(417d) Random Field Optimization

Random field theory is a branch of mathematics and statistics that provides theoretical and computational techniques for characterizing statistical maps [1]. A random field generalizes the notion of a multivariate (vector-valued) random variable (e.g., multivariate Gaussian variables) to that of random variables that live in infinite-dimensional domains (e.g., space and time) [2]. For instance, a time series is a multivariate random variable given by a collection of random variables defined at a discrete number of time points; a random field can be seen as the infinite-dimensional limit of the time series obtained by treating the time domain as continuous. In this talk, we observe that random field theory enables us to readily model a wide range of uncertainties that a system might exhibit over general problem domains. Random fields also provide a general modeling abstraction that captures stochastic processes, stochastic programs, and Gaussian processes and enables us to establish connections with areas such as spatio-temporal modeling, machine learning, topological data analysis, and Bayesian optimization [3, 4, 5, 6]. Example application areas include functional brain imaging [7], computer-generated imagery [8], weather forecasting [9], structural topological design [10], and robotic modeling [11]. However, random field theory has not been incorporated into optimization theory and it has shown limited applications within the process systems community.

Random field theory allow us to explore infinite-dimensional limits of traditional stochastic programming models. For instance, typical stochastic programs consider a multi-stage decision-making setting that involve a discrete-time stochastic process, decisions and decisions made at each time point [12,13]. Prominent application areas of this paradigm include model predictive control [14], process design [15], and flexibility analysis [16]. The probability space defined by the stochastic process is typically handled using a so-called scenario tree; unfortunately, the complexity of such trees increases exponentially with the number of time points [13]. This complexity can be mitigated by assuming stage-wise independence (i.e., it is I.I.D. over time) [17], but this assumption often does not hold for real-world systems. Furthermore, multi-stage problems are inherently designed for discrete time stages and do not readily generalize to decisions that need to be made in continuous time or over more general domains (e.g., space-time domains, as those found in optimization with stochastic PDEs). To address some of these challenges, we integrate modeling constructs of random field theory and stochastic programming, giving rise to a new class of problems that we call "random field optimization." The integration of these paradigms is readily motivated through lens of our unifying abstraction for infinite-dimensional optimization (e.g., problems that consider decision functions/policies that are indexed over continuous domains) [18]. We demonstrate that decision functions with uncertainty (e.g., random policies defined over space-time) can be treated as random fields. Moreover, we show that the correlations over general domains (e.g., space-time) can be modeled via random fields and directly incorporated into system optimization problems.

References:

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[14] Ranjeet Kumar, Michael J Wenzel, Mohammad N ElBsat, Michael J Risbeck, Kirk H Drees, and Victor M Zavala. Stochastic model predictive control for central HVAC plants. Journal of Process Control 90 (2020): 1-17.
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[17] Jikai Zou, Shabbir Ahmed, and Xu Andy Sun. Multistage stochastic unit commitment using stochastic dual dynamic integer programming. IEEE Transactions on Power Systems 34, no. 3 (2018): 1814-1823.
[18] Joshua L Pulsipher, Weiqi Zhang, Tyler J Hongisto, and Victor M Zavala. A unifying abstraction for modeling infinite-dimensional optimization problems. Under Review, 2021