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Extending the Realm of Optimization for Complex Systems: Uncertainty, Competition and Dynamics

Co-PIs Uday V. Shanbhag, Tamer Basar, Prashant G. Mehta (UIUC), and Sean Meyn (UF)

$1,000,000 (approx.)

Department of Energy Program Solicitation: DE-PS02-08ER08- 13, LAB 08-13 Category: Optimization

January 2009 – December 2013

Project Summary This project concerns the development of theory and algorithms to contend with planning and operational decisions within the next generation of engineering-economic systems. Such systems are expected to be massive in scale and encompass a variety of cooperative and competitive paradigms. As a result, conventional approaches that rely on linear or even convex static optimization hold little hope. Accordingly, our goal is to establish the foundations of a new framework that allows for possibly competitive decision-making in the presence of both dynamics and uncertainty, with the latter also including unmodeled adversarial behavior.

This work finds application on the simulation of a large-scale electricity network that captures the competitive, dynamic, stochastic, and worst-case aspects that are inherent to the workings of a real electricity market.

Our efforts have been organized as follows:

Planning under uncertainty: A majority of the multiperiod approaches in this context have been restricted to linear or possibly convex settings with little ability to address nonlinear nonconvex settings or more general bilevel (or multilevel) structures. We propose to develop globally convergent multiperiod approaches that rely on sequential quadratic programming and primal-dual frameworks which would lead to local minimizers (and not only local stationary points). It is widely recognized that obtaining exact solutions is generally impossible, and thus a focus of this work is in developing sampling-based bounds that result in rigorous confidence statements.

Decision-making in non-cooperative settings: We contend that in a majority of settings, the problem of interest is often an equilibrium problem (arising from a game) as opposed to a single-person optimization problem. Such a problem can take on a wealth of forms based on the specific information structures. A key question we address is the existence and uniqueness of equilibria in settings where coupled markets may sequentially clear (such as in forward and day-ahead markets in power networks). Algorithmic questions are addressed using efforts from (1) to develop convergent scalable schemes for solving equilibrium problems with both dynamics and uncertainty, with the latter admitting stochastic as well as adversarial (worst-case) modeling.

Optimal operations in complex systems: Operation-level problems have far different characteristics leading to massive dynamic programming problems. Through a variety of projection methods that are rooted in multiscale ideas, we aim at developing rigorous approaches with the final goal of obtaining approximately optimal policies and providing fundamental performance bounds. We also study how such ideas can be extended to the competitive realm. 



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