Selected Topics in Robust Convex Optimization

Transcription

1 Selected Topics in Robust Convex Optimization Arkadi Nemirovski School of Industrial and Systems Engineering Georgia Institute of Technology Optimization programs with uncertain data and their Robust Counterparts Tractability of Robust Counterparts Robust Optimization and Chance Constraints

2 Optimization programs with uncertain data and their Robust Counterparts Tractability of Robust Counterparts Robust Optimization and Chance Constraints

3 Robust Optimization is a methodology for processing uncertain optimization problems x R n is the decision vector min x {f(x, ζ) : F(x, ζ) K} ζ R d is the data (or data perturbation) f(x, ζ) : R n R d R and F(x, ζ) : R n R d R m are given functions, and K R m is a given set. f(, ), F(, ),K form the structure of the uncertain problem. In contrast to Stochastic Programming, RO does not assume stochastic nature of data ζ and uses instead uncertainbut-bounded uncertainty model: ζ runs through a given (typically, compact) uncertainty set Z R d.

4 RO, people: 1973: A.L. Soyster (LP) 1997: P. Kouvelis & G. Yu (IP) } L. El Ghaoui & H. Lebret & F. Oustry 1997 : (CP) A. Ben-Tal & A. Nemirovski 2000 : E. Adida, A. Atamturk, A. Beck, D. Bertsimas, C. Bhattacharyya, H.-G. Bock, S. Boyd, G. Calafiore, M. Diehl, Y. Eldar, E. Erdogan, L. Grate, E. Guslitzer, B. Golany, D. Goldfarb, C. Hol, G. Iyengar, M. Jordan, E. Kostina, O. Kostyukova, G. Lanckriet, A. Nilim, M. Sim, D. Pachamanova, C. Roos, C. Schrerer, A. Sood, A. Thiele, J.-Ph. Vial, M. Zhang,... RO, applications: Structural/Circuit/Network Design Control Signal Processing Machine Learning Portfolio Optimization Inventory...

5 Example of uncertain LP: Multi-product inventory with backlogged demand and shared warehouse capacity min C,x t,y t,w t,v t C [inventory management cost] s.t. Tt=1 [c h y t + c b w t + c p v t] C [cost description] x t+1 = x t + v t ζ t x t y t, 0 y t [bounds on [x t ] + ] x t w t, 0 w t v t v t v t q y t Q [balance equations] [bounds on backlogged demand] [bounds on orders] [warehouse capacity bound] x t R d : inventory state at time t q R d + : storage requirements v t R d : replenishment orders at time t c o R d + : ordering costs y t R d : stored items at time t c h R d + : holding costs w t R d : backlogged demand at time t c b R d + : backlog penalty Uncertain data: demands ζ = [ζ 1 ;...; ζ T ]

6 { min x {f(x, ζ) : F(x, ζ) K} : ζ Z } (Unc) Assume that our decision environment is such that All decisions x j should be made before ζ reveals itself and thus should be independent of ζ The constraints are hard : their violations cannot be tolerated We intend to take full care of all data ζ Z and do not care what happens when ζ Z. Under these assumptions, seemingly the only meaningful way to process (Unc) is to solve the Robust Counterpart min t,x of the uncertain problem. {t : ζ Z : f(x, ζ) t, F(x, ζ) K} (RC)

7 Example: To build the RC of the Inventory problem, we use balance equations to eliminate the states and pass to the RC of the resulting inequality constrained problem, thus arriving at min C,y t,w t,v t s.t. C Tt=1 [c h y t + c b w t + c p v t] C x 1 + t 1 τ=1 [v τ ζ τ ] y t, 0 y t x 1 t 1 τ=1 [x τ ζ τ ] w t, 0 w t v t v t v t, q y t Q ζ Z (RC) Note: When Z is a computationally tractable convex set, the semi-infinite problem (RC) is computationally tractable. E.g., when Z is polyhedral: Z = {ζ : u : Pζ + Qu + r 0}, RC can be converted into an explicit LP program of sizes polynomial in T, d and the sizes of the representation of Z.

8 Extending the notion of RC: Adjustable/Affinely Adjustable Robust Counterpart. Assumption All decisions x j should be independent of ζ is too restrictive in many applications: Some of x j are analysis variables which do not represent decisions at all and can therefore depend on the entire data. Examples: Converting the constraint i a T i x b i t with uncertain a i, b i into y i a T i x b i y i, i y i t, it is natural to allow the analysis variables y i to adjust themselves to the actual data. In the Inventory problem, the actual decisions are the replenishment orders v t and the inventory management cost C; the remaining variables x t, y t, w t are analysis ones, and we can allow these variables to adjust themselves to the actual data.

9 In dynamical decision-making, some of the decisions x j should be made when the actual data becomes partially known and thus can depend on the corresponding portions of the data Example: In the Inventory problem with uncertain demand, replenishment orders v t of day t usually can depend on the actual demands at days 1,..., t 1.

10 To account for adjustability, we allow for every x j to depend on a prescribed portion P j ζ of ζ: x j = X j (P j ζ), thus arriving at Adjustable Robust Counterpart min {t : ζ Z : f(x(ζ), ζ) t, F(X(ζ), ζ) K} t,{x j ( )} n j=1 [X(ζ) = {X j (P j ζ)}] (ARC) Note: ARC is infinite-dimensional and thus is typically heavily computationally intractable. Seemingly the only applicable technique is Dynamic Programming curse of dimensionality To overcome, to some extent, intractability of ARC, we restrict the decision rules to be affine: X j (P j ζ) = ξ 0 j + ξt j P jζ, thus arriving at the Affinely Adjustable Robust Counterpart min {t : ζ Z : f(x(ζ), ζ) t, F(X(ζ), ζ) K} t,{ξj 0,ξ j} n j=1 [X(ζ) = {ξj 0 + ξt j P jζ} n j=1 ] (AARC)

11 Example: The only actual decisions in the Inventory problem are orders v t. Assume that v t can depend on the preceding demands ζ t 1 = [ζ 1 ;...; ζ t 1 ]. To build the AARC, we introduce linear decision rules for the orders v t = v 0 t + V tζ t 1 make x t, y t, w t affine functions of ζ:x t = x 0 t + X tζ, y t = y 0 t + Y t ζ, w t = w 0 t + W tζ, thus ending up with min C C,vt 0,V t,...,wt 0,W t t T [c h [y0 t + Y tζ] + c b [w t + W t ζ] + c p[vt 0 + V tζ t 1 ]] C x 0 t+1 + X t+1ζ = x 0 t + X tζ + vt 0 + V tζ t 1 ζ t x 0 t + X tζ yt 0 + Y tζ, 0 yt 0 + Y tζ s.t. [x 0 t + X tζ] w 0 t + W tζ, 0 w 0 t + W tζ v t v 0 t + V tζ t 1 v t, q [y 0 t + Y tζ] Q ζ Z (AARC) Note: The AARC of the Inventory problem is computationally tractable provided that Z is so. E.g., when Z is a polyhedral set, (AARC) is equivalent to an explicit LP program.

12 Example (continued): Consider single-product Inventory with N = 10 and a box uncertainty set: (1 ρ)ζ n ζ (1 + ρ)ζ n. Here the ARC is well within the grasp of Dynamic Programming. How large are the gaps in the chain Opt(ARC) Opt(AARC) Opt(RC)? We built a sample of 768 Inventory problems with uncertainty of 10% 50% by picking at random cost coefficients, storage capacity and nominal demand trajectory ζ n and subsequent filtering out problems with infeasible ARC s. It turns out that Opt(ARC) = Opt(AARC) in every one of these 768 problems! Note: This phenomenon disappears when passing from minimizing the worst-case inventory management cost to minimizing the average of this cost.

13 Opt(RC) was typically essentially worse than Opt(ARC) = Opt(AARC): Opt(RC) Range of 1 (1, 2] (2, 10] (10, 1000] Opt(ARC) Frequency in the sample 38% 23% 14% 11% 15% In the RO context, affine decision rules not necessarily are bad!

14 Optimization programs with uncertain data and their Robust Counterparts Tractability of Robust Counterparts Robust Optimization and Chance Constraints

15 Robust Counterparts of uncertain problem are semi-infinite programs and thus can be intractable even when all instances of the uncertain problem are easy to solve. When Robust Counterparts are computationally tractable? What to do if it is not the case?

16 We focus on uncertain affinely perturbed LP/CQP/SDP problems { { min c T x ζ x + d ζ : A i ζ x + bi ζ K i, i = 1,..., m } : ζ Z } with fixed recourse: c ζ, d ζ, A i ζ, bi ζ : affine in ζ Fixed recourse [automatically valid for the RC]: All coefficients of the adjustable variables x j (those with P j 0) are certain (i.e., independent of ζ). K i : nonnegative rays/lorentz cones/semidefinite cones (uncertain LP, CQP, SDP, respectively). We always assume that Z is given by a strictly feasible semidefinite representation (P( ): affine in (ζ, u)). Z = {ζ : u : P(ζ, u) 0}

17 Investigating tractability of Robust Counterparts of uncertain affinely perturbed LP/CQP/SDP problems with fixed recourse reduces to investigating tractability of semi-infinite affinely perturbed conic inequalities ζ Z : A ζ x + b ζ K = nonnegative ray [Uncertain LP] Lorentz cone [Uncertain CQP] semidefinite cone [Uncertain SDP] [ Aζ, b ζ : affine in ζ ]

18 Tractability of a semi-infinite affinely perturbed conic inequality ζ Z : A ζ x + b ζ K depends on the tradeoff between the geometries of K and Z the more complicated is Z, the simpler should be K. Trivial case : Scenario-generated uncertainty set Theorem. The RC/AARC of an uncertain affinely perturbed LP/CQP/SDP problem { min x { c T ζ x + d ζ : A i ζ x + bi ζ K i, i = 1,..., m } : ζ Z } with fixed recourse and with scenario-generated uncertainty set Z = Conv{ζ 1,..., ζ N } is computationally tractable.

19 Solvable case : Uncertain LP Theorem. The RC/AARC of uncertain affinely perturbed LP problem { minx { c T ζ x + d ζ : A i ζ x + bi ζ K i, i = 1,..., m } : ζ Z } [K i : given by explicit lists of linear inequalities] with fixed recourse is computationally tractable. With Z given by a strictly feasible LP/CQP/SDP representation, the RC/AARC is an explicit LP/CQP/SDP program of sizes polynomial in the size of instances and the size of the representation of the uncertainty set.

20 Aside of a number of highly specific particular cases, semiinfinite conic quadratic/linear matrix inequalities are computationally intractable. Whenever it is the case, a natural course of actions in the RO context is to replace an intractable semiinfinite conic inequality with its safe tractable approximation.

21 Definition. Consider a semi-infinite conic inequality ζ Z : A ζ x + b ζ K (C) and let 0 Z. We embed (C) into the parametric family of semi-infinite conic inequalities (ρ 0: uncertainty level). (ζ ρz) : A ζ x + b ζ K (C ρ ) A system of convex constraints (S ρ ) in variables x and additional variables u is called a safe tractable approximation of (C ρ ) tight within factor ϑ 1, if [tractability] The constraints in (S ρ ) are efficiently computable [safety] Whenever x can be extended to a feasible solution of (S ρ ), x is feasible for (C ρ ) [tightness] Whenever x cannot be extended to a feasible solution of (S ρ ), x is not feasible for (C ϑρ ).

22 Safe tractable approximation of semi-infinite Conic Quadratic Inequality ζ = [ζ l ; ζ r ] ρz : A ζ lx b ζ l 2 c T ζ r x d ζ r [ Aζ l, b ζ l, c ζ r, d ζ r are affine in ζ ] (C ρ ) (C ρ ) is computationally tractable when: [simple ellipsoidal uncertainty] Z is an ellipsoid centered at the origin [side-wise uncertainty with unstructured norm-bounded lhs perturbations] Z = Z l Z r, Z l = {ζ l R p q : ζ l 1} and A ζ lx b ζ l P(x) + L T (x)ζ l R(x), where P(x), L(x), R(x) are affine in x and either L( ), or R( ) are constant.

23 Example: The semi-infinite Least Squares inequality (ζ R p q, ζ ρ) : [ [A n, b n ] + L T ζr ] [x;1] 2 t (C ρ ) admits exact SDP representation ti λl T L A n x + b n λi ρr[x; 1] [A n x + b n ] T ρ[r[x;1]] T t 0 (S ρ) in variables t, x, λ: (t, x) is feasible for (C ρ ) iff it can be extended to a feasible solution to (S ρ ).

24 ζ = [ζ l ; ζ r ] ρz : A ζ lx b ζ l 2 c T ζ r x d ζ r (C ρ ) (C ρ ) admits tight tractable safe approximation when the uncertainty is side-wise: Z = Z l Z r and [ -ellipsoidal lhs perturbation set] Z l = {ζ l : [ζ l ] T Q j ζ l 1, 1 j M} with Q j 0, j Q j 0 (C ρ ) admits an explicit safe SDP approximation tight within the factor ϑ = O(1) ln M. [structured norm-bounded lhs perturbations] Z l = {ζ l = {ζ l j }J j=1 : ζl j Rp j q j : ζ l j 1} A ζ lx b ζ l P(x) + j L T j (x)ζl j R j(x) where P(x), L j (x), R j (x) are affine in x and for every j either L j ( ), or R j ( ) are constant. (C ρ ) admits an explicit safe SDP approximation tight within the factor ϑ = π 2.

25 Example: [ Robust Least Squares Antenna/Filter Design ] The semi-infinite Least Squares inequality (ζ, ζ ρ) : A n (I + Diag{ζ})x b n 2 t (C ρ ) with implementation errors x j (1 + ζ j )x j admits safe SDP approximation t j λ j [A n x b n ] T A n x b n ti ρa n Diag{x} ρ[a n Diag{x}] T Diag{λ 1,..., λ n } which is tight within the factor π 2. 0 (S ρ )

26 Safe tractable approximation of semi-infinite Linear Matrix Inequality (ζ ρz) : A ζ (x) 0 [ Aζ (x) : bi-affine in x, ζ ] (C ρ ) Aside of scenario-generated uncertainty set, the only known case when (C ρ ) admits a tight tractable approximation is the case of structured norm-bounded uncertainty: Z = {ζ = {ζ j } M j=1 : ζ j R p j p j, ζ j 1, ζ j = ξ j I, j J } A ζ (x) = A n (x) + [ j L T j ζ j R j (x) + R T j (x)ζt j L j] [ A n (x), R j (x) are affine in x ]

27 Theorem. The semi-infinite LMI ( {ζ j } M j=1 ζ j ρ j, ζ j = ξ j I pj, j J ) : A n (x) + [ j L T j ζ j R j (x) + R T j (x)ζt j L (C ρ ) j] 0 with structured norm-bounded uncertainty admits a safe tractable SDP approximation. The tightness factor ϑ of this approximation depends solely on the largest size of the scalar perturbation blocks { 1, J = µ = max j J p j, J = and does not exceed ϑ(µ), where ϑ( ) is a universal function such that ϑ(1) = π 2, ϑ(2) = 2, ϑ(k) π k. In the case of M = 1 (single perturbation block), the approximation is exact: ϑ = 1. Applications in: Robust Structural Design, Lyapunov Stability Analysis/Synthesis under interval uncertainty, etc.

28 Optimization programs with uncertain data and their Robust Counterparts Tractability of Robust Counterparts Robust Optimization and Chance Constraints

29 RO does not assume stochastic nature of uncertain data and uses instead uncertain-but-bounded model of data perturbations. However: Stochastic nature of uncertainty, if any, can be utilized in the RO framework. Consider Uncertain LP with stochastic data. The entity of primary interest here is a randomly perturbed linear constraint w 0 (x) + d l=1 ζ l w l (x) 0 [ x: decision vector w0 (x),..., w d (x): affine ζ 1,..., ζ d R: primitive random perturbations ] (C) A natural way to process (C) is to pass to a chance version Prob { w 0 (x) + d l=1 ζ l w l (x) > 0 } ɛ (C ɛ ) of the constraint (A. Charnes, W. Cooper, G. Symonds, 1958).

30 π(w) Prob{ζ w w 0 + d l=1 ζ l w l > 0} ɛ (C ɛ ) There exists significant literature on chance constraints (T. Badics, D. Dentcheva, A. Dupačová, L. Miller, A. Prékopa, A. Ruszczynski, B. Vizvari, H. Wagner,...) However: In general, (C ɛ ) is difficult to process: In many cases, the feasible set of a chance constraint is nonconvex; Even when convex, the feasible set of (C ɛ ) can be computationally intractable : When ζ Uniform([0,1] d ), π(w) is quasi-convex (C. Lagoa et al., 2005). However, unless P=NP, it is impossible to compute π(w) with accuracy δ > 0 in time polynomial in d, total binary length of (rational) w and ln(1/δ) [L. Khachiyan, 1989].

31 π(w) Prob{ζ w w 0 + d l=1 ζ l w l > 0} ɛ (C ɛ ) When (C ɛ ) as it is is difficult to process, one can look for a safe tractable approximation of (C ɛ ) a computationally tractable convex set W ɛ such that W ɛ {w : π(w) ɛ} ( ) A way to build a safe tractable approximation of (C ɛ ): Take a convex function γ : R R with lim s γ(s) = 0, γ(0) = 1 and set Γ(w) = E {γ(ζ w )}. Note: π(w) Γ(αw) α > 0 and Γ( ) is convex, whence the set W ɛ = cl {w : α > 0 : Γ(αw) ɛ} is convex and satisfies ( ). Whenever Γ( ) is efficiently computable, W ɛ is a safe tractable approximation of (C ɛ ).

32 γ( ) : R R +, γ(0) = 1 : convex, lim s γ(s) = 0 Γ(w) E { γ ( w 0 + )} d l=1 ζ l w l : convex W ɛ cl {w : α > 0 : Γ(αw) ɛ} {w : Prob{ζ w > 0} ɛ} Note: W ɛ is a closed convex cone such that W ɛ = {w R d+1 : w 0 + d l=1 ζ lw l 0 ζ Z ɛ } for an appropriate convex compact set Z ɛ. The safe approximation w(x) W ɛ of the chance constraint Prob{w 0 (x) + d l=1 ζ l w l (x) > 0} ɛ is nothing but the Robust Counterpart ζ Z ɛ : w 0 (x) + ζ 1 w 1 (x) ζ d w d (x) 0 of the uncertain affinely perturbed linear constraint w 0 (x) + ζ 1 w 1 (x) ζ d w d (x) 0 with properly defined perturbation set Z ɛ.

33 How to choose γ( )? As far as the conservatism of the upper bound Prob{ζ w > 0} inf α>0 E{γ(ζαw )} [γ( ) 0 : convex nondecreasing, γ(0) = 1] is concerned, the best choice of γ( ) is γ(s) = max[1 + s,0]. This choice leads to the famous Conditional Value at Risk safe approximation of the chance constraint [ min β + 1 β R ɛ E{[ζw β] + } 0 }{{} CVaR ɛ (w) Prob{ζ w > 0} ɛ. However: Typically, CVaR ɛ ( ) is difficult to compute... ]

34 Ensuring computability: Bernstein approximation [Pinter, 1989; Nem.&Shapiro, 2005] Observation: Consider a chance constraint and assume that Prob{w 0 + d l=1 ζ l w l > 0} ɛ (C ɛ ) ζ 1,..., ζ d are independent We are smart enough to build efficiently computable convex bounds Φ l (r) ln ( E{e rζ l} ) on the logarithmic moment-generating functions of ζ l, l = 1,..., d. Choosing γ(s) = e s, one can set Γ(w) = exp{w 0 + d l=1 Φ l (w l )} thus arriving at a tractable safe approximation of (C ɛ ).

35 Example (one of many): Range and mean a priori information on ζ l. Consider a chance constraint Prob{w 0 (x) + d l=1 ζ l w l (x) > 0} ɛ ζ l [ 1,1]: independent E{ζ l } [µ l, µ+ l ] (C ɛ ) The associated Bernstein approximation of (C ɛ ) is φ l (s) = ζ Z ɛ : w 0 (x) + d l=1 ζ l w l (x) 0, Z ɛ = {ζ : d l=1 φ l (ζ l ) 2ln(1/ɛ)} ( (1 + s)ln 1+s 1+µ l ) ( + (1 s)ln 1 s 1 µ l ), 1 s µ l 0 ( ) ( ), µ l s µ+ l (1 + s)ln + (1 s)ln, µ + l s 1 1+s 1+µ + l 1 s 1 µ + l (Br)

36 ζ Z ɛ : w 0 (x) + d l=1 ζ lw l (x) 0, Z ɛ = {ζ : d l=1 φ l(ζ l ) 2ln(1/ɛ)}, φ l (s) =... (Br) }{{} Entropy uncertainty, Nem.&Shapiro, 2006 Prob{w 0 (x) + d l=1 ζ lw l (x) > 0} ɛ (C ɛ ) ζ l [ 1,1]: independent E{ζ l } [µ l, µ+ l ] (Br) can be further safely approximated by Explicit CP ζ Z ɛ + : w 0 (x) + l ζ lw l (x) 0 Z ɛ + = { ζ 1} [ { ζ 2 ] 2ln(1/ɛ)} +{µ ζ µ + } }{{} Ball-Box uncertainty, Ben-Tal & Nem., 2000 (BB) Explicit CQP

37 ζ Z ɛ + : w 0 (x) + l ζ lw l (x) 0 Z ɛ + = { ζ 1} [ { ζ 2 ] 2ln(1/ɛ)} (BB) +{µ ζ µ + } ζ Z ɛ : w 0 (x) + d l=1 ζ lw l (x) 0, Z ɛ = {ζ : d l=1 φ (Br) l(ζ l ) 2ln(1/ɛ)}, φ l (s) =... Prob{w 0 (x) + d l=1 ζ lw l (x) > 0} ɛ (C ɛ ) ζ l [ 1,1]: independent E{ζ l } [µ l, µ+ l ] (BB) is further safely approximated by Explicit CQP Explicit CP ζ Z ɛ ++ : w 0 (x) + l ζ lw l (x) 0 Z ɛ ++ = { ζ 1} [ { ζ 1 ] 2dln(1/ɛ)} +{µ ζ µ + } }{{} Budgeted uncertainty, Bertsimas & Sim, 2003 (Bd) Explicit LP

38 ζ Z : w 0 (x) + d l=1 ζ lw l (x) 0 (RC[Z]) Prob{w 0 (x) + d l=1 ζ lw l (x) > 0} ɛ (C ɛ ) ζ l [ 1,1]: independent E{ζ l } [µ l, µ+ l ] d = 64 d = 128 d = 256 Random 2D cross-sections of Entropy (red), Ball-Box (yellow) and Budgeted (green) uncertainty sets, ɛ = 0.005, µ ± l = 0. Black: cross-section of the support { ζ 1} of ζ.