Data-Driven Optimization under Distributional Uncertainty

Transcription

1 Data-Driven Optimization under Distributional Uncertainty Postdoctoral Researcher Electrical and Systems Engineering University of Pennsylvania

2 A network of physical objects - Devices - Vehicles - Buildings Allows objects to be - Sensed and controlled - Remotely across the network Growing rapidly, by 2020: - 50 billion devices devices per person Internet of Things (IoT) 2

3 What IoT Brings: More Sensing (and More Data) TLC: Taxi and Limousine Commission [Source: nyc.gov] Total size of dataset: 267 GB 1.1 billion taxi and Uber trips ( ) Pick-up and drop-off dates/times, locations, distances... 3

4 What IoT Brings: More Control Smart Home Appliances Connected and Autonomous Vehicles Wireless Traffic Light Control Smart Buildings 4

5 Smart Cities: IoT + Decision Support City Infrastructure Sensor Data Actions Control & Optimization Algorithm 5

6 Investment in Smart Cities The Smart Cities Initiative from the White House (Sep 2015)... an infrastructure to continuously improve the collection, aggregation, and use of data to improve the life of their residents by harnessing the growing data revolution, low-cost sensors, and research collaborations, and doing so securely to protect safety and privacy. TerraSwarm 6

7 TerraSwarm: Swarm at The Edge of The Cloud TerraSwarm How should we make use of data? How should we send data? How should we collect data? 7

8 Research Interests Research Topics Stochastic Systems Theory Convex Optimization Control Theory Statistics Applications Energy Multi-Agent Systems Network Dynamics Transportation 8

9 Research Overview Data-Driven Optimization Power generation am 12pm 6pm [ACC13], [SIOPT15] [CDC15], [TASE16], [ICCPS17] Privacy Solutions for Cyber-Physical Systems Pricing for Ridesharing [Allerton14], [TAC16] [ACC17] 9

10 Research Overview Data-Driven Optimization Power generation am 12pm 6pm [ACC13], [SIOPT15] [CDC15], [TASE16], [ICCPS17] Privacy Solutions for Cyber-Physical Systems Pricing for Ridesharing [Allerton14], [TAC16] [ACC17] 10

11 Motivation: Wind Energy Integration conventional power plant Power generation Wind Energy Data 6am 12pm 6pm [Source]: AESO Control Action: Allocation of energy storage wind power + storage devices How can we make use of the wind power generation data to maximally utilize wind power? 11

12 Motivation: On-Demand Ridesharing in Cities - Pick-up and drop-off times - Pick-up/drop-off locations - Travel distances Cause: Mismatch between supply and demand Control Action: Redistribution of empty vehicles How can we make use of the trip data to reduce the average wait time for passengers? 12

13 Background: Stochastic Programming minimize x E d [f(x, )] f : Objective function x : Decision variable - Ridesharing: Redirection of empty vehicles - Wind power integration: Allocation of storage : Stochastic phenomenon - Ridesharing: Future passenger demand - Wind power integration: Wind power generation d: Probability distribution of Probability distribution that models the stochastic phenomenon 13

14 Distribution is Not Always Available We often do not have: Instead, we have: Power generation am 12pm 6pm Question: How should these samples be used in a computationally tractable way with performance guarantees? 14

15 Using Sampled Data: Previous Methods Sample average approximation minimize x 1 n nx f(x, i ) i=1 Robust optimization minimize x max 2 f(x, ) i Weak guarantee on performance Can be extremely conservative Distributional Information + Uncertainty? 15

16 Using Sampled Data: Distributional Uncertainty Distributional uncertainty - An ambiguity set in the space of probability distributions - No assumption on the type (continuous vs discrete, Gaussian, uniform,...) of distributions - Contains the true distribution with high probability - Informally: Uncertainty of uncertainty D d (true distribution) Decision making problem: Distributionally robust optimization minimize x max d2d E d [f(x, )] (vs. minimize E d [f(x, )] ) x - Strong worst-case guarantees - Subsumes conventional robust optimization 16

17 Distributional Uncertainty Method 1: Based on certain (pseudo)metric - KL divergence - Wasserstein metric (earth mover s distance) Metric ball centered at the empirical distribution n D( ) = d: M(d, d) b o apple The ball contains d with high probability M d bd D (true distribution) Advantage: Nonparametric characterization Disadvantage: Complexity of decision making against D grows quickly with the number of samples empirical distribution 17

18 Distributional Uncertainty (cont d) Method 2: Based on generalized moments (this talk) D = {d: E d [g( )] 0} Assume: g is easily bounded D Examples - Moments: E[ ] =ˆ, cov[ ] = b - Tail probability: P( ) apple d (true distribution) Classical concentration inequalities can be used to compute the probability that D contains d! P 1 nx Example (Hoeffding s inequality): i E + t n i=1 apple exp( 2nt 2 ) 18

19 Challenges and My Contribution minimize x max d2d E d [f(x, )] Challenge: Finding the worst-case distribution - Infinite-dimensional optimization problem - Not numerically tractable Previous work on special instances - [Scarf, 1958]: Analytical solution for a special case - [Bertsimas, Popescu, 2005]: Optimal probability inequalities - [Vandenberghe, Boyd, Comanor, 2007]: Optimal Chebyshev bounds - [Delage, Ye, 2010]: Piecewise affine functions My contribution - Formulate equivalent convex optimization problem (under certain conditions) - Tractable numerical solutions - Conditions apply to many resource allocation and scheduling problems 19

20 Main Result: Equivalent Convex Optimization Problem Theorem: There exists an equivalent convex optimization problem for computing the worst-case distribution if f( ) The objective f( ) = max k f is piecewise concave f (k) ( ) f (k) concave The constraint g is piecewise convex g( ) g( ) =min l g (l) ( ) g (l) convex, Molei Tao, Ufuk Topcu, Houman Owhadi, Richard M. Murray, Convex optimal uncertainty quantification, SIAM Journal on Optimization, 25(3), ,

21 Piecewise Concave Functions concave piecewise affine 0-1 indicator max k2k {at k + b k } I( a) 1 resource allocation/scheduling 0 failure rate 21

22 Piecewise Convex Functions linear convex 0-1 indicator I( a) 1 mean covariance & higher moments tail probability 0 22

23 The Convex Optimization Problem X maximize {p kl, kl } k,l subject to k,l p kl f (k) ( kl /p kl ) X p kl =1 k,l p kl 0, 8k, l X p kl g (l) ( kl /p kl ) apple 0 k,l For: f( ) = g( ) = max f (k) ( ) k2{1,2,,k} min l2{1,2,,l} g(l) ( ) The worst case is always attained by a discrete distribution Total number of Dirac masses in the distribution: K L 23

24 Storage Allocation for Power Grid power flow f ij storage x i wind power (stochastic) i Storage Allocation Problem optimal power flow min. x max d E d applemin Wind Energy Wasted(x,,f) f {z } piecewise concave in 24

25 Numerical Example: IEEE 14-Bus Test Case Network with 5 generators Time: one day, 3-hour interval Mean and covariance obtained from real wind generation data Power generation am 12pm 6pm [Source]: AESO 25

26 The Influence of Information Constraints 50 Support Information Only Expected cost Total storage Support + Mean + Covariance Exact distribution, Ufuk Topcu, Molei Tao, Houman Owhadi, Richard M. Murray, Convex optimal uncertainty quantification: Algorithms and a case study in energy storage placement for power grids study, American Control Conference,

27 On-Demand Ridesharing Dispatch Center Distribution of Customer Demand Predicted Demand Dispatch Command Customer Vehicle min. X 1:T TX [J D (X t )+J E (X t,r t )] t=1 Demand Vehicle Flows Cost of Rebalancing Wait Time 27

28 Robust vs. Non-Robust Robust optimization against demand uncertainty min. X 1:T max r 1:T 2 TX [J D (X t )+J E (X t,r t )] t=1 Total requests number Boxplot of total requests number during one hour Region ID Number of experiments Costs distribution of dispatch solutions 40 non robust solutions 30 robust solutions Cost range NYC Dataset: 4 years, 100 GB Robust solution: 35.5% reduction Fei Miao,, Shan Lin, George J. Pappas, Taxi dispatch under model uncertainties, IEEE Conference on Decision and Control,

29 Conventional vs. Distributionally Robust Distributionally robust formulation min. X 1:T max d2d E r d ( X T ) [J D (X t )+J E (X t,r t )] t=1 Average cost Average cost comparison of DRO and RO 10 x 104 SOC Box 9 NR DRO ϵ confidence level (robust #1) (robust #2) (non-robust) (distr. robust) Note: Confidence level not optimized for distr. robust opt. Confidence level: Probability the true parameter/distribution lies outside ambiguity set Fei Miao,, Abdeltawab Hendawi, et al., Data-driven distributionally robust vehicle balancing with dynamic region partition, ACM/IEEE Intl. Conf. on Cyber-Physical Systems,

30 Future Directions Large Datasets Structured Models Online Optimization Distributed computation Approximate algorithms Markov properties Prior knowledge When to discard old data When to re-learn 30

31 Summary Distributional uncertainty: A new approach to data-driven optimization A rigorous way to make use of sampled data - Probabilistic guarantees - Worst-case analysis/design: Often required for engineering applications Computationally efficient - Convex formulation available for a large class of problems - Examples: Resource allocation and scheduling Power generation am 12pm 6pm D d (true distribution) 31