制作自适应Optimization及其在光子晶体设计中的应用
Fabrication-Adaptive Optimization, with an Application to Photonic Crystal Design
摘要：
通常的情况是，由于（i）技术限制，最优化问题的计算的最优解不能直接实现，而不管数据精度如何（如机器或过程的物理公差），（ii）故意简化模型，使其易于处理（通过忽略造成计算困难的某些类型的约束），和/或（iii）人为因素（让人们“做”最优解）。出于这一观察，我们提出了一个建模范式称为“制造自适应优化”的实施/制造处理的问题。我们开发的计算为重点的理论和算法，我们提出的计算结果，将考虑到实施/制造到约束最优化问题，出现在光子晶体设计。制造自适应优化框架源于函数的鲁棒正则化。当可行域不是赋范空间（如在应用设置中通常遇到的）时，制造自适应优化框架通常产生非凸优化问题。(In特殊情况下，可行域是一个有限维赋范空间，我们表明，制造自适应优化可以作为现代鲁棒优化的一个例子。我们研究了各种问题的特殊结构的功能，可行域，和规范，计算是易于处理的，并制定了一个算法方案来解决这些问题，尽管非凸性的挑战。我们应用我们的方法来计算制造自适应设计的二维光子晶体与各种规定的功能。
英文摘要：
It is often the case that the computed optimal solution of an optimization problem cannot be implemented directly, irrespective of data accuracy, due to either (i) technological limitations (such as physical tolerances of machines or processes), (ii) the deliberate simplification of a model to keep it tractable (by ignoring certain types of constraints that pose computational difficulties), and/or (iii) human factors (getting people to "do" the optimal solution). Motivated by this observation, we present a modeling paradigm called "fabrication-adaptive optimization" for treating issues of implementation/fabrication. We develop computationally-focused theory and algorithms, and we present computational results for incorporating considerations of implementation/fabrication into constrained optimization problems that arise in photonic crystal design. The fabrication-adaptive optimization framework stems from the robust regularization of a function. When the feasible region is not a normed space (as typically encountered in application settings), the fabrication-adaptive optimization framework typically yields a non-convex optimization problem. (In the special case where the feasible region is a finite-dimensional normed space, we show that fabrication-adaptive optimization can be re-cast as an instance of modern robust optimization.) We study a variety of problems with special structures on functions, feasible regions, and norms, for which computation is tractable, and develop an algorithmic scheme for solving these problems in spite of the challenges of non-convexity. We apply our methodology to compute fabrication-adaptive designs of two-dimensional photonic crystals with a variety of prescribed features.
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