Grey box optimization

优化领域的广度是广泛的。在一个极端，有结构是众所周知的和可利用的优化问题：例如线性规划或具有显式代数公式的凸优化。这不是我研究的目标问题。在另一个极端，黑盒优化（BBO）是指目标函数和约束的结构是未知的，并且在优化过程中无法利用的情况。当定义问题的函数通过耗时的模拟计算时，这些情况经常出现。 这些都是这个研究项目的核心问题。 求解最优化问题最有用的工具之一是导数。实际上，梯度向量是最陡的上升方向，并且可以在最大化上下文中遵循。无导数优化（Derivative-Free Optimization，DFO）是指函数的导数不可用或可能难以估计的情况。DFO和BBO之间的一个微妙区别是，在后者中，没有理由相信衍生物甚至存在，而在前者中，它们可能存在，但它们的表达不可用。在20世纪90年代之前，DFO和BBO油田只有准时的开发。但从那时起，人们对这些研究领域的兴趣稳步增加。这些都是非线性优化研究中发展最快的领域。这可以部分地解释为计算机现在能够在合理的时间内模拟复杂的工程过程，以及在工业环境中成功地利用算法解决真实的工程问题。 在这个建议中描述的研究项目建立在我以前的NSERC资助的DFO和BBO问题的直接搜索算法的工作。这个项目的目标是探索在广泛的优化领域的灰色地带：问题的结构的一部分，但不是全部，是可用的。例如，以前的工作集中在目标函数是黑盒函数的平方和的情况下，以及关于某些约束相对于某些变量的单调性的原始信息是明确已知的。本项目将研究其他类型的灰箱优化问题，其中高层次的定性或定量信息的约束，问题的性质和代理人。 本提案中概述的项目的所有开发将在真实的工程测试问题上进行测试，并将使用非光滑微积分工具进行严格的收敛分析。 这项研究的结果对我们在加拿大的工业合作者是有用的。 我们计划继续将我们的工作应用于水文科学，制药和生物信息产业，合金设计，超材料设计和航空等领域。

The breadth of the optimization field is wide. At one extremity, there are optimization problems for which the structure is well-known and exploitable: linear programming, or convex optimization with explicit algebraic formulations for example. This is not the target class of problems of my research. At the opposite extremity, blackbox optimization (BBO) refers to situations in which the structure of the objective function and of the constraints are unknown and cannot be exploited during the optimization process. These situations frequently arise when the functions defining the problem are computed through a time-consuming simulation. These are the problems at the center of this research project. One of the most useful tools for solving optimization problems is the derivative. Indeed, the gradient vector is the steepest ascent direction and can be followed in a maximization context. Derivative-free optimization (DFO) refers to the situation where the derivatives of the functions are unavailable, or potentially difficult to estimate. A subtle distinction between DFO and BBO is that in the latter there is no reason to believe that derivatives even exist, and in the former they might exist but their expression is not available. Prior to the 1990's, only punctual developments were made in the DFO and BBO fields. But since then, there has been a steady increase in the interest devoted to these research areas. These are among the most rapidly expanding areas of nonlinear optimization research. This may be explained in part by the fact that computers are now able to simulate complex engineering processes in reasonable time, and by the successful utilization of algorithms on real engineering problems in industrial environments. The research project described in this proposal builds on my prior NSERC-funded work on direct search algorithms for DFO and BBO problems. The objective of this project is to explore the grey zone in the wide optimization field: problems for which part of the structure, but not all, is available. For example, previous work has focused on situations where the objective function is the sum of the squares of blackbox functions, and where crude information with respect to the monotonicity of some constraints with respect to certain variables was explicitly known. The present project will study other types of grey box optimization problems in which high-level qualitative or quantitative information about the constraints, the nature of the problem and the surrogates are available. All developments in the projects outlined in this proposal will be tested on real engineering test problems and will be analyzed using tools from nonsmooth calculus for a rigorous convergence analysis. The outcome of this research is useful to our industrial collaborators in Canada. We plan to continue to apply our work in areas such as hydrological sciences, pharmaceutical and bioinformatic industry, alloy design, metamaterial design and aeronautics.