Distributionally Robust Optimization for Sequential Decision Making
Imperial College Business School, Imperial College London（帝国理工学院商学院）；Neuri Pte Ltd（Neuri私人有限公司）；Department of Industrial Systems Engineering and Management, National University of Singapore（新加坡国立大学工业系统工程与管理系）
The distributionally robust Markov Decision Process (MDP) approach asks for a distributionally robust policy that achieves the maximal expected total reward under the most adversarial distribution of uncertain parameters. In this paper, we study distributionally robust MDPs where ambiguity sets for the uncertain parameters are of a format that can easily incorporate in its description the uncertainty's generalized moment as well as statistical distance information. In this way, we generalize existing works on distributionally robust MDP with generalized-moment-based and statistical-distance-based ambiguity sets to incorporate information from the former class such as moments and dispersions to the latter class that critically depends on empirical observations of the uncertain parameters. We show that, under this format of ambiguity sets, the resulting distributionally robust MDP remains tractable under mild technical conditions. To be more specific, a distributionally robust policy can be constructed by solving a sequence of one-stage convex optimization subproblems.
Distributionally robust optimization (DRO) is a powerful tool for decision making under uncertainty. It is particularly appealing because of its ability to leverage existing data. However, many practical problems call for decision-making with some auxiliary information, and DRO in the context of conditional distribution is not straightforward. We propose a conditional kernel distributionally robust optimization (CKDRO) method that enables robust decision making under conditional distributions through kernel DRO and the conditional mean operator in the reproducing kernel Hilbert space (RKHS). I…
This paper considers a class of real-time decision making problems to minimize the expected value of a function that depends on a random variable $ξ$ under an unknown distribution $\mathbb{P}$. In this process, samples of $ξ$ are collected sequentially in real time, and the decisions are made, using the real-time data, to guarantee out-of-sample performance. We approach this problem in a distributionally robust optimization framework and propose a novel Online Data Assimilation Algorithm for this purpose. This algorithm guarantees the out-of-sample performance in high probability, and graduall…
The increasing share of renewables in the electricity generation mix comes along with an increasing uncertainty in power supply. In the recent years, distributionally robust optimization has gained significant interest due to its ability to make informed decisions under uncertainty, which are robust to misrepresentations of the distributional information (e.g., from probabilistic forecasts). This is achieved by introducing an ambiguity set that describes the potential deviations from an empirical distribution of all uncertain parameters. However, this set typically overlooks the inherent depen…
We consider the problem of look-ahead economic dispatch (LAED) with uncertain renewable energy generation. The goal of this problem is to minimize the cost of conventional energy generation subject to uncertain operational constraints. The risk of violating these constraints must be below a given threshold for a family of probability distributions with characteristics similar to observed past data or predictions. We present two data-driven approaches based on two novel mathematical reformulations of this distributionally robust decision problem. The first one is a tractable convex program in w…
Renewable energy generation is offered through electricity markets, quite some time in advance. This then leads to a problem of decision-making under uncertainty, which may be seen as a newsvendor problem. Contrarily to the conventional case for which underage and overage penalties are known, such penalties in the case of electricity markets are unknown, and difficult to estimate. In addition, one is actually only penalized for either overage or underage, not both. Consequently, we look at a slightly different form of a newsvendor problem, for a price-taker participant offering in electricity …
This paper focuses on solving a data-driven distributionally robust optimization problem over a network of agents. The agents aim to minimize the worst-case expected cost computed over a Wasserstein ambiguity set that is centered at the empirical distribution. The samples of the uncertainty are distributed across the agents. Our approach consists of reformulating the problem as a semi-infinite program and then designing a distributed algorithm that solves a generic semi-infinite problem that has the same information structure as the reformulated problem. In particular, the decision variables c…
Intensively studied in theory as a promising data-driven tool for decision-making under ambiguity, two-stage distributionally robust optimization (DRO) problems over Wasserstein balls are not necessarily easy to solve in practice. This is partly due to large sample size. In this article, we study a generic two-stage distributionally robust linear program (2-DRLP) over a 1-Wasserstein ball using an affine policy. The 2-DRLP has right-hand-side uncertainty with a rectangular support. Our main contribution is to show that the 2-DRLP problem has a tractable reformulation with a scale independent o…
The heating, ventilation and air condition (HVAC) system consumes the most energy in commercial buildings, consisting over 60% of total energy usage in the U.S. Flexible HVAC system setpoint scheduling could potentially save building energy costs. This paper first studies deterministic optimization, robust optimization, and stochastic optimization to minimize the daily operation cost with constraints of indoor air temperature comfort and mechanic operating requirement. Considering the uncertainties from ambient temperature, a Wasserstein metric-based distributionally robust optimization (DRO) …
We investigate the problem of risk averse robot path planning using the deep reinforcement learning and distributionally robust optimization perspectives. Our problem formulation involves modelling the robot as a stochastic linear dynamical system, assuming that a collection of process noise samples is available. We cast the risk averse motion planning problem as a Markov decision process and propose a continuous reward function design that explicitly takes into account the risk of collision with obstacles while encouraging the robot's motion towards the goal. We learn the risk-averse robot co…
Demand response (DR) programs aim to engage distributed demand-side resources in providing ancillary services for electric power systems. Previously, aggregated thermostatically controlled loads (TCLs) have been demonstrated as a technically viable and economically valuable provider of such services that can effectively compete with conventional generation resources in reducing load peaks and smoothing demand fluctuations. Yet, to provide these services at scale, a large number of TCLs must be accurately aggregated and operated in sync. This paper describes a Markov Decision Process (MDP) that…
On the Foundation of Distributionally Robust Reinforcement Learning
Department of Industrial and Systems Engineering, University of Southern California（工业与系统工程系，南加州大学）；Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology（工业工程与决策分析，香港科技大学）；Management Science and Engineering, Stanford University（管理科学与工程，斯坦福大学）；Stern School of Business, New York University（斯特恩商学院，纽约大学）
