Robust Optimization in AIMMS

Transcription

1 Robust Optimization in AIMMS Peter Nieuwesteeg Senior AIMMS specialist Online webinar July 20, 2010 Paragon Decision Technology Inc 5400 Carillon Point, Kirkland, WA, USA Internet: Tel: (425) Fax: (425)

2 Today s Roadmap Why Robust Optimization? Alternatives Basic Example Main Robust Optimization Concepts How to implement in AIMMS Robust Optimization Demo Advanced Features Adjustable variables Chance constraints Summary & Next Steps 2

3 Why Robust Optimization? Data Uncertainty! Do you have any of these sources of data uncertainty in your business? estimation errors: part of the data is measured/estimated; prediction errors: part of the data (e.g., future demands/prices) does not exist when problem is solved; implementation errors: some components of a solution cannot be implemented exactly as computed. Uncertainty is not considered in traditional (deterministic) optimization, but it can be handled by: Stochastic programming Robust optimization 3

4 Stochastic programming The stochastic programming approach: Generate scenarios to reflect possible realizations of uncertainty Assign a probability to these scenario s Create one optimization model that considers all these scenario s The down-side of this approach is: The number of scenarios grows rapidly The model becomes too large too quickly In many cases, it is hard to create the right scenarios In many cases, it is hard to come up with the right probabilities 4

5 Basic Example Max x + y st 3x + 4 y 12 x 0, y 0 (Optimal: x=4 and y = 0) What happens to the feasible space if 3 (C1) and 4 (C2) are not perfect, but can vary? For instance: C 1 2, C 2 2 C + C

6 Basic Example Original (C1,C2) = (3,4) Some: (C1,C2) = (2,5), (5,3), (3,5), (4,4) Most restrictive: (C1,C2) = (2,6), (6,2) Robust optimization will find the optimal solution for all combination of the (uncertain) coefficients (x = 1.5 & y = 1.5) 6

7 Robust Optimization Main Concepts uncertainty set U : the uncertain data belongs to certain ranges or regions, or depends on a certain (partially known) distribution Hard constraints : feasibility must be guaranteed Robust Optimization is the modeling framework of choice when the hard constraints must be satisfied for all data realizations within an uncertainty set U The robust optimal solution is the best solution against the worst possible data realization within the uncertainty set U. 7

8 Single Stage Linear RO Consider the linear program min c T x s.t. Ax b The constraints Ax b must be satisfied for all realizations of the data A within uncertainty set U The robust counterpart is (A. Ben-Tal and A. Nemirovski, 2000): min c T x s.t. Ax b A Є U 8

9 Uncertainty Set Main questions: How to specify a reasonable uncertainty set, i.e., meaningful for practical applications? When leads the uncertainty set to a computationally tractable (solvable) robust counterpart? 9

10 Computationally Tractable Box: L ij A ij U ij Robust counterpart: LP 10

11 Computationally Tractable Polyhedron: ij c ijm A ij δ m m = 1,, M Robust counterpart: LP 11

12 Computationally Tractable Ellipsoid: ij ((A ij c ij )/δ ij ) 2 1 Robust counterpart: Second Order Cone Problem (SOCP) 12

13 Computationally Tractable Convex Hull: Set of scenarios Convex Hull Scenarios Robust counterpart: LP 13

14 Refresh We have the robust counterpart: min c T x s.t. Ax b A Є U In order to formulate the problem, we need to define: The variable space: Traditional variables & constraints (Ax b) The parameter space: Uncertain variables & uncertainty constraint Based on this information, AIMMS is able to generate the robust counterpart and use existing solvers (XA, GUROBI, CPLEX & Mosek) to solve that model In close cooperation with Prof. Aharon Ben-Tal of the Technion Israel Institute of Technology 14

15 How to implement in AIMMS? Uncertainty Constraints 15

16 How to implement in AIMMS? Uncertainty attribute 16

17 How to implement in AIMMS? Box constraints Box: L ij A ij U ij 17

18 How to implement in AIMMS? Ellipsoid constraints Ellipsoid: ij ((A ij c ij )/δ ij )

19 How to implement in AIMMS? Convex Hull Convex Hull: Set of scenarios 19

20 Example Model Deterministic Version Goal of the example is to find the least cost investment in a chemical plant to reduce pollutants by at least given amount: 2 types of furnaces (Blast Furnaces & Open-Hearth furnaces) 3 types of pollutants (Particulates, Sulfur Oxides & Hydrocarbons) 3 type of modifications (Taller Smokestacks, Filters & Better Fuels) Each modification i has its own impact on pollutants and depends d on the furnace type Each modification has its own cost Each modification can be done partially Model and data taken from Introduction ti to Operations Research by Hillier & Lieberman 20

21 AIMMS Demo 1. Solve deterministic model 2. The improvement reductions are estimations. Introduce uncertainty to guarantee feasibility: -5 <= PR(ft,m,p) - PR(ft,m,p).level<= 5 3. Solve Robust Optimization Model This results in large increase in cost 4. Uncertainty (per pollutant) is not independent. Introducing an uncertainty constraint to limit the uncertainty: Sum((p), [PR(ft,m,p) - PR(ft,m,p).level]^2) <= Solve robust optimization model again This results in large increase in cost 21

22 DEMO Robust Optimization 22

23 Adjustable variables A non-adjustable variable reflects a here and now decision; they should get numerical values as a result of solving the problem before the actual data reveals itself; An adjustable variable reflects a decision made after (part of) the uncertain data has been revealed; Solution of adjustable variable depends on uncertain data. 23

24 Adjustable variable - Example Assume X(t) depends on Demand(s), s < t Linear decision rule: AIMMS naming: X(t) () = X 0 0( (t) + X s (t) () Demand(s) s<t X 0 (t) - X.adjustable.constant(t) Variables X s (t) - X.adjustable.Demand(t,s) 24

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26 Chance constraints The robust solution has to satisfy the constraint with probability at least 1- ε, where 0 ε 1, i.e. ε=0.05: Prob( a(ξ) T x b ) 95%; General characteristics of uncertain parameter ξ are needed; A chance constraint model is very difficult to solve, but there exists very good RO approximations; The more information about distribution is provided the better the approximation. 26

27 Distributions Bounded(m,s) mean m with support [m-s,m+s] Bounded(m,l,u) support [l,u] and mean m not in center Bounded(m_l,m_u,l,u) Bounded(m_l,m_u,l,u,v) Unimodal(c,s) Symmetric(c,s) Symmetric(c,s,v) Support(l,u) support [l,u] and mean in range [m_l,m_u] as above and variance bounded by v unimodal around c with support [c-s,c+s] symmetric around c with support [c-s,c+s] as above and variance bounded by v support [l,u] (and no info about the mean) Gaussian(m_l,m_u,v) Gaussian with mean in range [m_l,m_u], and variance bounded by v 27

28 Approximations Approximation Distribution Automatic Ball Box Ball-box Budgeted Bounded(m,s) linear conic linear conic linear Bounded(m,l,u) u) conic linear Unimodal(c,s) conic linear Symmetric(c,s) conic conic linear conic linear Support(l,u) linear linear Gaussian(l,u,v) conic 28

29 Chance constraint - AIMMS 29

30 Summary Features available in 3.11FR1: No need to reformulate the deterministic model Non-adjustable LP Mix of box, polyhedral and ellipsoidal uncertainty sets Uncertainty set defined by scenarios (Linear) adjustable LP Chance constraints Features available in future releases: MIP More decision rules for adjustable models 30

31 Pricing & Availability Pricing: $2,250 in 2010 (25% off regular price $3,000) Free academic use Availability: In 3.11 FR1 and higher only Commercial users with maintenance All academic users (need to Update license) Trial requestors Requirements: Any LP solver, for specific classes SOCP solver (CPLEX, MOSEK) 31

32 Additional information Try out RO: install AIMMS 3.11 Feature Release FR1 or higher Examples: Power System Expansion Portfolio Management (chance constraints) Article Dealing with uncertainty, downloadable from Contact us at 32