Data-Driven Optimization under Distributional Uncertainty Shuo Han Postdoctoral Researcher Electrical and Systems Engineering University of Pennsylvania
Internet of Things (IoT) • 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 - 6.58 devices per person UTC-IASE, Apr 2017 Shuo Han 2
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 (2009 - 2015) • Pick-up and drop-off dates/times, locations, distances... UTC-IASE, Apr 2017 Shuo Han 3
What IoT Brings: More Control Smart Home Appliances Connected and Autonomous Vehicles Wireless Traffic Light Control Smart Buildings UTC-IASE, Apr 2017 Shuo Han 4
Smart Cities: IoT + Decision Support City Infrastructure Sensor Data Actions Control & Optimization Algorithm UTC-IASE, Apr 2017 Shuo Han 5
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 UTC-IASE, Apr 2017 Shuo Han 6
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?” UTC-IASE, Apr 2017 Shuo Han 7
Research Interests Research Topics Applications Theory Convex Optimization Control Theory Statistics Stochastic Systems Energy Multi-Agent Network Transportation Systems Dynamics UTC-IASE, Apr 2017 Shuo Han 8
Research Overview Data-Driven Optimization 150 Power generation 100 50 0 6am 12pm 6pm [ACC13], [SIOPT15] [CDC15], [TASE16], [ICCPS17] Privacy Solutions for Cyber-Physical Pricing for Ridesharing Systems [Allerton14], [TAC16] [ACC17] UTC-IASE, Apr 2017 Shuo Han 9
Research Overview Data-Driven Optimization 150 Power generation 100 50 0 6am 12pm 6pm [ACC13], [SIOPT15] [CDC15], [TASE16], [ICCPS17] Privacy Solutions for Cyber-Physical Pricing for Ridesharing Systems [Allerton14], [TAC16] [ACC17] UTC-IASE, Apr 2017 Shuo Han 10
Motivation: Wind Energy Integration Wind Energy Data conventional 150 Power generation power plant 100 50 0 6am 12pm 6pm [Source]: AESO Control Action: Allocation of energy storage How can we make use of the wind + power generation data to maximally utilize wind power? storage devices wind power UTC-IASE, Apr 2017 Shuo Han 11
Motivation: On-Demand Ridesharing in Cities - Pick-up and drop-off times - Pick-up/drop-off locations - Travel distances Control Action: Redistribution of Cause: Mismatch between empty vehicles supply and demand How can we make use of the trip data to reduce the average wait time for passengers? UTC-IASE, Apr 2017 Shuo Han 12
Background: Stochastic Programming minimize E θ ∼ d [ f ( x, θ )] x Probability distribution that models the stochastic phenomenon • : Objective function f • : Decision variable x - Ridesharing: Redirection of empty vehicles - Wind power integration: Allocation of storage • : Stochastic phenomenon θ - Ridesharing: Future passenger demand - Wind power integration: Wind power generation • : Probability distribution of θ d UTC-IASE, Apr 2017 Shuo Han 13
Distribution is Not Always Available We often do not have: Instead, we have: 150 Power generation 100 50 0 6am 12pm 6pm Question : How should these samples be used in a computationally tractable way with performance guarantees? UTC-IASE, Apr 2017 Shuo Han 14
Using Sampled Data: Previous Methods Sample average approximation Robust optimization n 1 X minimize f ( x, θ i ) minimize max θ ∈ Θ f ( x, θ ) n x x i =1 Θ θ i • Weak guarantee on performance • Can be extremely conservative Distributional Information + Uncertainty ? UTC-IASE, Apr 2017 Shuo Han 15
Using Sampled Data: Distributional Uncertainty • Distributional uncertainty - An ambiguity set in the space of probability distributions - No assumption on the type (continuous vs discrete, D Gaussian, uniform, ...) of distributions - Contains the true distribution with high probability d (true distribution) - Informally: “Uncertainty of uncertainty” • Decision making problem: Distributionally robust optimization (vs. ) minimize max d ∈ D E θ ∼ d [ f ( x, θ )] minimize E θ ∼ d [ f ( x, θ )] x x - Strong worst-case guarantees - Subsumes conventional robust optimization UTC-IASE, Apr 2017 Shuo Han 16
Distributional Uncertainty • Method 1: Based on certain (pseudo)metric M - KL divergence - Wasserstein metric (earth mover’s distance) D • Metric ball centered at the empirical distribution d (true distribution) n o d : M ( d, b D ( ✏ ) = d ) ≤ ✏ b d • The ball contains with high probability d • Advantage: “Nonparametric” characterization empirical distribution • Disadvantage: Complexity of decision making against grows quickly with the number of D samples UTC-IASE, Apr 2017 Shuo Han 17
Distributional Uncertainty (cont’d) • Method 2: Based on generalized moments (this talk) D = { d : E θ ∼ d [ g ( θ )] � 0 } • Assume: is easily bounded g D • Examples d E [ θ ] = ˆ θ cov[ θ ] = b - Moments: , (true distribution) Σ P ( ✓ ≥ ¯ - Tail probability: ✓ ) ≤ ✏ • Classical concentration inequalities can be used to compute the probability that contains D d ! n 1 X Example (Hoeffding’s inequality): ≤ exp( − 2 nt 2 ) θ i ≥ E θ + t P n i =1 UTC-IASE, Apr 2017 Shuo Han 18
Challenges and My Contribution minimize max d ∈ D E θ ∼ d [ f ( x, θ )] 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 UTC-IASE, Apr 2017 Shuo Han 19
Main Result: Equivalent Convex Optimization Problem Theorem: There exists an equivalent convex optimization problem for computing the f ( θ ) worst-case distribution if • The objective is piecewise concave f f ( k ) f ( k ) ( θ ) concave f ( θ ) = max k • The constraint is piecewise convex g ( θ ) g g ( l ) g ( l ) ( θ ) g ( θ ) = min convex l Shuo Han, Molei Tao, Ufuk Topcu, Houman Owhadi, Richard M. Murray, “Convex optimal uncertainty quantification,” SIAM Journal on Optimization, 25(3), 1368–1387, 2015. UTC-IASE, Apr 2017 Shuo Han 20
Piecewise Concave Functions concave piecewise affine 0-1 indicator k ∈ K { a T k θ + b k } max I ( θ ≥ a ) 1 0 resource allocatio llocation/scheduling failure rate UTC-IASE, Apr 2017 Shuo Han 21
Piecewise Convex Functions linear convex 0-1 indicator I ( θ ≥ a ) 1 0 mean tail probability covariance & higher moments UTC-IASE, Apr 2017 Shuo Han 22
The Convex Optimization Problem X p kl f ( k ) ( γ kl /p kl ) For: maximize { p kl , γ kl } k,l k,l k ∈ { 1 , 2 , ··· ,K } f ( k ) ( θ ) f ( θ ) = max X subject to p kl = 1 k,l l ∈{ 1 , 2 , ··· ,L } g ( l ) ( θ ) g ( θ ) = min p kl ≥ 0 , ∀ k, l X p kl g ( l ) ( γ kl /p kl ) ≤ 0 k,l • The worst case is always attained by a discrete distribution • Total number of Dirac masses in the distribution: K · L UTC-IASE, Apr 2017 Shuo Han 23
Storage Allocation for Power Grid power flow f ij wind power θ i (stochastic) storage x i Storage Allocation Problem optimal power flow � min. max min Wind Energy Wasted ( x, θ , f ) E θ ∼ d x d f | {z } piecewise concave in θ UTC-IASE, Apr 2017 Shuo Han 24
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 150 Power generation 100 50 0 6am 12pm 6pm [Source]: AESO UTC-IASE, Apr 2017 Shuo Han 25
The Influence of Information Constraints Support Information Only 50 40 Expected cost 30 Support + Mean 20 + Covariance 10 Exact distribution 0 0 5 10 15 Total storage Shuo Han, 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, 2013. UTC-IASE, Apr 2017 Shuo Han 26
On-Demand Ridesharing Distribution of Customer Demand Dispatch Center Predicted Dispatch Demand Command Customer Vehicle Demand T X min. [ J D ( X t ) + J E ( X t , r t )] X 1: T t =1 Vehicle Flows Cost of Rebalancing Wait Time UTC-IASE, Apr 2017 Shuo Han 27
Recommend
More recommend
