Inertial search and asynchronous decompositions

In this dissertation, we present a new heuristic optimization algorithm, called Inertial Search. We also develop a new approach to the on-line contingency planning problem. This approach uses a particular instance of a more general class of decompositions, that we call asynchronous decompositions. Inertial Search does a coarse search of the decision space of nonconvex nonlinear programming (NLP) problems, and finds starting points for local optimization methods, like Sequential Quadratic Programming (SQP), which then find the corresponding minima. Inertial Search belongs to the family of trajectory methods, which map NLP problems to dynamical systems and solve the latter to obtain a trajectory that will lead to minima. It differs from other trajectory methods primarily in that its dynamical system does not have any equilibria, and so it does not stop at any local minimum. We use a multi-agent software organizational paradigm called Asynchronous Teams (A-Teams) to implement a scheme that enables several copies of Inertial Search and SQP, working asynchronously and in parallel, to find multiple minima for large NLPs. We provide experimental evidence of the efficacy of this scheme for difficult NLP test problems. The on-line contingency planning problem is to maintain and update control plans to guard against contingencies (disturbances that lead to a change in the operating configuration), while continuing to operate optimally (with respect to cost) in the current configuration. We study the on-line contingency planning problem, in detail, in the context of control of electric power networks. We first revise the traditional formulation of this problem using the concept of correction time, which is the time required to eliminate constraint violations caused by a contingency. In this way, we obtain a formulation that has a feasible solution regardless of the number of contingencies. We then develop an asynchronous decomposition for this revised formulation by using the notion of formations to keep the subproblems together. This decomposition is such that the subproblems corresponding to the current configuration, and the various contingency configurations, can be solved asynchronously and in parallel. We design and implement an A-Team for obtaining Pareto curves in the two objectives (cost and correction times). Using illustrative power networks, we show that our asynchronous decomposition is a good way to assist operators in control centers, in choosing a suitable compromise between multiple conflicting objectives, in real-time.