Strong Duality in Robust Semi-Definite Linear Programming under Data Uncertainty

Transcription

1 Strong Duality in Robust Semi-Definite Linear Programming under Data Uncertainty V. Jeyakumar and G. Y. Li March 1, 2012 Abstract This paper develops the deterministic approach to duality for semi-definite linear programming problems in the face of data uncertainty. We establish strong duality between the robust counterpart of an uncertain semi-definite linear programming model problem and the optimistic counterpart of its uncertain dual. We prove that strong duality between the deterministic counterparts holds under a characteristic cone condition. We also show that the characteristic cone condition is also necessary for the validity of strong duality for every linear objective function of the original model problem. In addition, we derive that a robust Slater condition alone ensures strong duality for uncertain semi-definite linear programs with affinely parameterized data. Finally, we present strong duality with computationally easy tractable dual programs for two classes of uncertainties: the scenario uncertainty and the spectral norm uncertainty. Key words. Robust optimization, semi-definite programming under uncertainty, strong duality, linear matrix inequality problems. AMS subject classification. 90C22,90C25,90C46 Research was partially supported by a grant from the Australian Research Counc. Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. E- ma: v.jeyakumar@unsw.edu.au Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. E- ma: g.li@unsw.edu.au 1

2 1 Introduction A semi-definite linear programming model problem SDP) in the absence of data uncertainty consists of minimizing a linear objective function over a linear matrix inequality constraint [21]. It is often expressed in the form x R n{ct 0 x i A i 0}, where the data consist of the objective vector c R n and the matrices A i S m, the space of m m) symmetric matrices. For A S m, A 0 resp., A 0) means that A is positive semi-definite resp., definite). Model problems of this form arise in a wide range of engineering applications [7, 21]. In practice, however, these data are subject to uncertainty due to modeling or forecasting errors [4, 5, 8, 18]. The robust optimization approach to examining semi-definite linear programming problems under data uncertainty, developed in this paper, is to treat uncertainty as deterministic via uncertainty sets which are closed and convex. On the other hand, stochastic approaches to optimization under uncertainty start by assuming the uncertainty has a probabistic description. The best known technique is based on stochastic programming [20]. We consider the semi-definite linear programming model problem with data uncertainty in the constraint UP ) x R n{ct 0 x i A i 0} and its Lagrangian dual program with the same uncertain data DP ) max F S m{ TrA 0F ) : TrA i F ) = c i, i = 1,..., n, F 0}, where, for each i = 0, 1,..., n, A i S m, is uncertain and belongs to the uncertainty set V i which is a closed and convex set in S m, and TrA i F ) is the trace of the matrix A i F. We study duality between the uncertain dual pair by examining their deterministic counterparts: the robust counterpart of UP) RP ) x R n{ct 0 x i A i 0, A i V i, i I}, where the index set I := {0, 1,..., n}, and the optimistic counterpart of the uncertain dual program DP) ODP ) sup A i V i,i I max F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0}. The strong duality in robust semi-definite programming states that x R n{ct 0 x i A i 0, A i V i, i I} = max A i V i,i I max F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0} 2

3 and it describes, in particular, that the value of the robust counterpart RP) primal worst value ) is equal to the value of the optimistic counterpart ODP) dual best value ) and the optimal value of optimistic counterpart is attained. Due to the importance of solving semi-definite programs under data uncertainty, a great deal of attention has recently been focussed on identifying and obtaining uncertaintyimmunized) robust solutions of uncertain semi-definite programs [3, 6, 8, 9, 15, 19]. More recently, strong duality has been established for robust convex programming problems with inequality constraints under uncertainty in [1]. For such inequality constrained problems, various characterizations of strong duality have also been given by the authors in [16]. In this paper, we aim to establish strong duality for linear optimization problems with uncertain linear matrix inequality constraints. We first prove strong duality between the deterministic dual pair RP) and ODP) under the condition that the robust characteristic cone D = { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0} A i V i,i I is closed and convex. We further prove that this robust characteristic cone condition is also necessary for the validity of robust strong duality for every linear objective function c T x. Related results for convex programming problems without uncertainty can be found in [12, 13, 14]. In particular, we show that a robust Slater strict feasibity condition [17] together with the convexity of the characteristic cone ensures strong duality. However, we lustrate by an example that the robust Slater condition is, in general, not necessary for strong duality. Second, in the case of uncertain semi-definite linear programs with affinely parameterized data [4], we show that the robust characteristic cone is a convex cone and that a robust Slater condition alone ensures strong duality. Third, we derive two classes of uncertain semi-definite linear programs for which strong duality holds under a robust Slater condition and the optimistic counterparts are computationally tractable semi-definite linear programs: the cases of scenario uncertainty [4] and spectral norm uncertainty [1, 4]. The organization of the paper is as follows. Section 2 presents strong duality results in terms of the robust characteristic cone. Section 3 establishes strong duality for uncertain semi-definite linear programs with affinely parameterized data under a robust Slater condition. Section 4 provides classes of uncertain semi-definite linear programs with computationally tractable optimistic counterparts. Section 5 concludes with a discussion on further research on the tractabity of optimistic counterparts. 3

4 2 Strong Duality for Robust SDPs In this Section, we present strong duality results in terms of the robust characteristic cone, D R n1, which is defined by D = { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0}. A i V i,i I We begin by showing that D is indeed a cone in R n1. Lemma 2.1. For each i I, let V i S m be closed and convex. Then, D is a cone. Proof. Clearly, 0 D. Let x D R n1 and µ > 0. Then, there exist A i V i, M 0 and r 0 such that x i = TrA i M), 1 i n and x n1 = TrA 0 M) r. So, letting M = µm 0 and r = µr 0, we obtain that µx i = TrA i M), 1 i n, and µx n1 = TrA 0 M) r. So, µx D, and hence, D is a cone. We now show that strong duality holds whenever the robust characteristic cone is closed and convex. Theorem 2.1. Strong Duality) Let V i S m be closed and convex, for i I, and let the robust feasible set F := {x R n : A 0 n x ia i 0, A i V i } =. Suppose that the robust characteristic cone D is closed and convex. Then, x R n{ct 0 x i A i 0, A i V i, i I} = max A i V i,i I max F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0}. Proof. [Weak Duality]. We first observe that, for each x R n with A 0 n x ia i 0, A i V i, i I, and for each F 0, A i V i, i I\{0}, with TrA if ) = c i and A 0 V 0, we have c T x = x i TrA if ) = Tr A 0 and so, the following weak duality holds: ) x i A i)f TrA 0F ) TrA 0F ), x R n{ct 0 x i A i 0, A i V i } max max A i V i,i I F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0}. [Reverse Inequality] To see the reverse inequality, we may assume that inf x R n{c T x : A 0 n x ia i 0, A i V i, i I} >. As the robust feasible set F, γ := inf x R n{ct x : A 0 x i A i 0, A i V i, i I} R. 4

5 Then, we see that the following implication holds: A 0 x i A i 0, A i V i, i I c T x γ ) [Homogenization]. Using 2.1), we now show, by the method of contradiction, that t 0 and ta 0 x i A i 0, A i V i, i I c T x tγ ) Suppose, to the contrary, that there exists x 0, t 0 ) R n R such that t 0 A 0 n x0 i A i 0, A i V i and c T x 0 t 0 γ < 0. If t 0 > 0, then we have A 0 n x 0 i t 0 A i 0, A i V i and c T x0 i t 0 ) γ < 0. This contradicts 2.1). If t 0 = 0, then n x0 i A i 0, A i V i, i I\{0} and c T x 0 < 0. Now, fix x F such that A i V i, A 0 n x ia i 0 and consider x α = x αx 0, α 0. Then, A 0 and x α i A i = A 0 x i αx 0 i )A i = A 0 x i A i α x 0 i A i 0, A i V i, i I c T x α γ = c T x γ) αc T x 0 as α. This also contradicts 2.1), and so, 2.2) holds. Now, fix A i V i. Define c R n1 by c = c, γ) and define à : Rn1 S m R by Ãz = ta 0 x i A i, t), for z = x, t) R n R. Then 2.2) can be equivalently rewritten as Now, we show that A i V i, Ãz S m R ) c T z ) c, γ) cod, 2.4) where cod denotes the closure of the convex hull of D. [Hyperplane separation]. Suppose that c, γ) / cod. Then, by the convex separation theorem, there exists z = x, t) R n R)\{0} such that As D = A i V i,i I is a cone, we obtain that u, x, t) < c, γ), x, t), u D. { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0} u, x, t) 0 < c, γ), x, t), u D. 5

6 This gives us that sup u, x, t) 0 and c, γ), x, t) < 0. u D On the other hand, it can be verified that, for each F, r) S m R, F, r), Ãx, t) = F, r), ta 0 n x ia i, t) = n x itra i F ) TrA 0 F ) r)t. Recall that for a linear mapping L : R p R q, p, q N, its conjugate mapping L : R q R p is defined by L y), x = y, Lx) for all x R p and y R q. Then, the conjugate mapping of Ã, denoted as Ã, is given by à F, r) = TrA 1 F ),..., TrA n F ), TrA 0 F ) r). So, letting w = x, t) R n R)\{0}, we have and sup A i V i,f 0,r 0 Ãw, F, r) = sup A i V i,f 0,r 0 à F, r), w = sup u, x, t) 0 u D c, w = c, γ), x, t) = c, γ), x, t) < 0. Hence, from the bipolar theorem, Ãw S m R ) and c, w < 0. This contradicts 2.3), and so 2.4) holds. [Dual Attainment]. By assumption, cod = D, and so, c, γ) D. Then, there exist A i V i, F 0 and r 0 such that So, TrA i F ) = c i, 1 i n and γ = TrA 0 F ) r TrA 0 F ). γ max max A i V i,i I F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0}, and hence the conclusion follows. We now show that the requirement for strong duality, that D is convex and closed, is in fact necessary and sufficient for the validity of strong duality for every linear objective function c T x. Theorem 2.2. Characterization of Strong Duality) Let V i S m be closed and convex and let the robust feasible set F := {x R n : A 0 n x ia i 0, A i V i }. Then, the following statements are equivalent: i) The robust characteristic cone D is closed and convex. ii) For each c R n, x R n{ct 0 x i A i 0, A i V i, i I} = max A i V i,i I max F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0}. 6

7 Proof. The conclusion wl follow from Theorem 2.1 if we show that [ii) i)]. Assume that ii) holds. Let z := z 1,..., z n, z n1 ) R n1 such that z cod. Let w = z 1,..., z n ). Then, we claim that inf{ w T x : A 0 x i A i 0, A i V i } z n1. 2.5) To see this, as z cod and D is a cone of dimension n 1, there exist z jk D and λ jk [0, 1], j = 1,..., n 1 with n1 λjk = 1 such that, as k, n1 λjk z jk z. Since z jk = z jk 1,..., zn jk, zn1) jk D = A i V i,i I { TrA 1F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0}, there exist A jk i V i, i = 1,..., n, F jk 0 and r jk 0 such that z jk i = TrA jk i F jk ), 1 i n, and z jk n1 = TrA jk 0 F jk ) r jk. Let w jk = z jk 1,..., z jk n ). Then, n1 λjk w jk w and n1 λ jk w jk ) T x = n1 = n1 λ jk λ jk z jk i x i Trx i A jk i F jk ). So, for each x = x 1,..., x n ) with A 0 n x ia i 0, A i V i, we have λ jk w jk ) T x = n1 = n1 λ jk Trx i A jk i F jk ) n1 λ jk Tr A jk 0 ) n1 x i A jk i )F jk λ jk TrA jk 0 F jk ) n1 n1 λ jk z jk n1 r jk ) λ jk zn1. jk Passing to limit and noting that n1 λjk w jk w = z 1,..., z n ) and n1 λjk z jk n1 z n1, we obtain that for each x = x 1,..., x n ) with A 0 n x ia i 0, A i V i w T x z n1. So, 2.5) follows. Now, applying ii) with c = w, where w = z 1,..., z n ), we obtain that max max A i V i,i I F S m{ TrA 0F ) : TrA i F ) = z i, i I\{0}, F 0} z n1. So, there exist F 0 and A i V i, i I with TrA i F ) = z i, such that z n1 TrA 0 F ). 7

8 This implies that z = z 1,..., z n, z n1 ) D := A i V i,i I { TrA 1 F ),..., TrA n F ), TrA 0 F )r) : F 0, r 0}. This shows that cod D and so, cod = D. So, D is closed and convex. Remark 2.1. Now, we show that the more general case, i.e., linear semi-definite problems where uncertainty occurs in both objective function and in the constraints can also be handled by our approach. Indeed, assume that the data c in the objective function c T x is also uncertain and c belongs to the uncertainty set C, then the corresponding uncertain semi-definite linear programs is given by inf c T x x 1,...,x n) T R n s.t. A 0 x i A i 0 where the datum c R n and A i S m are uncertain, and the data c, A i ) belongs to the uncertainty set C V i, i = 0, 1,..., n. This can be equivalently rewritten as inf x 1,...,x n1 ) T R n1 x n1 s.t. A 0 x i A i 0 x n1 c T x 0, where the datum c R n and A i S m are uncertain, and the data c, A i ) belongs to the uncertainty set C V i, i = 0, 1,..., n. Denote c = c 1,..., c n ) T R n and the n n) identity matrix by I n. Then, this situation can be modeled as the following uncertain linear semi-definite problem where uncertainty only occurs in the constraint inf x 1,...,x n1 ) T R n1 x n1 n1 s.t. A 0 x i A i 0 where the matrix data A i is uncertain, i = 0, 1,..., n 1, and the matrix A i belongs to the uncertainty set V i where { ) } { ) } 0 0 V 0 = : A 0 A 0 V 0, V i ci I = n 0 : A 0 0 A 0 V 0, i = 1,..., n i and { V n1 In 0 = 0 A i )}. 8

9 In the special case of UP) without data uncertainty, where each V i, i I, is a singleton set {A i }, we see that Theorem 2.2 yields the recent characterization of strong duality for semi-definite linear programs, given in [13]. Corollary 2.1. Let the feasible set F := {x R n : A 0 n x ia i 0}. Then, the following statements are equivalent: i) The cone D 0 := { TrA 1 F ),..., TrA n F ), TrA 0 F )r) : F 0, r 0} is closed. ii) For each c R n, x R n{ct 0 x i A i 0} = max F S m{ TrA 0F ) : TrA i F ) = c i, i = 1,..., n, F 0}. Proof. For each i I, let V i, be a singleton set {A i }. Then the cone D collapses to the cone D 0 := { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0}. Note that D 0 is always convex. So, the conclusion follows Theorem 2.2. We also show that under the Robust Slater Condition that F := {x R n : A 0 n x ia i 0, A i V i }, the characteristic cone D is closed and that strong duality is characterized by the convexity of D. Theorem 2.3. Let V i S m be compact and convex. Suppose that F = {x R n : A 0 n x ia i 0, A i V i }. Then the following statements hold i) The characteristic cone D is closed. ii) The cone D is convex if and only if, for each c R n, x R n{ct 0 x i A i 0, A i V i, i I} = max A i V i,i I max F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0}. Proof. i) Let {x k } R n1 such that {x k } D, x k = x k 1,..., x k n1) and x k x = x 1,..., x n1 ). Then, there exist A k i V i, i I, F k 0 and r k 0 such that and x k i = TrA k i F k ), 1 i n, 2.6) x k n1 = TrA k 0F k ) r k. 2.7) As V i is compact, by passing to a subsequence, if necessary, we may assume that A k i A i V i. We first show that { F k } is a bounded sequence. Otherwise, without of loss of F generality, we assume that F k. Then, k F F k Sm \{0}, where S m = {A S m A 0}. Dividing both sides of 2.6) and 2.7) by F k and passing to limit, we obtain that r k TrA i F ) = 0, 1 i n, and TrA 0 F ) = lim k F k 0. 9

10 Since F, there exists x 0 R n such that A 0 n x0 i A i 0, A i V i, and ) Tr A 0 x 0 i A i )F = TrA 0 F ) x 0 i TrA i F ) 0. On the other hand, because F S m \{0}, Tr A 0 ) n x0 i A i )F > 0. This is a contradiction, and so, { F k } is a bounded sequence. Hence, by 2.7), r k is also bounded. Hence, by passing to subsequence, if necessary, we can assume that F k F and r k r. Now, passing to limit in 2.6) and 2.7), we obtain that x i = TrA i F ), 1 i n, and x n1 = TrA 0 F ) r. 2.8) Thus, x D, and so, D is closed. ii). This wl follow from i) and the strong duality Theorem 2.2. We now provide two examples. The first example lustrate the situation where the robust Slater condition fas whe the cone D is closed and convex and strong duality holds. The second example shows that, in general, the characteristic cone can be nonconvex. Example 2.1. Consider the following robust semi-definite linear programming problem: where V 0 = E 1 ) inf x R 2 c 1 x 1 c 2 x 2 s.t. A 0 x 1 A 1 x 2 A 2 0, A i V i, i = 0, 1, 2,, V 1 = a 0 a 0 : a [0, 1] and V 2 = Clearly, for any A i V i, i = 0, 1, 2, A 0 x 1 A 1 x 2 A 2 = 1 x ax 1 0 ax 1 0. Note that, any positive definite matrix must have positive diagonal elements. So, for each A i V i, i = 0, 1, 2, A 0 x 1 A 1 x 2 A 2 is not positive definite, and so, the robust Slater condition fas. However, D = { TrA 1 F ), TrA 2 F ), TrA 0 F ) r) : F 0, r 0} A i V i = a [0,1] { 2af 5, f 1, f 1 r) : F = = R R R, 10 f 1 f 2 f 3 f 2 f 4 f 5 f 3 f 5 f 6 0, r 0}

11 which is closed and convex. On the other hand, the robust feasible set F is given by So, we have F = {x 1, x 2 ) : A 0 x 1 A 1 x 2 A 2 0, A i V i, i = 0, 1, 2} 1 x = {x 1, x 2 ) : 0 0 ax 1 0, a [0, 1]} 0 ax 1 0 = {x 1, x 2 ) : x 2 1, x 1 = 0}. inf x R 2{c 1x 1 c 2 x 2 : A 0 x 1 A 1 x 2 A 2 0, A i V i, i = 0, 1, 2} = Moreover, the optimistic counterpart of its uncertain dual is ODP 1 ) {, if c2 < 0, c 2, if c 2 0. max max A i V i,i=0,1,2 F S 3{ TrA 0F ) : TrA i F ) = c i, i = 1, 2, F 0} = max a [0,1] max F S 3{ f 1 : 2af 5 = c 1, f 1 = c 2, F = f 1 f 2 f 3 f 2 f 4 f 5 f 3 f 5 f 6 0}. Note that F 0 implies that f 1 0. So, if c 2 < 0, then the feasible set of ODP 1 ) is empty and hence the value of it is. Moreover, if c 2 0, then the feasible set is nonempty and hence maxodp 1 ) = c 2. Thus, inf x R 2{c 1x 1 c 2 x 2 : A 0 x 1 A 1 x 2 A 2 0, A i V i, i = 0, 1, 2} = max max A i V i,i=0,1,2 F S 3{ TrA 0F ) : TrA i F ) = c i, i = 1, 2, F 0} {, if c2 < 0, = c 2, if c 2 0. Example 2.2. Consider the following uncertain semi-definite linear program: min x 1 x 2 s.t. A 0 x 1 A 1 x 2 A 2 0 { )} 0 0 where the matrix data A 0, A 1, A 2 are uncertain and A 0 V 0 =, A { ) } { ) } 0 a 0 b V 1 = : a [0, 1] and A a 0 2 V 2 = : b [ 1, 0]. In this case, the b 0 characteristic cone D becomes D = = = { TrA 1 F ), TrA 2 F ), TrA 0 F ) r) : F 0, r 0} A i V i ) f1 f { 2af 2, 2bf 2, r) : F = 2 0, r 0} f 2 f 3 a [0,1], b [ 1,0] { 2af 2, 2bf 2, r) : f 1 0, f 3 0, f2 2 f 1 f 3, r 0} a [0,1], b [ 1,0] 11

12 It can be easy verified that 0, 2, 0) D by letting a = 0, b = 1, f 2 = 1, f 1 = f 3 = 2 and r = 0) and 2, 0, 0) D by letting a = 1, b = 0, f 2 = 1, f 1 = f 3 = 2 and r = 0). However, we now show that their mid point 1, 1, 0) / D. Otherwise, we have 2af 2 = 1 and 2bf 2 = 1 for some a [0, 1] and b [ 1, 0]. This implies that a 0, b 0 and f 2 0. So, we see that a = b. This enforces that a = b = 0 which is impossible. Thus, the characteristic cone is not convex. We now verify that the strong duality between the robust counterpart and the optimistic counterpart of the dual fas. To see this, note that the robust counterpart is This can equivalent written as min x 1 x 2 s.t. A 0 x 1 A 1 x 2 A 2 0 A i V i, i = 0, 1, 2 min x 1 x 2 0 ax s.t. 1 bx 2 ax 1 bx 2 0 ) 0 a [0, 1], b [ 1, 0]. Note that 0, 0) is the only feasible point for the robust counterpart, so the optimal value is 0. On the other hand, the optimistic counterpart of the dual is ) max max f1 f A i V i,i=0,1,2 F S m{ TrA 0F ) : TrA i F ) = 1, i = 1, 2, F = 2 0} f 2 f 3 = max max a [0,1],b [ 1,0] F S m{0 : 2af 1 = 1 and 2bf 2 = 1}. It can be seen that there is no a [0, 1] and b [ 1, 0] such that 2af 2 = 1 and 2bf 2 = 1. So, the feasible set of the optimistic dual is empty and hence, strong duality fas between the robust counterpart and the optimistic dual. From the above example, we see that the characteristic cone can be nonconvex in general. A sufficient condition ensuring the convexity of the characteristic cone can be found in the appendix. Finally, it is worth noting that, like the robust counterparts RP ), the optimistic counterparts ODP ) can, in general, be difficult to solve for certain classes of uncertainties [4]. This prompts us to examine classes of robust SDP where optimistic counterparts are easy solvable in the next section. 3 Duality with Tractable Optimistic Dual In this Section, we focus on classes of uncertain semi-definite linear programs for which optimistic counterparts can be computationally tractable and strong duality can be easy verified. It should be noted that the computational tractabity of primal robust SDPs may also be checked independently for these classes. We do not intend to identify cases of robust SDPs where duality and optimistic counterparts are easy solvable but the robust counterparts are not necessary tractable. 12

13 Let us first consider the uncertain semi-definite programming problem with spectral norm uncertainty: P s ) x R n{ct 0 x i A i 0} where A i Ṽi := {A i ρ i i : i S m, i spec 1}, A i S m, ρ i 0 and spec denotes the square root of the largest eigenvalue of the matrix T. In this case, the uncertainty set Ṽi is just a closed ball with center A i and radius ρ i in the matrix space S m associated with the spectral norm. The robust counterpart of this uncertain SDP problem is RP s ) x R n{ct 0 x i A i 0, A i {A i ρ i i : i S m, i spec 1}}. The optimistic counterpart of the dual of RP s ) is ODP s ) max max i spec 1, i I F S m{ Tr A 0 ρ 0 0 )F ) : Tr A i ρ i i )F ) = c i, i I\{0}, F 0}. In this case also we show that the robust characteristic cone D is a convex cone. Proposition 3.1. Let Ṽi = {A i ρ i i : i S m, spec 1}, A i S m. Then the robust characteristic cone D := { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0}, is convex. A i Ṽi,i I Proof. First, we note that D = { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0} = A i Ṽi i spec 1 { Tr A 1 ρ 1 1 )F ),..., Tr A n ρ n n )F ), Tr A 0 ρ 0 0 )F ) r) : F 0, r 0}. To see the convexity of D, let x = x 1,..., x n1 ) D and y = y 1,..., y n1 ) D. As D is a cone by Lemma 2.1), it suffices to show that x y D. Since x, y D, there exist H i, C i { : spec 1}, i I, M, N 0 and r, s 0 such that and x i = Tr A i ρ i H i )M ), 1 i n, x n1 = Tr A 0 ρ 0 H 0 )M ) r y i = Tr A i ρ i C i )N ), 1 i n, y n1 = Tr A 0 ρ 0 C 0 )N ) s. New, we show that, for each fixed F 0, [Tr A i ρ i I m )F ), Tr A i ρ i I m )F ) ] = 13 i spec 1 {Tr A i ρ i i )F ) }, i I. 3.9)

14 To see this, as I m, I m { i : i spec 1}, where I m is the m m) identity matrix, and for each fixed F 0, i Tr A i ρ i i )F ) is a continuous function, it follows by the intermediate value theorem that, for each fixed F 0, [Tr A i ρ i I m )F ), Tr A i ρ i I m )F ) ] {Tr A i ρ i i )F ) }, i I. i spec 1 To see the reverse inclusion, note that for each fixed F 0 see [2, Proposition 2.1]), sup Tr F ) = TrF ) and spec 1 So, the reverse inclusion follows as sup {Tr A i ρ i i )F ) } = Tr A i ρ i I m )F ) and i spec 1 Therefore, 3.9) gives us that inf Tr F ) = TrF ). spec 1 inf {Tr A i ρ i i )F ) } = Tr A i ρ i I m )F ). i spec 1 x i [Tr A i ρ i I m )M ), Tr A i ρ i I m )M ) ], 1 i n, x n1 r [Tr A 0 ρ 0 I m )M ), Tr A 0 ρ 0 I m )M ) ] and y i [Tr A i ρ i I m )N ), Tr A i ρ i I m )N ) ], 1 i n, y n1 s [Tr A 0 ρ 0 I m )N ), Tr A 0 ρ 0 I m )N ) ]. Then, and x i y i [Tr A i ρ i I m )M N) ), Tr A i ρ i I m )M N) ) ], 1 i n, x n1 y n1 r s) [Tr A 0 ρ 0 I m )M N) ), A 0 ρ 0 I m )M N) ) ]. Now, applying 3.9) we can find i with spec 1 such that x i y i = Tr A i ρ i i )MN) ), 1 i n, x n1 y n1 rs) = Tr A 0 ρ 0 0 )MN) ). Hence, x y D. We finally establish that strong duality holds for P s ) under the robust Slater condition and that the optimistic counterpart ODP s ) is a single semi-definite linear program. Theorem 3.1. For P s ), suppose that there exists x 0 = x 0 1,..., x 0 n) R n such that Then, A 0 x 0 i A i 0 A i {A i ρ i i : i spec 1}, i I\{0}. x R n{ct 0 x i A i 0, A i {A i ρ i i : i spec 1}} = max F S m{ Tr A 0 ρ 0 I)F ) : Tr A i ρ i I m )F ) c i, i I\{0} Tr A i ρ i I m )F ) c i, i I\{0}, F 0}. 14

15 Proof. From Proposition 3.1 and Theorem 2.3, we see that the cone D is closed and convex. Then, by Theorem 2.1, x R n{ct 0 x i A i 0, A i {A i ρ i i : i spec 1}} = max i spec 1, i I max F S m{ Tr A 0 ρ 0 0 )F ) : Tr A i ρ i i )F ) = c i, i I\{0}, F 0}. Since, 3.9) holds for each F 0, [Tr A i ρ i I m )F ), Tr A i ρ i I m )F ) ] = i spec 1 So, the optimistic counterpart can be simplified as {Tr A i ρ i i )F ) }, i I. 3.10) max max i spec 1, i I F S m{ Tr A 0 ρ 0 0 )F ) : Tr A i ρ i i )F ) = c i, i I\{0}, F 0} = max i spec 1, i I\{0} max F S m{ Tr A 0 ρ 0 I m )F ) : Tr A i ρ i i )F ) = c i, i I\{0}, F 0} = max F S m{ Tr A 0 ρ 0 I m )F ) : Tr A i ρ i I m )F ) c i, i I\{0}, Tr A i ρ i I m )F ) c i, i I\{0}, F 0}. Indeed, for each F 0 with Tr A i ρ i i )F ) = c i, we have Then, Tr A i ρ i I m )F ) c i and Tr A i ρ i I m )F ) c i, i I\{0}. max { Tr A 0 ρ 0 I m )F ) : Tr A i ρ i i )F ) = c i, i I\{0}, F 0} F S m i spec 1, i I\{0} max F S m{ Tr A 0 ρ 0 I m )F ) : Tr A i ρ i I m )F ) c i, i I\{0}, Tr A i ρ i I m )F ) c i, i I\{0}, F 0}. On the other hand, for any F 0 with Tr A i ρ i I m )F ) c i and Tr A i ρ i I m )F ) c i, i I\{0}, the equation 3.10) implies that there exist i with i spec 1 such that Tr A i ρ i i )F ) = c i. This gives us that max { Tr A 0 ρ 0 I m )F ) : Tr A i ρ i i )F ) = c i, i I\{0}, F 0} F S m i spec 1, i I\{0} max F S m{ Tr A 0 ρ 0 I m )F ) : Tr A i ρ i I m )F ) c i, i I\{0} Tr A i ρ i I m )F ) c i, i I\{0}, F 0}. Hence, the equality holds. 15

16 Remark 3.1. Comparison with the known result) It has been established for example see [4, Chapter 8]) that the robust counterpart of the uncertain semi-definite linear programming problem with spectral norm uncertainty can be reformulated as a new semi-definite linear programming, and so, is also computational tractable. Our approach provides an alternative way for identifying this tractable class of robust semi-definite linear program via a dual approach. Remark 3.2. Other simple tractable cases) Consider the semi-definite linear programming model problem with affinely parameterized diagonal matrix data: where for each i = 0, 1,..., n, P a ) inf x R n{ct x : A 0 A i V i = {diag a 0) i1 u j) i1 aj) i1,..., a0) a j) im x i A i 0} u j) im aj) im) : u = u 1,..., u k ) U, l = 1,..., m}, R for each l = 1,..., m and j = 1,..., k and U, l = 1,..., m, are convex compact sets in R k. Here, diagd 1,..., d n ) denotes a diagonal matrix with diagonal elements d 1,..., d n. The characteristic cone D collapses to D = u U { r m m λ l a 0) λ l a 0) 1l u j) 1l aj) 1l ),..., m u j) aj) ) ) : λ l 0, r 0} Let v l = u 1),..., uk), u1) 1l,..., uk) 1l, u1) nl,..., uk) nl ) and g l x, v l ) = a 0) u j) aj) ) x i a 0) λ l a 0) nl u j) a j) ). u j) nl aj) nl ), Then, direct verification gives us that D = v l Q n i=0 U,λ l 0 epi m λ lg l, v l ) ) where epif is the epigraph of a function f given by epif = {x, s) : s fx)} and f denotes the usual convex conjugate of a convex function f. Clearly, g l, v l ) is a linear function and g l x, ) is also linear. Thus [16, Proposition 2.3] implies that v l Q n i=0 U,λ l 0 epi m λ lg l, v l ) ) is convex, and so, the characteristic cone D is convex. 1) If for each i = 0, 1,..., n, l = 1,..., m, U = co{z 1),..., z p) } which is the convex hull of a given finitely generated scenarios z t) = z 1t),..., z kt) ) R k, t = 1,..., p, and p N, then the characteristic cone D is closed and so, strong duality between the robust counterpart and the optimistic dual always holds see [16, Theorem 4.1]). Moreover, the 16

17 optimistic dual can be simplified as max max A i V i,i I F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0} = max { u U,λ l 0 m = max { m λ l a 0) λ l 0 m λ l a 0) λ l a 0) p t=1 µ jt) p t=1 u j) aj) µ jt) ) : m z jt) a j) ) : z jt) a j) ) = c i, p t=1 λ l a 0) µ jt) = 1, µ jt) 0} u j) a j) ) = c i, i = 1,..., n} Letting λ jt) = λ l µ jt) 0, then p t=1 λjt) = λ l, i = 1,..., n, j = 1,..., k, l = 1,..., m, and the optimistic dual can be further simplified as max λ l 0,λ jt) 0 { m λ l a 0) p t=1 λ jt) z jt) a j) : m λ l a 0) p t=1 λ jt) z jt) a j) = c i, p t=1 λ jt) = λ l } which is a linear programming problem. 2) If for each i = 0, 1,..., n, l = 1,..., m, U = {z R k : z 1} and the robust Slater condition holds, strong duality between the robust counterpart and the optimistic dual always holds by Theorem 2.3. Moreover, the optimistic dual can be simplified as max max A i V i,i I F S m{ TrA 0F ) : TrA i F ) = c i, i I\{0}, F 0} = max { u U,λ l 0 = max u 1) m,...,u k) ) 1,λ l 0 λ l a 0) { m u j) aj) λ l a 0) ) : m u j) aj) λ l a 0) ) : m λ l a 0) u j) a j) ) = c i, i = 1,..., n} u j) a j) ) = c i, i = 1,..., n} Letting λ j) = λ l u j), then λ l λ 1),..., λ k) ) and the optimistic dual can be further simplified as max λ l λ 1),...,λ k) ) { m λ l a 0) λ j) aj) : m λ l a 0) which is a second-order cone linear programming problem. 4 Conclusion and Further Research λ j) a j) = c i, i = 1,..., n} In this paper we considered semi-definite linear programs subject to data uncertainty. We have presented a deterministic approach for studying such uncertain programs by 17

18 restricting the uncertain data to closed and convex uncertainty sets. We have established necessary as well as sufficient conditions for strong duality for uncertain SDPs by proving strong duality between the associated deterministic counterparts, namely, the robust counterpart and the optimistic counterpart. Our study presented in this paper raises interesting questions for further research. For instance, the tractabity in robust semi-definite programming often relates to whether the robust counterpart of an uncertain semi-definite program is computationally tractable or not. On the other hand, from the dual point of view, the investigation of the tractabity of optimistic counterpart of the dual of an uncertain semi-definite program would be of great interest, particularly, when strong duality is avaable. We have provided two classes of uncertain semi-definite linear programs for which strong duality holds under a robust Slater condition and the optimistic counterparts are computationally tractable semi-definite linear programs. It would be useful to examine this dual tractabity for other specially structured problems with broad classes of uncertainty sets. Especially, the duality approach, presented in this paper, could be applied to uncertain second-order cone programming problems. This would lead to identifying other broad classes of computationally tractable optimistic counterparts and wl be presented elsewhere. Appendix: Convexity of the Characteristic Cone In this appendix, we provide a sufficient condition for the convexity of the characteristic cone. Consider the uncertain semi-definite programming model problem P a ) x R n{ct 0 x i A i 0} where for each i I, the matrix A i is uncertain and the uncertain data matrix is affinely parameterized, i.e., the matrix data A i belongs to the uncertainty set V i = {A 0) i u j i Aj) i : u 1 i,..., u k i ) U i }, where U i, i = 0, 1,..., n, is a compact convex set in R k and A j) i S m i = 0, 1,..., n. The robust counterpart of this uncertain SDP problem is RP a ) x R n{ct 0 x i A i 0, A i {A 0) i The optimistic counterpart of its dual problem is ODP a ) max u 1 i,...,uk i ) U i i I max { Tr A 0) F S m 0 Tr A 0) i 18 u j 0 Aj) 0 )F ) : u j i Aj) i : u 1 i,..., u k i ) U i }}. u j i Aj) i )F ) = c i, i I\{0}, F 0}.

19 We show in this case that the robust characteristic cone D is a convex cone. Proposition 4.1. Convexity of the Characteristic Cone: Sufficient Condition) For each i I, A i V i = {A 0) i k uj i Aj) i : u 1 i,..., u k i ) U i }, where U i, i = 0, 1,..., n, is a compact convex set in R k, A j) 0 S m and A j) i 0 i = 1,..., n, j = 1,..., k. Then the robust characteristic cone is convex. D := A i V i,i I { TrA 1 F ),..., TrA n F ), TrA 0 F ) r) : F 0, r 0}, Proof. Let x, y D and let λ [0, 1]. As x, y D, there exist H i, C i V i, i I, M, N 0 and r, s 0 such that and x i = TrH i M), i I\{0}, x n1 = TrH 0 M) r, y i = TrC i N), i I\{0}, x n1 = TrC 0 N) s. As H i, C i V i, there exist u 1 i,..., u k i ) U i and vi 1,..., vi k ) U i such that, for each i I, H i = A 0) i u j i Aj) i and C i = A 0) i v j i Aj) i. Now, fix an i I\{0}. We see that λx i 1 λ)y i = TrλH i M 1 λ)c i N) = Tr λa 0) i u j i Aj) i )M 1 λ)a 0) i ) = Tr A 0) i λm 1 λ)n) u j i Tr λa j) i M ) v j i Tr 1 λ)a j) i N )) ) v j i Aj) i )N Define, for each i I and j = 1,..., k, ) u j w j i λa Tr j) i M )v ji Tr 1 λ)a j) i N i = ) if Tr λm 1 λ)n)a j) ) Tr λm1 λ)n)a j) i 0 i u j i if Tr λm 1 λ)n)a j) ) i = 0. Clearly, w 1 i,..., w k i ) U i, i I. Moreover, for each i I\{0} and j = 1,..., k w j i Tr λm 1 λ)n)a j) ) i = u j i Tr λa j) i M ) v j i Tr 1 λ)a j) i N ). 19

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