Optimization Problems with Quasi-Equilibrium Constraints: Control, Identification, and Design

具有准平衡约束的优化问题：控制、辨识和设计

基本信息

批准号：
2012391
负责人：
Carlos Rautenberg
金额：
$ 19.99万
依托单位：
George Mason University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2012391&HistoricalAwards=false
关键词：
Optimization Problems Quasi Equilibrium Constraints

项目摘要

A wide range of problems in applied sciences involve constraints on variables of interest. These naturally arise in modeling of complex physical phenomena but also appear as a result of hierarchy or competition. Two different classes of these constraints can be described: explicit, where the bounds are known in advance, and implicit, where the bounds depend on the solution of the problem itself. One simple example of an implicitly constrained problem is that of finding the position of an elastic membrane with an obstacle that deforms upon the action of the membrane. In this example, the membrane position is the variable of interest, and the position of the obstacle is the implicit bound or constraint. The control and parameter identification for this class of implicitly constrained problems represent a significant challenge for a large variety of problems. Some possible applications include the design of composite materials that sustain large forces without plastic deformation, the manufacture of multilayer organic light emitting diodes (OLEDs), and the detection of subsurface cracks in buildings that may compromise structural integrity and lead to catastrophic failure.An increasing number of challenging problems in applied sciences involve non-differentiable structures as well as partial differential operators, thus leading to nonsmooth distributed parameter systems. Many of these problems have, directly in the problem formulation, an additional form of implicit constraint resulting in a quasi-variational inequality (QVI). This is commonly found in elastoplasticity, friction mechanics, superconductivity, and also arises as the result of competition of a finite resource in generalized Nash games. Structurally speaking, QVIs are nonconvex and nonsmooth problems that possess a variational formulation with a constraint not known a priori and depending on the state itself. Many design, control or identification problems involve QVIs. In particular, these are formulated as an optimization problem with the QVI as constraint, and where the design variable or the unknown parameter is of piecewise constant nature. This significantly increases the difficulty of the overall problem but it links it directly to real life applications. This proposal focuses on a class of optimization problems with quasi-variational constraints. The formulation is wide enough to include problems associated to water accumulation in real topographical data, non-isothermal elastoplasticity, and current flow on organic multilayer structures (LEDs). We aim at the development of solution algorithms of QVIs, and optimization thereof. The approaches will include an appropriate form of Moreau-Yosida regularization of the implicit constraint, and novel forms of regularization to guarantee a piecewise constant nature of solution parameters. The new methods developed here will enable the solution of problems that are currently intractable.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

应用科学中的各种问题涉及对感兴趣的变量的限制。这些自然出现在建模复杂的物理现象时，但由于层次结构或竞争而出现。可以描述这些约束的两个不同类别的类别：显式，即预先知道界限的情况，而界限则取决于问题本身的解决方案。隐式约束问题的一个简单示例是找到弹性膜的位置，其障碍物在膜的作用上变形。在此示例中，膜位置是感兴趣的变量，障碍物的位置是隐式约束或约束。该类别的隐式约束问题的控制和参数识别代表了各种问题的重大挑战。一些可能的应用包括设计的复合材料，这些复合材料在没有塑性变形的情况下维持大力，多层有机光发射二极管（OLEDS）（OLEDS）的制造以及建筑物中的地下裂纹的检测可能会损害结构完整性，并损害灾难性失败，从而导致灾难性的越来越多的疾病，从而导致了不可或缺的疾病，因此具有较大的成分，并构成了良好的疾病，又构成了良好的疾病，并构成了良好的疾病，并分别构成了良好的疾病。分布式参数系统。这些问题中有许多直接在问题制定中，是隐性约束的另一种形式，导致了准差异不平等（QVI）。这通常是在弹性性，摩擦力学，超导性中发现的，并且由于广义纳什游戏的有限资源的竞争而产生。从结构上讲，QVI是非概念和非平滑问题，具有差异表述，并取决于状态本身，并取决于状态本身。许多设计，控制或识别问题涉及QVI。特别是，这些被称为QVI作为约束的优化问题，以及在何处设计变量或未知参数具有分段恒定性质。这大大增加了整体问题的难度，但它将其直接与现实生活应用联系起来。该提案重点介绍了一系列具有准差异约束的优化问题。该公式足够宽，包括在实际地形数据，非等温弹性性和有机多层结构（LED）上流动的流动中与水积累相关的问题。我们旨在开发QVI的解决方案算法及其优化。这些方法将包括对隐式约束的适当形式的莫罗 - yosida正则化，以及新型的正则化形式，以保证解决方案参数的分段恒定性质。此处开发的新方法将使目前棘手的问题解决方案。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来支持。