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

应用科学中的许多问题都涉及对感兴趣的变量的限制。这些自然地出现在复杂物理现象的建模中，但也作为等级或竞争的结果出现。可以描述这些约束的两种不同类别：显式约束和隐式约束，前者的边界是预先已知的，后者的边界取决于问题本身的解。隐式约束问题的一个简单的例子是求一个弹性膜的位置，该弹性膜具有一个在膜的作用下变形的障碍物。在本例中，膜位置是感兴趣的变量，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辨识对于许多类型的问题来说是一个巨大的挑战。一些可能的应用包括设计承受较大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故障的亚表面裂缝。在应用科学中，越来越多的挑战性问题涉及不可微结构和偏微分算子，从而导致非光滑分布参数系统。这些问题中的许多直接在问题表述中具有附加形式的隐式约束，从而导致拟变分不等式(QVI)。这在弹塑性力学、摩擦力学、超导电性中很常见，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、非光滑的问题，它具有一个变分公式，约束条件未知，并且依赖于状态本身。许多设计、控制或识别问题都涉及合格投资者。具体地说，这些都被描述为一个以QVI为约束的优化问题，其中设计变量或未知参数是分段常量性质的。这大大增加了整个问题的难度，但它将其直接链接到现实生活中的应用程序。本文主要研究一类具有拟变分约束的优化问题。该公式的范围足够广泛，足以包括与实际地形数据中的水积累、非等温弹塑性以及有机多层结构(LED)上的电流流动相关的问题。我们致力于发展QVI的求解算法，并对其进行优化。这些方法将包括隐式约束的适当形式的Moreau-Yosida正则化，以及保证解参数的分段恒定性质的新形式的正则化。这里开发的新方法将使目前难以解决的问题得到解决。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.