Distributionally Robust Optimization with ROME (part 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 2009 1 / 58

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 2009 2 / 58

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 2009 3 / 58

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 2009 4 / 58

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 2009 5 / 58

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 2009 6 / 58

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 2009 7 / 58

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 2009 8 / 58

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 2009 9 / 58

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 2009 10 / 58

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 2009 11 / 58

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 2009 12 / 58

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 2009 13 / 58

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 2009 14 / 58

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 2009 15 / 58

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 2009 16 / 58

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 2009 17 / 58

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 2009 18 / 58

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 2009 19 / 58

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 2009 20 / 58

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 2009 21 / 58

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 2009 22 / 58

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 2009 23 / 58

General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 24 / 58

General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 25 / 58

General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 26 / 58

General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 27 / 58

General Segregations (Pictorial) J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 28 / 58

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 2009 29 / 58

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 2009 30 / 58

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 2009 31 / 58

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 2009 32 / 58

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 2009 33 / 58

( ( ) ) + 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 2009 34 / 58

(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 2 2 0 + r ˆζ) 2 + y Σy J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 35 / 58

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 2009 36 / 58

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 2009 37 / 58

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 2009 38 / 58

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 2009 39 / 58

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 2009 40 / 58

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 2009 41 / 58

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 2009 42 / 58

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 0 1 + yz ) + ŷ(z) = ( y 0 + yz ) + ( y 0 1 + 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 2 2 2 + 1 ( y 0 1 y 2 2 y s.t. u 0 v 0 = y 0 u v = y 1 ) 2 1 2 Objective = 0.707 (better!) Aim to find algorithm to automatically predict this decision rule J. Goh, M. Sim (NUS) DRO with ROME (part 1) 18 Jun 2009 43 / 58

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 2009 44 / 58

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 2009 45 / 58

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 2009 46 / 58

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 2009 47 / 58

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 2009 48 / 58

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 2009 49 / 58

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 2009 50 / 58

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 2009 51 / 58

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 2009 52 / 58

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 2009 53 / 58

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 2009 54 / 58

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 2009 55 / 58

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 2009 56 / 58

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 2009 57 / 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 2009 58 / 58