发表机构
George Mason University（乔治梅森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对算子复合信赖域方法，提出相对原始-对偶间隙证书，保证收敛性并给出复杂度界，通过半线性椭圆控制问题验证了方法的有效性。
AI 中文摘要
我们研究信赖域最小化问题，其中目标为光滑（可能非凸）泛函加上一个凸泛函与有界线性算子的复合。相对原始-对偶间隙条件同时控制近似近端梯度步的误差及其线性模型下降量。结合一个可计算的绝对平稳性检验，该条件产生有限的柯西搜索、近端平稳性度量收敛到零，以及外部试验次数为$O(\varepsilon^{-2})$的界。外部分析允许线性算子取值于Banach空间，且不要求对偶达到。当算子取值于Hilbert空间且正则项有限且Lipschitz时，对偶近端梯度方法产生趋于零的有限间隙，前提是所需的近端映射和泛函值可计算。我们证明了恢复和平均原始候选的$O(j^{-1})$间隙界，并给出精确恢复点的原始误差的更尖锐界。一个带有未平滑全变差正则化和$L^2$控制成本的半线性椭圆控制问题在全$H^1$度量下说明了该方法。在五个网格上，外部和状态牛顿计数保持恒定，而内点迭代次数变化轻微。
英文摘要
We study trust-region minimization of a smooth, possibly nonconvex functional plus a convex functional composed with a bounded linear operator. A relative primal--dual gap condition controls both the error in an approximate proximal-gradient step and its linear-model decrease. Together with a computable absolute stationarity test, it yields a finite Cauchy search, convergence of the proximal stationarity measure to zero, and an $O(\varepsilon^{-2})$ bound on outer trials. The outer analysis allows the linear operator to take values in a Banach space and does not require dual attainment. When the operator takes values in a Hilbert space and the regularizer is finite and Lipschitz, the dual proximal-gradient method produces finite gaps tending to zero, provided the required proximal maps and functional values can be evaluated. We prove $O(j^{-1})$ gap bounds for both recovered and averaged primal candidates and give a sharper bound on the primal error for exactly recovered points. A semilinear elliptic control problem with unsmoothed total-variation regularization and an $L^2$ control cost illustrates the method in the full $H^1$ metric. Across five meshes, outer and state Newton counts remain constant, while interior-point iteration counts vary mildly.
