Robust Conic Optimization - Semantic Scholar

Robust Conic Optimization Dimitris Bertsimas

Melvyn Sim y

March 2004

Abstract In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust linear programming problems (LPs) become second order cone problems (SOCPs), robust SOCPs become semidenite programming problems (SDPs), and robust SDPs become NP-hard. We propose a relaxed robust counterpart for general conic optimization problems that (a) preserves the computational tractability of the nominal problem specically the robust conic optimization problem retains its original structure, i.e., robust LPs remain LPs, robust SOCPs remain SOCPs and robust SDPs remain SDPs moreover, when the data entries are independently distributed, the size of the proposed robust problem especially under the 2 norm is practically the same as the nominal problem, and (b) allows us to provide a guarantee on the probability that the robust solution is feasible, when the uncertain coe cients obey independent and identically distributed normal distributions. l

Boeing Professor of Operations Research, Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-363, Cambridge, MA 02139, [email protected]. The research of the author was partially supported by the Singapore-MIT alliance. y NUS Business School, National University of Singapore, [email protected].

1

1 Introduction The general optimization problem under parameter uncertainty is as follows:

~ 0) max f0 (x D

~ i) 0 subject to fi (x D

x 2 X

i 2 I

(1)

~ i), i 2 f0g  I are given functions, X is a given set and D~ i  i 2 f0g  I is the vector of where fi (x D uncertain coecients. We de ne the nominal problem to be Problem (1) when the uncertain coecients D~ i take values equal to their expected values D0i . In order to address parameter uncertainty Problem (1) Ben-Tal and Nemirovski 1, 3] and independently by El Ghaoui et al. 11, 12] propose to solve the following robust optimization problem max Dmin f0(x D0) 0 2U0 s:t: Dmin fi (x Di ) 0 i 2 I 2U i

i

(2)

x 2 X

where Ui , i 2 f0g  I are given uncertainty sets. The motivation for solving Problem (2) is to nd a solution x 2 X that \immunizes" Problem (1) against parameter uncertainty. By selecting appropriate uncertainty sets Ui , we can address the tradeo between robustness and optimality. In designing such an approach two criteria are important in our view:

(a) Preserving the computational tractability both theoretically and most importantly practically of

the nominal problem. From a theoretical perspective it is desirable that if the nominal problem is solvable in polynomial time, then the robust problem is also polynomially solvable. More speci cally, it is desirable that robust conic optimization problems retain their original structure, i.e., robust linear programming problems (LPs) remain LPs, robust second order cone problems (SOCPs) remain SOCPs and robust semide nite programming problems (SDPs) remain SDPs.

(b) Being able to nd a guarantee on the probability that the robust solution is feasible, when the

uncertain coecients obey some natural probability distributions. This is important, since from these guarantees we can select parameters that aect the uncertainty sets Ui that allows to control the tradeo between robustness and optimality. Let us examine whether the state of the art in robust optimization has the two properties mentioned above: 2

1. Linear Programming: A uncertain LP constraint is of the form a~0 x ~b, for which a~ and ~b are subject to uncertainty. When the corresponding uncertainty set U is a polyhedron, then the robust counterpart is also an LP (see Ben-Tal and Nemirovski 3, 4] and Bertsimas and Sim 8, 9]). When U is ellipsoidal, then the robust counterpart becomes an SOCP. For linear programming there are probabilistic guarantees for feasibility available ( 3, 4] and 8, 9]) under reasonable probabilistic assumptions on data variation. 2. Quadratic Constrained Quadratic Programming (QCQP): An uncertain QCQP constraint ~ k22 + ~b0x + c~  0, where A~ , ~b and c~ are subject to data uncertainty. The robust is of the form kAx counterpart is an SDP if the uncertainty set is a simple ellipsoid, and NP -hard if the set is polyhedral (Ben-Tal and Nemirovski 1, 3]). To the best of our knowledge, there are no available probabilistic bounds. 3. Second Order Cone Programming (SOCP): An uncertain SOCP constraint is of the form ~ + b~k2  c~0x + d~, where A~ , ~b, c~ and d~ are subject to data uncertainty. The robust counterpart kAx ~ , b~ belong in an ellipsoidal uncertainty set U1 and c~, d~ belong in another ellipsoidal is an SDP if A ~ , b~, c~, d~ vary together in a common ellipsoidal set. set U2 . The problem is NP -hard, however, if A To the best of our knowledge, there are no available probabilistic bounds.

P

4. Semidenite Programming (SDP): An uncertain SDP constraint of the form nj=1 A~j xj  B~ , where A~1  ::: A~n and B~ are subject to data uncertainty. The robust counterpart is NP -hard for ellipsoidal uncertainty sets, while there are no available probabilistic bounds.

P

5. Conic Programming: An uncertain Conic Programming constraint of the form nj=1 A~j xj K B~ , where A~1 ::: A~n and B~ are subject to data uncertainty. The cone K is closed, pointed and with a nonempty interior. To the best of our knowledge, there are no results available regarding tractability and probabilistic guarantees in this case. Our goal in this paper is to address (a) and (b) above for robust conic optimization problems. Speci cally, we propose a new robust counterpart of Problem (1) that has two properties: (a) It inherits the character of the nominal problem for example, robust SOCPs remain SOCPs and robust SDPs remain SDPs (b) under reasonable probabilistic assumptions on data variation we establish probabilistic guarantees for feasibility that lead to explicit ways for selecting parameters that control robustness. The structure of the paper is as follows. In Section 2, we describe the proposed robust model and in Section 3, we show that the robust model inherits the character of the nominal problem for LPs, 3

QCQPs, SOCPs and SDPs. In Section 4, we prove probabilistic guarantees for feasibility for these classes of problems. In Section 5, we show tractability and give explicit probabilistic bounds for general conic problems. Section 6 concludes this paper.

2 The Robust model In this section, we outline the ingredients of the proposed framework for robust conic optimization.

2.1 Model for parameter uncertainty The model of data uncertainty we consider is

D~ = D0 + X Dj z~j  j 2N

(3)

where D0 is the nominal value of the data, Dj , j 2 N is a direction of data perturbation, and z~j  j 2 N are independent and identically distributed random variables with mean equal to zero, so ~ ] = D0 . The cardinality of N may be small, modeling situations involving a small collection that E D of primitive independent uncertainties (for example a factor model in a nance context), or large, ~ are potentially as large as the number of entries in the data. In the former case, the elements of D ~ are weakly dependent or even independent strongly dependent, while in the latter case the elements of D (when jN j is equal to the number of entries in the data). The support of z~j  j 2 N can be unbounded or bounded. Ben-Tal and Nemirovskii 4] and Bertsimas and Sim 8] have considered the case that jN j is equal to the number of entries in the data.

2.2 Uncertainty sets and related norms In the robust optimization framework of (2), we consider the uncertainty set U as follows:

8 9 < = X U = :D j 9u 2 0 (see 4]). This norm is used in modeling bounded and symmetrically distributed random data.

The l1 \ l1 norm: maxf ;1 kuk1 kuk1g, ; > 0 (see 8, 7] ). Note that this norm is equal to l1 if ; = jN j, and l1 if ; = 1. This norm is used in modeling bounded and symmetrically distributed

random data, and has the additional property that the robust counterpart of an LP is still an LP (Bertsimas et al. 7]).

Given a norm k:k we consider the dual norm k:k de ned as

ksk = kmax s0x: xk1 We next show some basic properties of norms satisfying Eq. (5), which we will subsequently use in our development.

Proposition 1 If the norm k k satises Eq. (5), then we have (a) kwk = kw+ k : (b) For all v w such that v+  w+ kvk  kwk : (c) For all v w such that v+  w+ kvk  kwk: Proof (a) Let y 2 arg maxkxk1 w0x, and for every j 2 N , let zj = jyj j if wj 0 and zj = ;jyj j, otherwise. Clearly, w0z = (w+ )0y + w0y . Since, kzk = kz + k = ky+ k = kyk  1, and from the optimality of y, we have w0z  w0 y, leading to w0 z = (w+ )0y + = w0 y. Since kwk = kw+ k, we obtain kwk = kmax (w)0 x = max (w+ )0 x+ = max (w+ )0 x = kw+ k : xk1 kxk1 kxk1

(b) Note that If v +  w+ ,

+ 0 kwk = kmax (w+ )0 x+ = kmax xk1 (w ) x: xk1 x0

(v+ )0 x  kmax (w+ )0 x = kwk : kvk = kmax xxk 1 xxk 1 0 0 (c) We apply part (b) to the norm k:k . From the self dual property of norms k:k = k:k, we obtain part (c).

5

2.3 The class of functions (x D) f



We impose the following restrictions on the class of functions f (x D) in Problem (1) (we drop index j for clarity):

Assumption 1 The function f (x D) satises: (a) The function f (x D) is concave in D for all x 2 0. From Assumption 1(b) f (x 0) = 0, contradicting the concavity of f (x D) (Assumption 1(a)). Suppose that x is feasible in Problem (12). De ning t = s and y = ksk , we can easily check that (x t y ) are feasible in Problem (13). Conversely, suppose, x is infeasible in (12), that is,

f (x D0 ) < ksk : Since, tj sj = maxf;f (x Dj ) ;f (x ;Dj )g 0 we apply Proposition 1(b) to obtain ktk ksk . Thus, f (x D0) < ksk  ktk  y i.e., x is infeasible in (13). (b) It is immediate that Eq. (12) can be written in the form of Eq. (13). In Table 2, we list the common choices of norms, the representation of their dual norms and the corresponding references.

3.2 Representation of the function maxf; (x D) ; (x ;D)g The function g (x Dj ) = maxf;f (x Dj ) ;f (x ;Dj )g naturally arises in Theorem 1. Recall that a norm satis es kAk 0 kkAk = jkj kAk, kA + B k  kAk + kB k, and kAk = 0 implies that f

10





f



lp, p 1

kuk kuk2 kuk1 kuk1 kukp

l2 \ l1 norm

maxfkuk2  kuk1 g

Norms l2 l1 l1

ktk  y References ktk2  y 4] tj  y 8j 2 N 7] P t y 7] j 2N j P q q;q 1 q;1 y 7] j 2N tj ks ; tk + 1 P s  y 2 jN j +

s2
0

4]

p 2 0. From Assumption 1(b) f (x 0) = 0, contradicting the concavity of f (x A) (Assumption 1(a)). (b) For k 0, we apply Assumption 1(b) and obtain g(x kA) = maxf;f (x kA) ;f (x ;kA)g = k maxf;f (x A) ;f (x ;A)g = kg (x A): Similarly, if k < 0 we have

g (x kA) = maxf;f (x ;k(;A)) ;f (x ;k(A))g = ;kg(x A):

(c) Using Eq. (6) we obtain

g(x A + B ) = g (x 21 (2A + 2B ))  12 g (x 2A) + 12 g (x 2B ) = g(x A) + g (x B): 11

Note that the function g (x A) does not necessarily de ne a norm for A, since g (x A) = 0 does not necessarily imply A = 0. However, for LP, QCQP. SOCP(1), SOCP(2) and SDP, and speci c direction of data perturbation, Dj , we can map g (x Dj ) to a function of a norm such that

g(x Dj ) = kH(x Dj )kg  where H(x Dj ) is linear in Dj and de ned as follows (see also the summary in Table 3):

(a) LP:

f (x D) = a x ; b, where D = (a b) and Dj = (aj  bj ). Hence, 0

g(x Dj ) = maxf;(aj )0 x + bj  (aj )0 x ; bj g = j(aj )0x ; bj j:

(b) QCQP:

q

;



f (x D) = (d ; (b0x + c))=2 ; kAxk22 + (d + b0x + c)=2 2 , where D = (A b c d) and Dj = (Aj  bj  cj  0). Therefore,

8 < (bj )0x+ cj s j 2  (bj )0x+ cj 2 j + kA xk2 +  g(x D ) = max : 2 2 s  j 0 j 2 9 = j 0 j ; (b )2x+ c + kAj xk22 + (b )2x+ c   j 0 j  s  j 0 j 2 =  (b )2x+ c  + kAj xk22 + (b )2x+ c :

(c) SOCP(1):

f (x D) = c0x + d ; kAx + bk22, where D = (A b c d) and Dj = (Aj  bj  0 0). Therefore, g(x Dj ) = kAj x + bj k2:

(d) SOCP(2):

f (x D) = c0x+d;kAx+bk22, where D = (A b c d) and Dj = (Aj  bj  cj  dj ). Therefore,

n

g(x Dj ) = max ;(cj )0x ; dj + kAj x + bj k2 (cj )0x + dj + kAj x + bj k2 = j(cj )0x + dj j + kAj x + bj k2:

(e) SDP:

o

f (x D) = min (Pnj=1 Ai xi ; B), where D = (A1 ::: An  B ) and Dj = (Aj1 ::: Ajn Bj ). 12

Type LP QCQP SOCP(1) SOCP(2) SDP

h i r = rr , r1 = 1 0

r = H(x Dj ) r = (aj )0x ; bj h i Aj x

g (x Dj ) = krkg jr j , r0 = (( bj )0x + cj )=2 kr1k2 + jr0j

bj )0 x+ cj )=2 r = Aj x + bj h i r = rr01 , r1 = Aj x + bj , r0 = (cj )0x + dj R = Pni=1 Aji xi ; Bj ((

krk2 kr1k2 + jr0j kRk2

Table 3: The function H(x Dj ) and the norm k kg for dierent conic optimization problems. Therefore,

n

 

o

g (x Dj ) = max ;min (Pnj=1 Aji xi ; B j ) ;min ; Pnj=1 Aji xi ; Bj n  P  o P = max max ; nj=1 Aji xi ; B j  max( nj=1 Aji xi ; B j )

 n  X   =  Aji xi ; B j  :   j =1

2

3.3 The nature and size of the robust problem In this section, we discuss the nature and size of the proposed robust conic problem. Note that in the ~ ) we add at most jN j + 1 new proposed robust model (13) for every uncertain conic constraint f (x D variables, 2jN j conic constraints of the same nature as the nominal problem and an additional constraint involving the dual norm. The nature of this constraint depends on the norm we use to describe the uncertainty set U de ned in Eq. (4). When all the data entries of the problem have independent random perturbations, by exploiting sparsity of the additional conic constraints, we can further reduce the size of the robust model. Essentially, we can express the model of uncertainty in the form of Eq. (3), for which z~j is the independent random variable associated with the j th data element, and Dj contains mostly zeros except at the entries corresponding to the data element. As an illustration, consider the following semide nite constraint,

0 1 0 1 0 1 B@ a1 a2 CA x1 + B@ a4 a5 CA x2  B@ a7 a8 CA  a2 a3

a5 a6 a8 a9 such that each element in the data d = (a1 : : : a9)0 has an independent random perturbation, that is a~i = a0i + ai z~i and z~i are independently distributed. Equivalently, in Eq. (3) we have

X d~ = d0 + diz~i 9

i=1

13

l1-norm l1 -norm l1 \ l1 -norm l2-norm l2 \ l1 -norm Num. Vars. n+1 1 2jN j + 2 1 2jN j + 1 Num. linear Const. 2n + 1 2n + 1 4jN j + 2 0 3jN j Num SOC Const. LP QCQP SOCP(1) SOCP(2) SDP

0 LP SOCP SOCP SOCP SDP

0 LP SOCP SOCP SOCP SDP

0 LP SOCP SOCP SOCP SDP

1 SOCP SOCP SOCP SOCP SDP

1 SOCP SOCP SOCP SOCP SDP

Table 4: Size increase and nature of robust formulation when each data entry has independent uncertainty. where d0 = (a01 : : : a09)0 and di is a vector with ai at the ith entry and zero, otherwise. Hence, we can simplify the conic constraint in Eq. (13), f (x d1 ) + t1 0 or

00 1 0 1 0 11 a 0 0 0C 0 0 CC min B @B @ 1 CA x1 + B@ A x2 ; B@ A A + t1 0  0

0

0 0

0 0

as t1 ; minfa1x1  0g or equivalently as linear constraints t1 ;a1 x1  t1 0. In Appendix A we derive and in Table 4 we summarize the number of variables and constraints and their nature when the nominal problem is an LP, QCQP, SOCP (1) (only A b vary), SOCP (2) (A b c d vary) and SDP for various choices of norms. Note that for the cases of the l1 , l1 and l2 norms, we are able to collate terms so that the number of variables and constraints introduced is minimal. Furthermore, using the l2 norm results in only one additional variable, one additional SOCP type of constraint, while maintaining the nature of the original conic optimization problem of SOCP and SDP. The use of other norms comes at the expense of more variables and constraints of the order of jN j, which is not very appealing for large problems.

4 Probabilistic Guarantees In this section, we derive a guarantee on the probability that the robust solution is feasible, when the uncertain coecients obey some natural probability distributions. An important component of our analysis is the relation among dierent norms. We denote by h  i the inner product on a vector space, 14

0 such that 1 krk  qhr ri   krk  2 g  g 1

for all r in the relevant space.

(16)

Proposition 4 For the norm k kg dened in Table 3 for the conic optimization problems we consider, Eq. (16) holds with the following parameters:

(a) (b) (c) (d)

LP: 1 = 2 = 1:

p

QCQP, SOCP(2): 1 = 2 and 2 = 1. SOCP(1): 1 = 2 = 1:

p

SDP: 1 = 1 and 2 = m:

Proof (a) LP: For r 2 < and krkg = jrj, leading to Eq. (16) with 1 = 2 = 1: (b) QCQP, SOCP(2): For r = (r1 r0)0 2 0, using the p pp inequality a + b  2 a2 + b2 and a2 + b2  a + b, we have p p1 (kr1k2 + jr0j)  r0r = krk2  kr1k2 + jr0j 2 p

leading to Eq. (16) with 1 = 2 and 2 = 1: (c) SOCP(1): For all r, Eq. (16) holds with 1 = 2 = 1: q q (d) Let j , j = 1 : : : m be the eigenvalues of the matrix A. Since kAkF = trace(A2) = Pj 2j and kAk2 = maxj jj j, we have

kAk2  kAkF  pmkAk2 p leading to Eq. (16) with 1 = 1 and 2 = m:

The central result of the section is as follows.

15

Theorem 2 (a) Under the model of uncertainty in Eq. (3), and given a feasible solution x in Eq. (7), then 0 1 X ~ ) < 0)  P @k rj z~j kg > ksk A  P(f (x D j 2N

where

rj = H(x Dj )

sj = krj kg  j 2 N:

(b) When we use the l2-norm in Eq. (8), i.e., ksk = ksk2, and under the assumption that zj are normally and independently distributed with mean zero and variance one, i.e., z~  N (0 I ), then 0 1 p   2! s X X   e  B C P @ r j z~j  >  (17) krj k2g  A   exp ; 22  j2N g j 2N where  = 1 2 , 1 , 2 derived in Proposition 4 and  > .

Proof

We have

~ ) < 0) P(f (x D

0 1 X  P @f (x D0) + f (x Dj z~j ) < 0A (From (6)) j 2N    P f (x Pj2N Dj z~j ) < ;ksk (From (12), sj = kH(x Dj )kg ) 0 0 1 1 X X  P @min @f (x Dj z~j ) f (x ; Dj z~j )A < ;ksk A j 2N 0 1 j2N X = P @g (x Dj z~j ) > ksk A 0 j2N 1 X = P @kH(x Dj z~j )kg > ksk A j 2N 0 1 X = P @k H(x Dj )~zj kg > ksk A (H(x D) is linear in D ) j 2N 0 1 X = P @k rj z~j kg > ksk A : j 2N

16

(b) Using, the relations krkg  1phr ri and krkg 12 phr ri from Proposition 4, we obtain 0 1  s X X   PB krj k2gCA @ rj z~j  >  j 2N j 2N g 0 1 2 X X   = PB @ rj z~j  > 2 krj k2g CA j 2N j 2N 0 * g 1 + X X X  P @2122 rj z~j  rk z~k > 2 hrj  rj iA j 2N k2N j 2N 0 1 X X X = P @2 hrj  rk iz~j z~k > 2 hrj  rj iA 0 j2N k2N 1 j 2N X = P @2 z~0 Rz~ > 2 hrj  rj iA  j 2N

where Rjk = hrj  rk i. Clearly, R is a symmetric positive semide nite matrix and can be spectrally decomposed such that R = Q0 Q, where is the diagonal matrix of the eigenvalues and Q is the P corresponding orthonormal matrix. Let y~ = Qz~ so that z~0 Rz~ = y~0 ~y = j 2N j y~j2. Since z~  N (0 I ), we also have y~  N (0 I ), that is, y~j , j 2 N are independent and normally distributed. Moreover,

X

j 2N

Therefore,

j = trace(R) =

X

j 2N

hrj  rj i:

0 1 X P @2 z~0 Rz~ > 2 hrj  rj iA j 2N 0 1 X X = P @2 j y~j2 > 2 j A j 2N  j2N 2 P  E exp  j 2N j y~j2  P  (From Markov's inequality, > 0)  exp 2 j 2N j Q E exp  2 y~2 j j  = j 2N  2 P (~yj2 are independent) exp  j 2N j   2 2 j ! Q y~ j 2N E

=

Q 

exp j

 2P  for all > 2 and 2 j  1 8j 2 N exp  j 2N j    2 2 j  !

j 2N

E exp



P

y~j 

exp 2 j 2N j



 17

where the last inequality follows from Jensen inequality, noting that x2 j  is a concave function of x if 2 j 2 0 1]. Since y~j  N (0 1),



 y~2 !! j

E exp

Thus, we obtain

Q

j 2N

 

E exp





 y~2 2 j  !

P

!

s

Z1 2  ; 2

: y 1 =p exp ; 2 dy =

;2 2 1 j



exp 2 j 2N j





Q







 1 2 j 2N exp  j 2 ln  ;2   = P exp 2 j 2N j    P  exp 2 12 ln  ;2 j 2N j  P  = : exp 2 j 2N j

We select = 1=(2  ) where  = maxj 2N j , and obtain



P



P



   2 !! exp 2 12 ln  ;2 j 2N j

1  P  = exp 2 ln ; 2 ; 2  exp 2 j 2N j

where = ( j 2N j )= : Taking derivatives and choosing the best , we have 2

= 22 ; 2  for which  > . Substituting and simplifying, we have

 

 2 1 exp 2 ln ; 2 ; 2

!!  p =

!

e exp(; 2 )   pe exp(; 2 )  22  22 p

where the last inequality follows from 1, and from e exp(; 2 22 ) < 1 for  > . ~ ) < 0, implies that kz~k > . Thus, when z~  N (0 I) Note that f (x D

~ ) < 0)  P(kz~k > ) = 1 ; 2jN j(2) P(f (x D

(18)

where 2jN j ( ) is the cdf of a -square distribution with jN j degrees of freedom. Note that the bound (18) ~ ) in contrast to bound (17) that depends on f (x D~ ) does not take into account the structure of f (x D via the parameter . To illustrate this, we substitute the value of the parameter  from Proposition 4 in Eq. (17) and report in Table 6 the bound in Eq. (17). To amplify the previous discussion, we show in Table 6 the value of  in order for the bound (17) to be less than or equal to . The last column shows the value of  using bound (18) that is independent of the structure of the problem. We choose jN j = 495000 which is approximately the maximum number 18

Type

Probability bound of infeasibility pe exp(; 2 ) LP 2 qe 2 QCQP 2  exp(; 4 ) pe exp(; 2 ) SOCP(1) 2 qe 2 SOCP(2) 2  exp(; 4 ) qe 2 SDP m  exp(; 2m ) ~ ) < 0) for z~  N (0 I ). Table 5: Probability bounds of P(f (x D



10;1 10;2 10;3 10;6

LP QCQP SOCP(1) SOCP(2) SDP Eq. (18)

2:76 3:57 4:21 5:68

3:91 5:05 5:95 7:99

2:76 3:57 4:21 5:68

3:91 5:05 5:95 7:99

27:6 35:7 42:1 56:8

704:5 705:2 705:7 706:9

Table 6: Sample calculations of  using Probability Bounds of Table 5 for m = 100, n = 100, jN j = 495 000. of data entries in a SDP constraint with n = 100 and m = 100. Although the size jN j is unrealistic for constraints with less data entries such as LP, the derived probability bounds remain valid. Note that p bound (18) leads to  = O( jN j ln(1=)). For LP, SOCP, and QCQP, bound (17) leads to  = O(ln(1=)), which is independent of the p dimension of the problem. For SDP it leads to we have  = O( m ln(1=)). As a result, ignoring the structure of the problem and using bound (18) leads to very conservative solutions.

5 General cones In this section, we generalize the results in Sections 2-4 to arbitrary conic constraints of the form, n X A~j xj K B~ 

j =1

19

(19)

where fA~1  ::: A~n  B~ g = D~ constitutes the set of data that is subject to uncertainty, and K is a closed, convex, pointed cone with nonempty interior. For notational simplicity, we de ne

X A(x D~ ) = A~j xj ; B~ n

j =1

so that Eq. (19) is equivalent to

A(x D~ ) K 0:

(20)

We assume that the model for data uncertainty is given in Eq. (3) with z~  N (0 I ). The uncertainty set U satis es Eq. (4) with the given norm satisfying kuk = ku+ k: Paralleling the earlier development, starting with a cone K and constraint (20), we de ne the function f (  ) as follows so that f (x D) > 0 if and only if A(x D) K 0.

Proposition 5 For any V K 0, the function f (x D) = max s:t: A(x D) K V 

(21)

satises the properties:

(a) (b) (c) (d)

f (x D) is bounded and concave in x and D. f (x kD) = kf (x D) 8k 0. f (x D) y if and only if A(x D) K y V . f (x D) > y if and only if A(x D) K y V .

Proof (a) Consider the dual of Problem (21): z = min hu A(x D)i s:t: hu V i = 1 u K 0 where K is the dual cone of K . Since K is a closed, convex, pointed cone with nonempty interior, so is K (see 5]). As V K 0, for all u K 0 and u 6= 0, we have hu V i > 0, hence, the dual problem is bounded. Furthermore, since K has a nonempty interior, the dual problem is strictly feasible, i.e., there exists u K 0 hu V i = 1. Therefore, by conic duality, the dual objective z has the same nite objective as the primal objective function f (x D). Since A(x D) is a linear mapping of D and an 20

ane mapping of x, it follows that f (x D) is concave in x and D.

(b) Using the dual expression of f (x D), and that A(x kD) = kA(x D), the result follows. (c) If = y is feasible in Problem (21), we have f (x D) = y. Conversely, if f (x D) y, then A(x D) K f (x D)V K yV . (d) Suppose A(x D) K yV , then there exists  > 0 such that A(x D) ; yV K V or A(x D) K ( + y )V . Hence, f (x D)  + y > y . Conversely, since V K 0, if f (x D) > y then (f (x D) ; y)V K 0. Hence, A(x D) K f (x D)V K yV . Remark : With y = 0, (c) establishes that A(x D) K 0 if and only if f (x D) 0 and (d) establishes that A(x D) K 0 if and only if f (x D) > 0.

The proposed robust model is given in Eqs. (7) and (8). We next derive an expression for g(x D) = maxf;f (x D) ;f (x ;D)g.

Proposition 6 Let g(x D) = maxf;f (x D) ;f (x ;D)g. Then g(x D) = kH(x D)kg where H(x D) = A(x D) and

kSkg = min fy : yV K S K ;yV g :

Proof

We observe that

g(x D) = maxf;f (x D) ;f (x ;D)g = minfy j ; f (x D)  y ;f (x ;D)  y g = minfy j A(x D) K ;y V  ;A(x D) K ;y V g (From Proposition 5(c)) = kA(x D)kg : We also need to show that k:kg is indeed a valid norm. Since V K 0, then kS kg 0. Clearly, k0kg = 0 and if kS kg = 0, then 0 K S K 0, which implies that S = 0. To show that kkS kg = jkjkS kg , we observe that for k > 0, kkS kg = min fy j yV K kS K ;yV g  y y y = k min k j k V K S K ; k V = kkS kg :

21

Likewise, if k < 0

kkS kg = min fy j yV = min fy j y V = k ; k S kg = ;kkS kg :

K kS K ;yV g K ;kS K ;yV g

Finally, to verify triangle inequality,

kS kg + kT kg = min fy j yV K S K ;yV g + min fz j zV K T K ;zV g = min fy + z j y V K S K ;y V  z V K T K ;z V g min fy + z j (y + z )V K S + T K ;(y + z )V g = kS + T kg : For the general conic constraint, the norm, k kg is dependent on the cone K and a point in the interior of the cone V . Hence, we de ne k kKV := k kg . Using Proposition 5 and Theorem 1 we next show that the robust counterpart for the conic constraint (20) is tractable and provide a bound on the probability that the constraint is feasible.

Theorem 3 We have (a) (Tractability) For a norm satisfying Eq. (5), constraint (7) for general cones is equivalent to A(x D0) K yV  tj V K A(x Dj ) K ;tj V  j 2 N ktk  y y 2