Structured Blackbox Optimization

最优化是对函数最小化或最大化的研究，几乎在科学的每一个领域都是自然而然地出现的。在一些应用中，优化的使用是显而易见的，例如在设计新的道路时将成本降到最低。在其他应用中，优化的使用更加微妙，例如医学图像中的去噪。在优化中，黑盒是任何分析上不可用的函数。当在一个点上求值时，黑盒返回一个目标函数值。此外，一些黑盒返回(子)渐变向量。黑盒优化问题是定义问题的部分或全部函数由黑盒给出的任何优化问题。黑盒函数的一种常见情况是计算机模拟的输出。在给定一些输入参数的情况下，模拟将执行并返回函数值。如果模拟是用开放源码语言实现的，那么可以使用自动微分来进一步获得(子)梯度向量。随着计算机模拟在现代研究中变得无处不在，黑盒优化是解决未来现实应用的重要研究领域之一，在一些应用中，黑盒具有一些可见的结构。结构化黑盒优化问题是指一些或全部底层函数由黑盒给出，但问题本身具有某种可见的数学结构的任何优化问题。结构化黑盒优化的一个简单例子是最小化“最坏结果”。在这种情况下，每个场景都通过一个黑盒提供，最终目标是最小化所有黑盒函数的最大值。这可以通过将所有黑盒函数的最大值视为单个黑盒函数来实现(通常也是这样)。然而，如果我们认识到极大值函数的结构，我们就可以设计出更快更准确的算法来解决这个问题。我的研究重点是结构化黑盒优化。我的工作包括开发结构化黑盒优化的新算法，应用算法解决现实世界的优化问题，以及提高结构化黑盒优化背后的数学知识。对算法设计感兴趣的HQP将在开发收敛分析、实现算法和算法的数值测试方面进行培训。对优化应用感兴趣的HQP将接受培训，以确定优化问题中的结构，考虑开发这种结构的方法，并在考虑求解时间和质量的同时选择适当的算法来解决问题。对工作理论分析感兴趣的HQP将接受优化理论的广泛领域的培训，包括在泛函分析和变分分析方面的坚实基础。

Optimization, the study of minimizing or maximizing a function, arises naturally in virtually every area of science. In some applications the use of optimization is obvious, such as minimizing the cost when designing a new road. In other applications the use of optimization is more subtle, such as denoising in medical imaging.In optimization, a blackbox is any function that is not analytically available. When evaluated at a point, a blackbox returns an objective function value. In addition, some blackboxes return a (sub)gradient vector. A blackbox optimization problem is any optimization problem where some, or all, of the functions defining the problem are given by blackboxes.One common occurrences of blackbox functions is the output of a computer simulation. Given some input parameters, the simulation executes and returns a function value. If the simulation is implemented in an open source language, then automated differentiation could be employed to further obtain a (sub-)gradient vector. As computer simulations have become ubiquitous in modern research, blackbox optimization represents one of the most important areas of research for solving future real-world applications.In some applications, the blackbox has some visible structure. A structured blackbox optimization problem is any optimization problem where some, or all, of the underlying functions are given by blackboxes, but the problem itself has some visible mathematical structure. A simple example of structured blackbox optimization is minimizing the `worst-case outcome'. In this case, each scenario is provided through a blackbox, and the final objective is to minimize the maximum of all the blackbox functions. This can be (and often has been) approached by considering the maximum of all the blackbox functions as a single blackbox function. However, if we recognize the structure of the max function, we can design algorithms that are faster and more accurate for this problem.My research focuses on structured blackbox optimization. My work includes the development of novel algorithms for structured blackbox optimization, the application of algorithms to solve real-world optimization problems, and the advancement of knowledge in the mathematics behind structured blackbox optimization.HQP interested in working in algorithm design will be trained in developing convergence analysis, implementing algorithms, and numerical testing of algorithms. HQP interested in working in optimization applications will be trained in determining the structures within optimization problems, considering methods to exploit this structure, and selecting the appropriate algorithm to solve problems while considering solution time and quality. HQP interested in working theoretical analysis will be trained in the broad field of optimization theory, including strong foundations in functional analysis and variational analysis.