复合Optimization中的一致逼近
Consistent Approximations in Composite Optimization
摘要：
在计算过程和灵敏度分析中出现了最优化问题的近似。结果对解决方案的影响可能是显著的，即使是一个问题的组成部分的小近似值也会转化为解决方案中的大错误。我们指定的条件下，近似表现良好的意义上的最小化，不动点，和水平集，这导致了一致的近似框架。该框架是为一类广泛的复合问题，这是既不凸也不光滑。我们使用随机最优化，基于神经网络的机器学习，分布鲁棒优化，惩罚和增广拉格朗日方法，邻域点方法，同伦方法，平滑方法，扩展非线性规划，差分凸规划和多目标最优化的例子来演示该框架。一个增强的近端方法说明了算法的可能性。定量分析通过提供收敛率来补充发展。
英文摘要：
Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.
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