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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.

应用科学中的一系列问题都涉及到对感兴趣的变量的约束。这些自然出现在复杂物理现象的建模中，但也出现在等级或竞争中。这些约束可以分为两类：显式约束，即边界是事先已知的;隐式约束，即边界取决于问题本身的解。隐式约束问题的一个简单例子是找到弹性膜的位置，该弹性膜具有在膜的作用下变形的障碍物。在该示例中，膜位置是感兴趣的变量，并且障碍物的位置是隐式边界或约束。这类隐式约束问题的控制和参数识别是一个重大的挑战，各种各样的问题。一些可能的应用包括设计能够承受较大作用力而不发生塑性变形的复合材料、制造多层有机发光二极管（OLED）以及检测建筑物中可能危及结构完整性并导致灾难性失效的地下裂缝。应用科学中越来越多的挑战性问题涉及不可微结构以及偏微分算子，从而导致非光滑分布参数系统。许多这些问题，直接在问题的制定，一个额外的形式的隐式约束导致准变分不等式（QVI）。这在弹塑性、摩擦力学、超导性中很常见，也是广义纳什博弈中有限资源竞争的结果。从结构上讲，QVI是非凸和非光滑的问题，拥有一个变分公式与约束未知的先验和依赖于状态本身。许多设计、控制或识别问题都涉及QVI。特别是，这些被制定为一个优化问题的QVI作为约束，并且其中的设计变量或未知参数是分段恒定的性质。这大大增加了整个问题的难度，但它直接将其与真实的生活应用联系起来。本文研究一类具有拟变分约束的优化问题。该配方是足够广泛的，包括与水积累在真实的地形数据，非等温弹塑性，和有机多层结构（LED）上的电流相关的问题。我们的目标是开发QVI的求解算法，并对其进行优化。这些方法将包括适当形式的Moreau-Yosida正则化的隐式约束，和新形式的正则化，以保证分段常数性质的解决方案参数。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。