Thomas Edmunds

Algorithms for nonlinear bilevel mathematical programs

The bilevel programming problem (BLPP) is a model of a leader-follower game in which play is sequential and cooperation is not permitted. Some basic properties of the general model are developed, and a conjecture relevant to solution procedures is presented. Two algorithms are presented for solving various versions of the game when certain convexity conditions hold. One algorithm relies upon a hybrid branch-and-bound scheme and does not guarantee global optimality. Another is based on objective function cuts and, barring numerical stability problems with the optimizer, is guaranteed to converge to an epsilon -optimal solution. The performance of the two algorithms is examined using randomly generated test problems. The computational performance of the branch-and-bound algorithm is explored, and the cutting-plane algorithm is used to determine whether or not the branch-and-bound algorithm is uncovering global optima. >

Trilevel Optimization in Power Network Defense

We present a trilevel optimization model of resource allocation in electric power network defense. This model identifies the most critical network components to defend against possible terrorist attacks. The goal of defense is to minimize the economic cost that the attacks may cause, subject to resource constraints. We describe a decomposition approach for finding an optimal solution to the trilevel model, which is based on iteratively solving smaller nested bilevel problems. Our testing results demonstrate the advantages of trilevel optimization over bilevel optimization in network defense.

A Multiattribute Utility Analysis of Alternatives for the Disposition of Surplus Weapons-Grade Plutonium

This paper describes an application of multiattribute utility theory to support the selection of a technology for the disposition of surplus weapons-grade plutonium by the Department of Energy (DOE). This analysis evaluated 13 alternatives, examined the sensitivity of the recommendations to the weights and assumptions, and quantified the potential benefit of the simultaneous deployment of several technologies. The measures of performance that were identified through the creation of a hierarchy of objectives helped to organize the information collected during the evaluation process, and the results of the analysis were presented to DOE on several occasions. This analysis supported the final DOE recommendation to pursue a strategy of the parallel development of two of the most preferred technologies.

An algorithm for the mixed-integer nonlinear bilevel programming problem

The bilevel programming problem (BLPP) is a two-person nonzero sum game in which play is sequential and cooperation is not permitted. In this paper, we examine a class of BLPPs where the leader controls a set of continuous and discrete variables and tries to minimize a convex nonlinear objective function. The followers objective function is a convex quadratic in a continuous decision space. All constraints are assumed to be linear. A branch and bound algorithm is developed that finds global optima. The main purpose of this paper is to identify efficient branching rules, and to determine the computational burden of the numeric procedures. Extensive test results are reported. We close by showing that it is not readily possible to extend the algorithm to the more general case involving integer follower variables.

A Multiclass Hybrid Production Center in Heavy Traffic

This paper presents an analysis of a single-stage hybrid production system that makes multiple types of products, some of which are made to-order while others are made to-stock. The analysis begins with a formal heavy traffic limit theorem of the production system, which is modeled as a mixed queueing network. Taking insights from the limit theorem, the analysis continues with the development of an approximation procedure. Numerical experiments indicate that this procedure provides good estimates for performance measures such as fill rates and average inventory levels.

Evaluation of Alternatives for the Disposition of Surplus Weapons-usable Plutonium

The Department of Energy Record of Decision (ROD) selected alternatives for disposition of surplus, weapons grade plutonium. A major objective of this decision was to prevent the proliferation of nuclear weapons. Other concerns addressed included economic, technical, institutional, schedule, environmental, and health and safety issues. The analysis reported here was conducted in parallel with technical, environmental, and nonproliferation analyses; it uses multiattribute utility theory to combine these considerations in order to facilitate an integrated evaluation of alternatives. This analysis is intended to provide additional insight regarding alternative evaluation and to assist in understanding the rationale for the choice of alternatives recommended in the ROD. Value functions were developed for objectives of disposition, and used to rank alternatives. Sensitivity analyses indicated that the ranking of alternatives for the base case was relatively insensitive to changes in assumptions over reasonable ranges. The analyses support the recommendation of the ROD to pursue parallel development of the vitrification immobilization alternative and the use of existing light water reactors alternative. 27 refs., 109 figs., 20 tabs.

Time-axis decomposition of large-scale optimal control problems

Continuous-time optimal control problems can rarely be solved directly but have to be approximated with discrete analogues. Shorter time steps lead to more accurate approximations, but result in formulations that are often too big for computer memory. This paper presents a technique for decomposing the problem along the time axis and iterating toward a solution in a leader-follower framework.In the model, the leader controls a set of coordination parameters, which he passes to the followers, who then solve their individual subproblems. State and sensitivity information is returned to the leader, who attempts to minimize an unconstrained problem in the coordination space. Parameters are updated and the process continues until improvement ceases. Two advantages of this technique are that feasible solutions to the original problem are available at each iteration and that the optimal coordination parameters obtained provide some measure of feedback control. Computational results are presented for a comprehensive set of test problems.

Exploitation of Ambiguous Cues to Infer Terrorist Activity

To aid intelligence analysts in processing ambiguous data regarding nuclear terrorism threats, we develop a methodology that captures and accounts for the uncertainty in new information and incorporates prior beliefs on likely nuclear terrorist activity. This methodology can guide the analyst when making difficult decisions regarding what data are most critical to examine and what threats require greater attention. Our methodology is based on a Bayesian statistical approach that incorporates ambiguous cues to update prior beliefs of adversary activity. We characterize the general process of a nuclear terrorist attack on the United States and describe, using a simplified example, how this can be represented by an event tree. We then define hypothetical cues for the example and give notional strengths to each cue. We also perform sensitivity analysis and show how cue strengths can affect inference. The method can be used to help support decisions regarding resource allocation and interdiction.

Integrated stochastic weather and production simulation modeling

High penetration of intermittent renewable generators can substantially increase the variability and uncertainty in power system operations. Energy storage and demand response have been proposed as resources that can be used to mitigate this uncertainty and variability. This paper describes planning system that couples a stochastic weather model, renewable generation models that are driven by the weather, a stochastic production simulation model, and a system stability model. The system is used to simulate operation of the California grid with 33% variable renewable generation in the year 2020. The values of energy storage and demand response are estimated by identifying the avoided costs of the conventional hydro and fossil resources that they displace when providing regulation, load following, and energy arbitrage functions. The impacts on system stability are also assessed.

High-performance computing for electric grid planning and operations

High-performance computing (HPC) is having a profound impact on scientific discovery and engineering in a variety of areas, and researchers are beginning to demonstrate how HPC can impact problems in energy grid planning and operations. Contemporary supercomputers can perform over 1015 floating point operations per second and have more than 1.4 petabytes of memory - roughly 5 orders of magnitude greater than a commodity PC workstation. This level of computing power changes the very nature of problems that can be solved. Researchers at LLNL have used HPC systems to accelerate execution of a renewables planning study, by solving a thousand unit commitment and dispatch problems in parallel; this generated new insights and allowed for a more detailed study than would have been otherwise achievable. Ongoing work at LLNL includes the development and testing of new parallel algorithms for unit commitment problems, including multi-scenario stochastic unit commitment. These algorithms will enable greater grid and time resolution and provide more accurate solutions because of the increase in model fidelity.