(234a) New Measures for Shaping Trajectories in Dynamic Optimization

Dynamic optimization (DO) is sub-field of infinite-dimensional optimization where decision variables are defined on (indexed over) temporal domains [1]. DO encompasses a wide breadth of problem classes such as optimal control [2], model predictive control [3], and parameter estimation of dynamic systems [4]. Problems in DO typically employ the Bolza objective which minimizes the integral of the cost trajectory [5]. This has the effect of uniformly penalizing the cost function over the span of the temporal domain. However, we can envision more advanced DO formulations that aim to shape the cost trajectory with other measures (e.g., peak or excursion costs). For example, Risbeck and Rawlings recently proposed an MPC objective that penalizes the total and peak costs [6].

Risk measures denote modeling constructs in stochastic optimization (SO), another sub-field of infinite-dimensional optimization, that are summarizing statistics (e.g., average, variance, quantiles, worst-case values) that are used to shape the probability density of random cost and constraint functions [7]. These were originally motivated by financial SO problems that sought to manipulate random cost functions in desirable ways (e.g., minimize the impact of extreme events). Following our unifying abstraction for infinite-dimensional optimization (InfiniteOpt) problems in [1], we have shown that DO problems are analogous to (and special cases of) two-stage stochastic programs [8]. Here, a key observation is that the Bolza objective integral from DO and risk measures from SO can be commonly abstracted as measure operators that scalarize infinite-dimensional variables/functions. This connection presents the opportunity for us to transfer risk measures (and their rich theory) from SO to a DO context.

In this talk, we propose a new class of measures for shaping time-dependent trajectories in DO that arise from time-valued analogues to risk measures used in SO [8]. We show that this extensive collection of measures provides us with significant flexibility in modeling DO problems, and we show how these can be applied in DO for computing and manipulating interesting features of time-dependent trajectories (e.g., excursion costs and quantiles). Moreover, we establish properties for risk measures in a DO context. We also discuss how to implement these measures in the Julia modeling package InfiniteOpt.jl which we demonstrate via a case study in optimal pandemic isolation policy design.

References:

[1] Pulsipher, Joshua L., Weiqi Zhang, Tyler J. Hongisto, and Victor M. Zavala. "A unifying modeling abstraction for infinite-dimensional optimization." Computers & Chemical Engineering 156 (2022): 107567.

[2] Lewis, Frank L., Draguna Vrabie, and Vassilis L. Syrmos. “Optimal control.” John Wiley & Sons (2012).

[3] Rawlings, James Blake, David Q. Mayne, and Moritz Diehl. “Model predictive control: theory, computation, and design.” Vol. 2. Madison: Nob Hill Publishing (2017).

[4] Shin, Sungho, Ophelia S. Venturelli, and Victor M. Zavala. "Scalable nonlinear programming framework for parameter estimation in dynamic biological system models." PLoS computational biology 15, no. 3 (2019): e1006828.

[5] Bolza, Oskar. "Über den „Anormalen Fall” beim Lagrangeschen und Mayerschen Problem mit gemischten Bedingungen und variablen Endpunkten." Mathematische Annalen 74, no. 3 (1913): 430-446.

[6] Risbeck, Michael J., and James B. Rawlings. "Economic model predictive control for time-varying cost and peak demand charge optimization." IEEE Transactions on Automatic Control 65, no. 7 (2019): 2957-2968.

[7] Ruszczyński, Andrzej, and Alexander Shapiro. "Optimization of risk measures." Probabilistic and randomized methods for design under uncertainty, Springer, London, (2006): 119-157.

[8] Pulsipher, Joshua L., Benjamin R. Davidson, and Victor M. Zavala. "New Measures for Shaping Trajectories in Dynamic Optimization." arXiv preprint arXiv:2110.07041 (2021).