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

George Mason University, Fairfax VA

Investigators

Abstract

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.