Morgan Jones — Page 2

arXiv2018

Using SOS for Optimal Semialgebraic Representation of Sets: Finding Minimal Representations of Limit Cycles, Chaotic Attractors and Unions

In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinantlike objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.

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arXiv2018

A Dynamic Programming Approach to Evaluating Multivariate Gaussian Probabilities

We propose a method of approximating multivariate Gaussian probabilities using dynamic programming. We show that solving the optimization problem associated with a class of discrete-time finite horizon Markov decision processes with non-Lipschitz cost functions is equivalent to integrating a Gaussian functions over polytopes. An approximation scheme for this class of MDPs is proposed and explicit error bounds under the supremum norm for the optimal cost to go functions are derived.

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arXiv2017

Solving Dynamic Programming with Supremum Terms in the Objective and Application to Optimal Battery Scheduling for Electricity Consumers Subject to Demand Charges

In this paper, we consider the problem of dynamic programming when supremum terms appear in the objective function. Such terms can represent overhead costs associated with the underlying state variables. Specifically, this form of optimization problem can be used to represent optimal scheduling of batteries such as the Tesla Powerwall for electrical consumers subject to demand charges - a charge based on the maximum rate of electricity consumption. These demand charges reflect the cost to the utility of building and maintaining generating capacity. Unfortunately, we show that dynamic programming problems with supremum terms do not satisfy the principle of optimality. However, we also show that the supremum is a special case of the class of forward separable objective functions. To solve the dynamic programming problem, we propose a general class of optimization problems with forward separable objectives. We then show that for any problem in this class, there exists an augmented-state dynamic programming problem which satisfies the principle of optimality and the solutions to which yield solutions to the original forward separable problem. We further generalize this approach to stochastic dynamic programming problems and apply the results to the problem of optimal battery scheduling with demand charges using a databased stochastic model for electricity usage and solar generation by the consumer.

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