta )/(p+\beta -1)$和${\omega }_p=\epsilon$如果p  = 1，则${\omega }_p=\mathrm{\Phi }(X^0)$如果p > 1。此外，如果导数满足 Lipschitz 条件并且一致的正则性假设成立，则两种复杂性都减少到$○(|\mathrm{日志}(\epsilon )|)$，独立于 p。