Grey box optimization

优化领域的广度很大。在一个极端，存在结构众所周知且可利用的优化问题：例如线性规划或带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）是指目标函数的结构和约束是未知的，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，经常会出现这些情况。这些是这个研究项目的核心问题。求解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，可以在最大化环境中遵循。无导数优化（DFO）是指函数的导数不可用，或者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，甚至没有理由相信衍生品存在，而在前者中，衍生品可能存在，但无法得到表达。在20世纪90年代之前，只有在DFO和BBO领域取得了准时的发展。但从那以后，人们对这些研究领域的兴趣一直在稳步增长。这些都是非线性优化研究中发展最为迅速的领域。部分原因可能是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实际工程问题。本提案中描述的研究项目建立在我之前nserc资助的关于DFO和BBO问题的直接搜索算法的工作基础上。这个项目的目的是探索在广泛的优化领域的灰色地带：问题的部分结构，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箱函数平方和的情况下，以及关于某些变量的某些约束的单调性的粗糙信息是明确已知的。本项目将研究其他类型的灰盒优化问题，其中关于约束、问题性质和替代的高级定性或定量信息是可用的。本提案中概述的所有项目的发展都将在实际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。我们计划继续在水文科学、制药和生物信息工业、合金设计、超材料设计和航空等领域应用我们的工作。

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.