Distributionally Robust Optimization with ROME (part 1)

Transcription

1 Distributionally Robust Optimization with ROME (part 1) Joel Goh Melvyn Sim Department of Decision Sciences NUS Business School, Singapore 18 Jun 2009 NUS Business School Guest Lecture J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

2 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

3 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

4 Classical Robust Optimization Robust optimization (RO) has been traditionally used to immunize an uncertain optimization problem from infeasibility. Classical approach: define a support set for the uncertainties, and use duality arguments to construct the robust counterpart (RC). RC: a deterministic equivalent of the uncertain optimization problem that ensures feasibility with probability 1. E.g. Soyster (1973), Ben-Tal and Nemirovski (1998), Bertsimas and Sim (2004), El Ghaoui, Oustry, and Lebret (1998). J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

5 Affinely Adjustable Robust Counterpart (AARC) Ben-Tal, Goryashko, Guslitzer, and Nemirovski (2004) introduced Affinely-Adjustable RC (AARC), to allow delayed decision-making. (aka LDR) Decisions are affine functions of uncertainties. Can model two-stage and multi-stage problems. Advantage: Tractable approximation of many stochastic programs, and is distribution-free. Disadvantage: Can be too conservative for some problems. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

6 Using Moments for RO A different perspective on RO: Moment Information Goes back to Scarf (1958) Distribution-free newsvendor with known mean and variance. El Ghaoui, Oks, and Oustry (2004): Use mean and covariance to compute WVaR for portfolio of assets. Popescu (2007) Use mean-covariance information to robustly optimize expected utility. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

7 DRO Combines These Ideas Distributionally Robust Optimization (DRO) is at the intersection of these research threads. Existing work: Chen, Sim, Sun, and Zhang (2008) LDR extensions Chen and Sim (2009) Goal driven optimization Natarajan, Sim, and Uichanco (2009) Expected utility J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

8 Design Goals of DRO To design a unified theoretical framework for DRO To build a software modeling tool to solve DRO problems within this framework Key ingredients: Multi-stage decisions, but more flexible compared to LDRs. Distribution-free, but partially characterized by support, moments, and directional deviations (Chen, Sim, and Sun, 2007). J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

9 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

10 Notation z R N : uncertainties with distribution P F. x R n : here-and-now decision variables. y k ( z) R m k : wait-and-see decision rules, k [K]. Each decision rule need not depend on the full uncertainty vector. We introduce K information index sets, to capture the dependency structure, denoted by I k [N], k [K]. y k Y(m k, N, I k ) k [K] { ( Y(m, N, I) f : R N R m : f z + ) } λ i e i = f(z), λ R N i/ I I [N] {1, 2,..., N} J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

11 Information Index Set Example Suppose you have a Decision Rule, y 1 ( z) which has information index set I 1, i.e. y 1 Y(m, N, I 1 ). The information index set contains two elements, I 1 = {2, 3}. Then, the Decision Rule can be equivalently expressed as y 1 ( z) = y 1 ( z 2, z 3 ). J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

12 Information Index Sets Allows us to handle more general non-anticipative scenarios than multi-stage problems. For a multistage problem with sequential decisions, we have I 1 I 2... I K Reflects the requirement that we only can make use of uncertainties in the past. Other set inclusions (especially occuring in network problems) lead to more general non-anticipative constraints. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

13 Model of Uncertainty Actual uncertainty distribution P lies in a family of distributions F Partially characterized by distributional properties: Support W: Tractable conic representable, full-dim. Mean Support Ŵ: Tractable conic representable. Covariance matrix Σ: Positive definite Upper bounds on Directional Deviations for stochastically independent components (H σ, σ f, σ b). The original uncertainties may not have independent components, but we allow the possibility that a linear transformation of the uncertainties by H σ may have independent components. i.e. w = H σ z might have independent components. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

14 General Model Z GEN = min x,{y k ( )} K k=1 s.t. c 0 x + sup P F c l x + sup P F T ( z)x + ( K ) E P d 0,k y k ( z) ( k=1 K ) E P d l,k y k ( z) b l k=1 K U k y k ( z) = v( z) k=1 y k y k ( z) y k x 0 y k Y(m k, N, I k ) l [M] k [K] k [K] Convention: All (in)equalities involving uncertainties are taken to hold with probability 1 over all distributions P F. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

15 General Model (data in red) Z GEN = min x,{y k ( )} K k=1 s.t. c 0 x + sup P F c l x + sup P F T ( z)x + ( K ) E P d 0,k y k ( z) ( k=1 K ) E P d l,k y k ( z) b l k=1 K U k y k ( z) = v( z) k=1 y k y k ( z) y k x 0 y k Y(m k, N, I k ) l [M] k [K] k [K] We assume T ( z), v( z), are affine functions of z, i.e. T ( z) = T 0 + N i=1 T i z i, v( z) = v 0 + N i=1 vi z i J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

16 Example: Piecewise-linear terms Common to find E P (( ) +) in terms in optimization problems in the OM setting. E.g. Newsvendor / inventory models. A distributionally robust version can be cast in our framework too: sup P F ( sup E P y( z) + ) b P F y Y(m, N, I) E P (s( z)) b s( z) 0 s( z) y( z) s, y Y (m, N, I) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

17 Using Linear Decision Rules (LDRs) General model is intractable. Intuitively, the space of all functions is too big. Following Ben-Tal et. al. (2004), we restrict ourselves to LDRs, where the recourse decisions are affinely dependent on the model uncertainties. Define the space of LDRs { L(m, N, I) = f : R N R m : ( y 0, Y ) R m R m N : f(z) = y 0 + Y z Y e i = 0, i / I Use this approximate Y(m, N, I) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

18 LDR Approximation of General Model Z LDR = min x,{y k ( )} K k=1 s.t. c 0 x + sup P F c l x + sup P F T ( z)x + ( K ) E P d 0,k y k ( z) ( k=1 K ) E P d l,k y k ( z) b l k=1 K U k y k ( z) = v( z) k=1 y k y k ( z) y k x 0 y k L(m k, N, I k ) l [M] k [K] k [K] J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

19 LDR Approx. of General Model (Explicit) Z LDR = ( K K ) min c 0 x + d 0,k y 0,k + sup d 0,k Y k ẑ x,{y 0,k,Y k } K k=1 s.t. c l x + T 0 x + T j x + k=1 ẑ Ŵ ( k=1 K K ) d l,k y 0,k + sup d l,k Y k ẑ b l k=1 K U k y 0,k = v 0 k=1 K U k Y k e j = v j k=1 ẑ Ŵ k=1 l [M] j [N] y k y 0,k + Y k z y k z W k [K] Y k e j = 0 j / I k, k x 0 J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

20 Is LDR Approximation too Conservative? LDR approximation of General Model is tractable, but is it too conservative? will proceed by using the basic LDR approximation as a starting point, and aim to find other more complex decision rules which are better. Z GEN?????? Z LDR J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

21 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

22 Segregating the Uncertainties Idea: to re-map the uncertainties into a higher-dimensional space, and apply an LDR on the new uncertainties, for more flexibility. E.g. for a scalar uncertainty, we can segregate into positive and negative parts (as was done by Chen, Sim, Sun, and Zhang, 2008). z = }{{} z + }{{} z ζ 1 ζ 2 Can segregate more generally into distinct segments of the real line. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

23 General Segregations Consider L + 1 partition points on the extended real line Ordered, denoted by {ξ k } L+1 k=1, ξ 1 =, ξ L+1 = +. Segregating mapping function: M : R R L, ζ = M(z) z if ξ k z ξ k+1 ζ k = ξ k if z ξ k ξ k+1 if z ξ k+1 Define image of supports under the mapping: V {{M(z) : z W} } ˆV M(ẑ) : ẑ Ŵ J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

24 General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

25 General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

26 General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

27 General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

28 General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

29 Segregations are Affinely-infertible L ζ i = z + i=1 L ξ i z = i=2 In general, for vector uncertainties, we have: L ζ i i=1 L i=2 z = F M(z) + g z R N F = [ I I... I ] g = constant ξ i We can now define a new set of LDRs on the (higher-dimensional) segregated uncertainties, i.e. y k ( z) = r k (M( z)) = r 0,k + R k M( z), k [K] J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

30 General Model under SLDR (Exact) Z SLDR = c 0 K ( K ) x + d 0,k r 0,k + sup d 0,k R k ˆζ k=1 ˆζ ˆV k=1 ( K K ) s.t. c l x + d l,k r 0,k + sup d l,k R k ˆζ b l l [M] k=1 ˆζ ( ˆV k=1 K ) T (F ζ + g)x + U k r 0,k + U k R k ζ = v(f ζ + g) ζ V min x,{r k ( )} K k=1 k=1 y k r 0,k + R k ζ y k ζ V, k [K] x 0 r k M Y(m k, N, I k ) k [K] J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

31 General Model Under SLDR (Approximated) Z SLDR = min x,{r 0,k,R k } K k=1 c 0 x + ( K K ) d 0,k r 0,k + sup d 0,k R k ˆζ ˆζ ˆV k=1 k=1 ( K K ) s.t. c l x + d l,k r 0,k + sup d l,k R k ˆζ b l k=1 ˆζ ˆV k=1 K T 0 x + U k r 0,k = ν 0 T j x + k=1 K U k R k e j = ν j j [N E] k=1 l [M] y k r 0,k + R k ζ y k ζ V k [K] R k e j = 0 j / Φ k, k [K] x 0 J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

32 Properties of SLDR Theorem For the general problem, Z SLDR Z LDR Increases the size of the problem (and computational effort needed), but retains the linear structure. Can we do better? Z GEN??? Z SLDR Z LDR Will base further improvement on the SLDR. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

33 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

34 ( ( ) ) + Robust Bounds on sup E P r 0 + r ζ P F We will need this later for the BDLDR Different pairs of distributional properties lead to different upper bounds. Mean + Support Mean + Covariance Mean + Directional Deviations Innermost argument: Scalar-valued SLDR J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

35 (Mean + Support) & (Mean + Covariance) ( ( ) ) + To bound sup E P r 0 + r ζ P F F characterized by ˆV and V (Mean, Support) π 1 ( r 0, r ) ( { } s ˆζ inf s R N E sup ˆζ ˆV ( { + sup max r 0 + r ζ s ζ, s ζ })) ζ V F characterized by ˆV and Σ (Mean, Covariance) { π 2 ( r 0, r ) inf y {y:f y=r} sup ˆζ ˆV { 1 ( ) r 0 + r ˆζ + 1 } } (r r ˆζ) 2 + y Σy J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

36 Mean + Directional Deviation F characterized by ˆV and (H σ, σ f, σ b ) (Mean, Directional Devations) π 3 ( r 0, r ) inf s 0,s,x 0,x x 0 +x g σ =r 0 F σ x=r (r 0 s 0 s g σ ) + sup(r s F σ )ˆζ ˆζ ˆV + ψ(s 0 x 0, s x) + ψ(s 0, s) { ( )} 2 Where ψ(x 0 λ, x) = inf λ>0 e exp 1 λ sup { x 0 + x } u ẑ σ + 2 ẑ 2λ 2 σ Ŵσ u j = max {x j σ f,j, x j σ b,j } F σ = H σ F g σ = H σ g J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

37 F characterized by all of these properties Use infimal convolution to combine bounds Resulting bound is better than individual bounds But not tight π ( r 0, r ) = min π 1 ( r 0,1, r 1) + π 2 ( r 0,2, r 2) + π 3 ( r 0,3, r 3) s.t. r 0 = r 0,1 + r 0,2 + r 0,3 r = r 1 + r 2 + r 3 J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

38 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

39 Review of Deflected LDR (DLDR) Originally introduced by Chen, Sim, Sun, and Zhang (2008) to circumvent restrictiveness imposed by the LDR. The structure of the set of constraints might allow some piecewise-linear decision rules to be used instead. Apply bounds on expected positive part of an LDR to exploit piecewise-linearity. Bi-deflected LDR seeks to improve and extend the DLDR. Improve: Reduce the objective even further. Extend: Explore modeling scenarios (non-anticipativity, expectations in constraints) which the DLDR didn t handle. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

40 Motivating Example Scalar uncertainty z with unknown distribution P in a family F. F defined by infinite support, zero mean, and unit variance. Consider the following problem: min y( ) sup E P ( y( z) z ) P F 0 y( z) 1 y Y (1, 1, {1}) Solution is pretty straightforward: piecewise-linear recourse function: z if 0 z 1 y sol ( z) = 0 if z 0 1 if z 1 J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

41 Example Reformulated into DRO Framework Apply the identities x = x + x, x = x + + x. Model reformulates into DRO framework: min y( ) min y( ),u( ),v( ) s.t. sup E P ( y( z) z ) P F 0 y( z) 1 y Y (1, 1, {1}) sup P F E P (u( z) + v( z)) u( z) v( z) = y( z) z 0 y( z) 1 u( z), v( z) 0 y, u, v Y (1, 1, {1}) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

42 Using Different Decision Rules Using LDR, problem is infeasible, objective = +. Using DLDR (of Chen et. al. 2008), we get the piecewise-linear decision rule: û D (z) = ( u 0 + uz ) + + ( v 0 + vz ) ˆv D (z) = ( v 0 + vz ) + + ( u 0 + uz ) y(z) = y 0 + yz After applying bounds, becomes an SOCP ( ) ( Z DLDR = min u 0 + v 0 u 0,u,v 0,v u v 2 s.t. u v = 1 0 u 0 v 0 1 ) 2 Objective = 1. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

43 Different Decision Rules (II) Consider this hypothetical decision rule, which satisfies the model constraints: û(z) = ( u 0 + uz ) + + ( v 0 + vz ) + ( y 0 + yz ) ˆv(z) = ( v 0 + vz ) + + ( u 0 + uz ) + ( y yz ) + ŷ(z) = ( y 0 + yz ) + ( y yz ) + Applying robust bounds, problem transforms into the SOCP: ( ) ( ) min u 0 y 0,y,u 0,u,v 0 + v 0,v u + 1 ( ) y 0 v ( y 0 1 y 2 2 y s.t. u 0 v 0 = y 0 u v = y 1 ) Objective = (better!) Aim to find algorithm to automatically predict this decision rule J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

44 Two-stage Bi-deflected LDR Problem Setup Consider a simpler two-stage model first: min x,r( ) s.t. ) c x + sup E P (d r( ζ) P F T ( ζ)x + Ur( ζ) = ν( ζ) y r( ζ) y r L(m, N E, [N E ]) Define the index sets of non-infinite bounds: { } J = i [m] : y i > J = {i [m] : y i < + } J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

45 Two-stage BDLDR Sub-problems We solve a series of sub-problems: i J min p d p s.t. Up = 0 p i = 1 p j 0 j J p j 0 j J \ {i} i J min q d q s.t. Uq = 0 q i = 1 q j 0 j J q j 0 j J \ {i} Feasible i J Feasible i J J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

46 Two-stage BDLDR Consider an LDR which satisfies the relaxed constraints: T ( ζ)x + Ur( ζ) = ν( ζ) r j ( ζ) y j r j ( ζ) y j j J \ J j J \ J The two-stage BDLDR is defined as: ˆr( ζ) r( ζ) + ( ) r i ( ζ) y }{{} i i J Linear Part p i }{{} + ( r i ( ζ) y i ) + i J Optimal solution of the i th sub-problem (Deflected Part) q i }{{} J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

47 Two-stage BDLDR Properties Theorem The Bi-Deflected Linear Decision Rule, ˆr( ζ), satisfies the following properties 1. U ˆr( ζ) = Ur( ζ) 2. y ˆr( ζ) y For Reference: ˆr( ζ) r( ζ) + }{{} i J Linear Part ( r i ( ζ) y i ) p i }{{} + ( r i ( ζ) y i ) + i J Optimal solution of the i th sub-problem (Deflected Part) q i }{{} J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

48 BDLDR Approximation of Two-stage Problem Using the BDLDR, the problem becomes: Z BDLDR = min x,r( ) s.t. ) c x + sup E P (d r( ζ) P F + ( ( ) ) sup E P r i ( ζ) y i d p i i J P F R + sup P F i J R E P ( ( r i ( ζ) y i ) + ) d q i T ( ζ)x + Ur( ζ) = ν( ζ) r j ( ζ) y j r j ( ζ) y j r L(m, N E, [N E ]) j J \ J j J \ J J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

49 Performance of Two-stage BDLDR Theorem For the two-stage problem, Z BDLDR Z DLDR Z LDR. Proof is a bit tedious, but the basic idea is that the BDLDR includes the DLDR as a special case, and hence is more flexible and therefore yields a lower objective. Next, to generalize the BDLDR for use together with non-anticipativity requirements and expectation constraints. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

50 BDLDR for General Model Define the following sets k [K]: Similarly, we have two sub-problems: N + (k) = {j [K] : Φ k Φ j } N (k) = {j [K] : Φ j Φ k } k [K], i J k j N + (k) U j p i,k,j = 0 p i,k,j l 0 l J j j N + (k) p i,k,j l 0 l J j j N + (k) \ {k} p i,k,k l 0 l J k \ {i} p i,k,k i = 1 k [K], i J k j N + (k) U j q i,k,j = 0 q i,k,j l 0 l J j j N + (k) q i,k,j l 0 l J j j N + (k) \ {k} q i,k,k l 0 l J k \ {i} q i,k,k i = 1 J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

51 Constructing the BDLDR Consider an LDR which satisfies the relaxed constraints K T ( ζ)x + U k r k ( ζ) = ν( ζ) k=1 r k j ( ζ) y k j k [K], j J k \ J k rj k( ζ) y k j r k L(m k, N E, Φ k ) k [K], j J k \ J k k [K] The non-anticipative BDLDR is defined as: ˆr k ( ζ) r k ( ζ) + ( ) r j i }{{} ( ζ) y j i p i,j,k }{{} + ( ) r j i ( ζ) y j + i q i,j,k }{{} j N Linear Part (k) i J j i J j Optimal solution of i th sub-problem (Deflected Part) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

52 BDLDR Properties Theorem Each non-anticipative BDLDR, ˆr k ( ζ) satisfies the following properties: 1. K k=1 U kˆr k ( ζ) = K k=1 U k r k ( ζ) 2. y k ˆr k ( ζ) y k, k [K] For reference: ˆr k ( ζ) r k ( ζ) + }{{} Linear Part j N (k) i J j ( ) r j i ( ζ) y j i p i,j,k }{{} + ( r j i ( ζ) y j i i J j Optimal solution of i th sub-problem (Deflected Part) ) + q i,j,k }{{} J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

53 Explicit BDLDR Applied to General Problem Z BDLDR = min x,{r 0,k,R k } K k=1 c 0 x + K k=1 K + d 0,k r 0,k + sup ˆζ ˆV { K } d 0,k R k ˆζ k=1 ( π r 0,j i + y j i, Rj e i) d 0,k p i,j,k k=1 j N (k) i J 0,j,k K ( + π r 0,j i y j i, Rj e i) d 0,k q i,j,k k=1 j N (k) i J 0,j,k { K K } s.t. c l x + d l,k r 0,k + sup d l,k R k ˆζ k=1 ˆζ ˆV k=1 K ( + π r 0,j i + y j i, Rj e i) d l,k p i,j,k k=1 j N (k) i J l,j,k K ( + π r 0,j i y j i, Rj e i) d l,k q i,j,k b l l [M] k=1 j N (k) i J l,j,k J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

54 Explicit BDLDR Applied to General Problem (cont.) s.t. T 0 x + K U k r 0,k = ν 0 k=1 K T j x + U k R k e j = ν j k [K], j [N E ] k=1 r 0,k j + e j R k ζ y k j ζ V k [K], j J k \ J k r 0,k j + e j R k ζ y k j ζ V k [K], j J k \ J k x 0 J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

55 BDLDR Performance Theorem For the general problem, we have. Z GEN Z BDLDR Z SLDR Z LDR The k th BDLDR sums over j N (k), These are the indices which are contained by the current (k th ) information index set, BDLDR does not violate non-anticipative of the LDR which it is based upon. J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

56 Outline Introduction DRO Framework Segregated LDR ( Robust bounds on sup E P ( ) +) P F Bi-Deflected LDR Conclusion J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

57 Summary Presented a framework for using different decision rules for Distributionally Robust Optimization. LDRs: Tractable approximation Poor performance SLDRs: Improves performance Retains linear structure BDLDRs: Improves performance beyond SLDR Requires robust bounds Problem becomes complex J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58

58 ROME: Robust Optimization Made Easy Solves problems within our DRO framework Comprehensive modeling examples J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun / 58