Pareto Efficiency in Robust Optimization

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Transcription:

Pareto Efficiency in Robust Optimization Dan Iancu Graduate School of Business Stanford University joint work with Nikolaos Trichakis (HBS) 1/26

Classical Robust Optimization Typical linear optimization problem under uncertainty max xpr pt x n a T i x ď b i, @i P I max min xpr n upu ppuqt x a i puq T x ď b i puq, @u P U, @i P I For many classes of U, results in tractable formulations Soyster [1973], Ben-Tal and Nemirovski [1998, 2002], El-Ghaoui et al. [1998], Bertsimas and Sim [2003, 2004],... Framework successfully adopted in many applications However, approach suffers from several shortcomings... 2/26

Conservativeness Worst-case focus Ñ limited potential upside. Absolute or relative regret [Savage, 1972], soft-robustness [Ben-Tal et al., 2010], light-robustness [Fischetti and Monaci, 2009], bw-robustness [Roy, 2010, Gabrel et al., 2011], α-robustness [Kalai et al., 2012],... Typically same (or slightly decreased) modeling flexibility and same (or slightly increased) computational complexity Trade off robustness (worst-case performance) for potential upside 3/26

Multiplicity of solutions Classical RO framework can lead to multiple solutions Typically seen as a benefit [Bertsimas et al., 2010], [Iancu et al., 2012] 4/26

Multiplicity of solutions Classical RO framework can lead to multiple solutions Typically seen as a benefit [Bertsimas et al., 2010], [Iancu et al., 2012] Which solution to pick? Minimax/maximin criteria can lead to Pareto inefficiencies Idea recognized in other areas [Young, 1995], [Bertsimas et al., 2012], [Ogryczak, 1997], [Suh and Lee, 2001] Typical fix (lexicographic max-min fairness) only works for a finite set of scenarios 4/26

In this talk... Adapt and formalize the property of Pareto Efficiency in RO Illustrate that RO need not produce solutions with this property Provide basic characterization of Pareto solutions Extend RO framework to produce Pareto solutions, at essentially no extra computational cost Illustrate benefits in three popular applications 5/26

Setup max xpx min ppu pt x Feasible set of solutions X x P R n : Ax ď b ( Uncertainty set of objective coefficients U tp P R n : Dp ě du 6/26

Setup max xpx min ppu pt x Feasible set of solutions X x P R n : Ax ď b ( Uncertainty set of objective coefficients U tp P R n : Dp ě du Classical RO framework results in Optimal value z RO Set of robustly optimal solutions X RO x P X : Dy ě 0 such that D T y x, y T d ě z RO( 6/26

Set of robustly optimal solutions X RO x P X : Dy ě 0 such that D T y x, y T d ě z RO( x P X RO guarantees that no other solution exists with higher worst-case objective value p T x 7/26

Set of robustly optimal solutions X RO x P X : Dy ě 0 such that D T y x, y T d ě z RO( x P X RO guarantees that no other solution exists with higher worst-case objective value p T x What if an uncertainty scenario materializes that does not correspond to the worst-case? Are there any guarantees that no other solution exists that, apart from protecting us from worst-case scenarios, also performs better overall, under all possible uncertainty realizations? 7/26

Pareto Robustly Optimal solutions max xpx min ppu pt x (1) Definition A solution x is called a Pareto Robustly Optimal (PRO) solution for Problem (1) if (a) it is robustly optimal, i.e., x P X RO, and (b) there is no x P X such that p T x ě p T x, @p P U, and p T x ą p T x, for some p P U. 8/26

Pareto Robustly Optimal solutions max xpx min ppu pt x (1) Definition A solution x is called a Pareto Robustly Optimal (PRO) solution for Problem (1) if (a) it is robustly optimal, i.e., x P X RO, and (b) there is no x P X such that p T x ě p T x, @p P U, and p T x ą p T x, for some p P U. X PRO Ď X RO : set of all PRO solutions 8/26

Some questions Given a RO solution, is it also PRO? How can one find a PRO solution? Can we optimize over X PRO? Can we characterize X PRO? Is it non-empty? Is it convex? When is X PRO X RO? How does the notion generalize in other RO formulations? 9/26

Finding PRO solutions Theorem Given a solution x P X RO and an arbitrary point p P ripuq, consider the following linear optimization problem: Then, either maximize p T y subject to y P U x `y P X. the optimal value is zero and x P X PRO, or the optimal value is strictly positive and x x `y P X PRO, for any optimal solution y. U def ty P R n : y T p ě 0, @p P Uu is the dual of U 10/26

Remarks Finding a point p P ripuq can be done efficiently using LP techniques Testing whether x P X RO is no harder than solving the classical RO problem in this setting Finding a PRO solution x P X PRO is no harder than solving the classical RO problem in this setting 11/26

Corollaries If p P ripuq, all optimal solutions to the problem below are PRO: maximize p T x subject to x P X RO If 0 P ripuq, then X PRO X RO If p P ripuq, then X PRO X RO if and only if the optimal value of the LP below is zero: maximize p T y subject to x P X RO y P U x `y P X 12/26

Optimizing over / Understanding X PRO Secondary objective r: can we solve maximize r T x subject to x P X PRO? Interesting case: X RO X PRO 13/26

Optimizing over / Understanding X PRO Secondary objective r: can we solve Proposition X PRO is not necessarily convex. maximize r T x subject to x P X PRO? X tx P R 4` : x 1 ď 1, x 2 `x 3 ď 6, x 3 `x 4 ď 5, x 2 `x 4 ď 5u ei ( U conv, i P t1,...,4u z RO 1, and X RO tx P X : x ě 1u x 1 1 2 4 1 T, x 2 1 4 2 1 T P X PRO 0.5x 1 `0.5x 2 is Pareto dominated by 1 3 3 2 T P X RO. 13/26

Optimizing over / Understanding X PRO Secondary objective r: can we solve maximize r T x subject to x P X PRO? Proposition If X RO X PRO, then X PRO X ripx RO q H. Whether the solution to a nominal RO is PRO depends on algorithm used for solving LP Simplex vs. interior point 13/26

Optimizing over X PRO by MILP Proposition For any r P R n and any p P ripuq, let px,µ,η,z q be an optimal solution of the following MILP Then, x P argmax xpx PRO r T x. maximize r T x subject to x P X RO µ ď Mp1 zq b Ax ď Mz DA T µ dη ě D p µ ě 0, η ě 0, z binary. 14/26

Optimizing over X PRO by sampling Algorithm 1 Sampling heuristic for solving the problem max xpx PRO r T x 1: ˆXPRO Ð H 2: for i Ð 1,...,N do 3: Sample a point p P ripuq. 4: ˆXPRO Ð ˆX PRO Y argmax xpx RO p T x. 5: end for 6: Solve max xp ˆXPRO r T x. Theorem x P X PRO if and only if D p P ripuq such that argmax xpx RO p T x. 15/26

Generalizations Robust MILP Main results all readily extend! Testing, finding and optimizing over PRO solutions is as hard as solving the classical robust counterpart 16/26

Generalizations Robust MILP Main results all readily extend! Testing, finding and optimizing over PRO solutions is as hard as solving the classical robust counterpart Only change: when X RO X PRO, there may exist x P X PRO X ripx RO q.! X px 1,x 2,x 3 q P Z 2` 1 ˆR : 2 x 1 ` 1 ) 5 x 2 ď 1, x 3 ě 1, x 3 ď 0 U conv`te 1, e 2, e 3 u X RO x P X : x 3 0 ( X PRO r0 2 0s T, r5 0 0s T, r1 2 0s T ( x r1 2 0s T P ri`convpx RO q 16/26

Generalizations uncertainty in the constraints minimize c T x subject to Ax ě b, @A P U A, with U A Ă R mˆn a bounded polyhedral uncertainty set. 17/26

Generalizations uncertainty in the constraints minimize c T x subject to Ax ě b, @A P U A, with U A Ă R mˆn a bounded polyhedral uncertainty set. Vector of slacks spx,aq def Ax b, @x P R n, A P U A RO solutions X RO x P R n : c T x ď z RO, spx,aq ě 0, @A P U A ( RO solution guarantees that no other solution exists yielding nonnegative slacks for all A, and at a lower cost than z RO 17/26

PRO solutions under constraint uncertainty minimize c T x subject to Ax ě b, @A P U A, (3) To compare two slack vectors: slack value vector v quantifies the relative value of slack in each constraint E.g., if (3) comes from epigraph formulation, choose v e1 Definition A solution x is called a Pareto Robustly Optimal (PRO) solution for Problem (3) if x P X RO, and there is no other x P X RO such that v T sp x,aq ě v T spx,aq, @A P U A, and v T sp x,āq ą v T spx,āq, for some Ā P U A. All previous results directly apply 18/26

Numerical experiments 19/26

Numerical experiments Example (Portfolio) n `1 assets, with returns r i r i µ i `σ i ζ i, i 1,...,n, r n`1 µ n`1 ζ unknown, U tζ P R n : 1 ď ζ ď 1, 1 T ζ 0u Objective: select weights x to maximize worst-case portfolio return 19/26

Numerical experiments Example (Portfolio) n `1 assets, with returns r i r i µ i `σ i ζ i, i 1,...,n, r n`1 µ n`1 ζ unknown, U tζ P R n : 1 ď ζ ď 1, 1 T ζ 0u Objective: select weights x to maximize worst-case portfolio return Example (Inventory) One warehouse, N retailer where uncertain demand is realized Transportation, holding costs and profit margins differ for each retailer Demand driven by market factors d i d 0 i `qt i z, i 1,...,N Market factors z are uncertain z P U tz P R N : b 1 ď z ď b 1, B ď 1 T z ď Bu 19/26

Numerical experiments Example (Project management) A PERT diagram given by directed, acyclic graph G pn,eq N are project events, E are project activities / tasks 5 7 S b B a d c A e f C g F 2 3 S 4 6 8 F 20/26

Numerical experiments Example (Project management) A PERT diagram given by directed, acyclic graph G pn,eq N are project events, E are project activities / tasks Task e P E has uncertain duration τ e τ 0 e `δ e δ P δ P R E ` : δ ď b 1, 1T δ e ď B ( Task e P E can be expedited by allocating a budgeted resource x e τ e τ 0 e `δ e x e 1 T x ď C Goal: find resource allocation x to minimize worst-case completion time 20/26

Methodology Generate 10,000 random problem instances For every problem, find x P X RO using simplex method Test whether x P X PRO If not, find x P X PRO that Pareto dominates x Record ˆp T p x xq ˆp T x, where ˆp is nominal scenario Record maxppu p T p x xq p T x. 21/26

Results finance and inventory examples # of occurrences 600 400 200 0 0 10 20 30 40 50 600 400 200 0 10 20 30 40 50 # of occurrences 600 400 200 0 0 600 400 200 10 20 30 40 50 0 10 20 30 40 nominal gain (%) maximum gain (%) 50 Figure: TOP: portfolio example. BOTTOM: inventory example. LEFT: performance gains in nominal scenario. RIGHT: maximal performance gains. 22/26

Results project management example # of occurrences # of occurrences 200 150 100 50 50 0 10 20 30 40 0 10 20 30 40 400 (b) 400 (b) 300 200 100 (a) 200 (a) 150 100 300 200 100 0 10 20 30 40 0 10 20 30 40 nominal performance gain (in %) maximum performance gain (in %) Figure: TOP: configuration (a). BOTTOM: configuration (b). LEFT: performance gains in nominal scenario. RIGHT: maximal performance gains 23/26

Conclusions Adapted the well known concept of Pareto efficiency in the context of the RO methodology By focusing exclusively on worst-case outcomes, the classical RO paradigm can lead to inefficiencies and sub-optimal performance in practice Extended the RO framework via practical methods that verify Pareto optimality, and generate PRO solutions Preserved complexity of the underlying robust problems ñ PRO solutions have a significant upside, at no extra cost or downside D. Iancu, N. Trichakis. Pareto Efficiency in Robust Optimization, 2012 - Optimization Online. 24/26

References I A. Ben-Tal and A. Nemirovski. Robust convex optimization. Mathematics of Operations Research, 23(4):769 805, 1998. A. Ben-Tal and A. Nemirovski. Robust optimization - methodology and applications. Mathematical Programming, 92(3): 453 480, 2002. A. Ben-Tal, D. Bertsimas, and D. B. Brown. A soft robust model for optimization under ambiguity. Operations Research, 58: 1220 1234, 2010. D. Bertsimas and M. Sim. Robust discrete optimization and network flows. Mathematical Programming, 98(1-3):49 71, 2003. D. Bertsimas and M. Sim. The price of robustness. Operations Research, 52(1):35 53, 2004. D. Bertsimas, D. A. Iancu, and P. A. Parrilo. Optimality of affine policies in multistage robust optimization. Mathematics of Operations Research, 35(2):363 394, 2010. D. Bertsimas, V. F. Farias, and N. Trichakis. On the efficiency-fairness tradeoff. Management Science, 2012. To appear. L. El-Ghaoui, F. Oustry, and H. Lebret. Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9 (1):33 52, 1998. M. Fischetti and M. Monaci. Light robustness. In R. Ahuja, R. Möhring, and C. Zaroliagis, editors, Robust and Online Large-Scale Optimization, volume 5868 of Lecture Notes in Computer Science, pages 61 84. Springer Berlin / Heidelberg, 2009. V. Gabrel, C. Murat, and L. Wu. New models for the robust shortest path problem: Complexity, resolution and generalization. Annals of Operations Research, pages 1 24, 2011. D. A. Iancu, M. Sharma, and M. Sviridenko. Supermodularity and affine policies in dynamic robust optimization. Submitted for publication, 2012. R. Kalai, C. Lamboray, and D. Vanderpooten. Lexicographic α-robustness: An alternative to min max criteria. European Journal of Operational Research, 220(3):722 728, 2012. W. Ogryczak. On the lexicographic minimax approach to location problems. European Journal of Operational Research, 100(3): 566 585, 1997. B. Roy. Robustness in operational research and decision aiding: A multi-faceted issue. European Journal of Operational Research, 200(3):629 638, 2010. 25/26

References II L. J. Savage. The Foundations of Statistics. Dover Publications, Inc., second edition, 1972. A. L. Soyster. Technical Note Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming. Operations Research, 21(5):1154 1157, 1973. M.-h. Suh and T.-y. Lee. Robust optimization method for the economic term in chemical process design and planning. Industrial & Engineering Chemistry Research, 40(25):5950 5959, 2001. H. P. Young. Equity: In Theory and Practice. Princeton University Press, 1995. 26/26

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