Robust linear semi-in nite programming duality under uncertainty?

Transcription

1 Mathematical Programming (Series B) manuscript No. (will be inserted by the editor) Robust linear semi-in nite programming duality under uncertainty? Robust linear SIP duality Goberna M.A., Jeyakumar V. 2, Li G. 2, López M.A. Department of Statistics and Operations Research, University of Alicante, 38 Alicante, Spain 2 Department of Applied Mathematics, University of New South Wales, Sydney 252, Australia The date of receipt and acceptance will be inserted by the editor Abstract In this paper, we propose a duality theory for semi-in nite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-in nite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then show that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and su cient for robust duality in the sense that robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with a nely parameterized data uncertainty, we establish that robust duality is easily satis ed under a Slater type constraint quali cation. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-in nite linear inequalities. Key words Robust optimization - semi-in nite linear programming - parameter uncertainty - robust duality - convex programming.? This research was partially supported by ARC Discovery Project DP2 of Australia and by MICINN of Spain, Grant MTM C3-.

2 2 Goberna M.A. et al. Introduction Duality theory has played a key role in the study of semi-in nite programming [3, 5, 6, 22] which traditionally assumes perfect information (that is, accurate values for the input quantities or system parameters), despite the reality that such precise knowledge is rarely available in practice for realworld optimization problems. The data of real-world optimization problems more often than not are uncertain (that is, they are not known exactly at the time of the decision) due to estimation errors, prediction errors or lack of information [3 6]. Robust optimization [2] provides a deterministic framework for studying mathematical programming problems under uncertainty. It is based on a description of uncertainty via sets, as opposed to probability distributions which are generally used in stochastic approaches [7, 24]. A successful treatment of robust optimization approach for linear programming problems as well as convex optimization problems under data uncertainty has been given by Ben-Tal and Nemirovski [3 5], and El Ghaoui [2]. The present work was motivated by the recent development of robust duality theory [, 2] for convex programming problems in the face of data uncertainty. To set the context of this work, consider a standard form of linear semi-in nite programming (SIP) problem in the absence of data uncertainty: (SP ) inf hc; xi s.t. ha t ; xi b t ; 8t 2 T; where T is an arbitrary (possible in nite) index set, c; a t 2 R n, and b t 2 R; t 2 T. The linear SIP problem in the face of data uncertainty in the constraints can be captured by the parameterized model problem (USP ) inf hc; xi s.t. ha t (v t ); xi b t (w t ); 8t 2 T; where a t : V t! R n ; b t : W t! R; V t R q ; W t R q2 ; q ; q 2 2 N: The uncertain set-valued mapping U : T R q ; q = q + q 2 ; is de ned as U t := V t W t for all t 2 T: We represent by u t := (v t ; w t ) 2 V t W t an arbitrary element of U t or a variable ranging on U t : So, gph U = f(t; u t ) : u t 2 U t ; t 2 T g : Throughout the paper, u 2 U means that u is a selection of U; i.e., that u : T! R q and u t 2 U t for all t 2 T (u can be also represented as (u t ) ). In stochastic programming [7,24] each set U t is equipped with a probability distribution and each selection of U determines a scenario for (SP ). As an illustration of the model, consider the uncertain linear SIP problem: inf (x ;x 2)2R 2fx : a t x + a 2 t x 2 b t ; t 2 T g

3 Robust linear semi-in nite programming duality under uncertainty 3 where the data a t ; a 2 t are uncertain, and for each t 2 T := [; ], a t 2 [ 2t; + 2t], a 2 t 2 [=(2 + t); =(2 t)] and b t. Then, this uncertain problem can be captured by our model as inf x s.t. ( + 2vt )x + 2+v 2 t x 2 ; 8t 2 T; where u t = (v t ; v 2 t ; w t ) 2 R 3 is the uncertain parameter and u t 2 U t := V t W t with V t := [ t; t] [ t; t] and W t = fg. In this case, a t (v t ; v 2 t ) = + 2v t ; =(2 + v 2 t ) and b t (w t ). The robust counterpart of (USP ) [,3,5] is (RSP ) inf hc; xi () s.t. ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph U: The problem (RSP) is indeed a linear SIP problem. We assume that (RSP ) is a consistent problem with a closed and convex feasible set F := fx : ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph Ug 6= ;: Now, for each xed selection u 2 U, the Lagrangian dual of (USP ) is ( ) (DP ) t a t (v t ) = ; sup 2R (T ) + t b t (w t ) : c + where R (T ) + denotes the set of mappings : T! R + (also denoted by ( t ) ) such that t = except for nitely many indexes, and sup ; = by convention. The optimistic counterpart [, 2] of the Lagrangian dual of (DP ) is given by ( ) (ODP ) By construction, If inf(rsp ) = sup u=(v t;w t) 2U 2R (T ) + t b t (w t ) : c + inf(rsp ) sup(odp ): t a t (v t ) =, then (ODP ) has no feasible solution, i.e. [ c =2 co conefa t (v t ); t 2 T g; u=(v t;w t) 2U where co conefa t (v t ); t 2 T g denotes the convex cone generated by fa t (v t ); t 2 T g: We say that robust duality holds whenever inf(rsp ) = max(odp ), i.e. ( ) inffhc; xi : x 2 F g = t a t (v t ) = max u=(v t;w t) 2U 2R (T ) + t b t (w t ) : c + :

4 4 Goberna M.A. et al. whenever inf(rsp ) is nite. In this paper, we make the following key contributions: Firstly, we establish that robust duality holds for (USP) whenever the robust moment cone, M := [ u=(v t;w t) 2U co conef(a t (v t ); b t (w t )); t 2 T ; ( n ; )g; is closed and convex. We further show that the closed-convex robust moment cone condition in the case of constraint uncertainty is in fact necessary and su cient for robust duality in the sense that robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. We also derive strong duality between the robust counterpart (RSP) and its standard (or Haar) dual problem [5] in terms of a robust characteristic cone, illustrating the link between the Haar dual and the optimistic dual. Secondly, for the important case of a nely parametrized data uncertainty [2], we show that the convexity of the robust moment cone always holds and that it is closed under a robust Slater constraint quali cation together with suitable topological requirements on the index set and the uncertainty set of the problem. Thirdly, we derive robust forms of Farkas lemma [,, 7] for systems of uncertain semi-in nite linear inequalities in terms of the robust moment cone and the robust characteristic cones. Finally, we extend our results to deal with uncertain linear SIP problems where data uncertainty occurs at both objective function and at constraints. The organization of the paper is as follows. Section 2 provides robust duality theorems under geometric conditions in terms of robust moment and characteristic cones. Section 3 shows that these cone conditions are satis ed if and only if robust duality holds for every linear objective function. Section 4 presents robust Farkas lemma for uncertain semi-in nite linear inequalities. Section 5 extends the results to deal with uncertain linear SIP problems where data uncertainty occurs at both objective function and constraints. 2 Robust Duality Let us introduce the necessary notation. We denote by kk and B n the Euclidean norm and the open unit ball in R n : By n we represent the null vector of R n. For a set C R n ; we de ne its convex hull co C and conical hull cone C as co C = f P m i= ic i : i ; P m i= i = ; c i 2 C; m 2 Ng and cone C = S C; respectively. The topological closure of C is cl C: Given h: R n! R [ f+g; such that h 6= +, the epigraph of h is epih := f(x; r) 2 R n+ : h(x) rg

5 Robust linear semi-in nite programming duality under uncertainty 5 and the conjugate function of h is h : R n! R [ f+g such that h (v) := supfhv; xi h(x) j x 2 domhg: The indicator and the support functions of C are denoted respectively by C and C: Let C R n be a convex set and h: C! R: Identifying h with its extension to R n by de ning h (x) := + for any x =2 C; h is called convex when epih is convex, concave when h is convex, and a ne when it is both convex and concave on C: We de ne the robust moment cone of (RSP ) as [ M = co conef(a t (v t ); b t (w t )); t 2 T ; ( n ; )g: (2) u=(v t;w t) 2U The next academic example shows that M can be neither convex nor closed. Example Consider the simple uncertain linear SIP problem (SP ) inf x x 2 s.t. ha t ; xi b t ; 8t 2 T; where T = [; ]; a is uncertain on the set V = f(cos ; sin ) : 2 [; 2] \ Qg (a dense subset of the circle fx : kxk = g) whereas the remaining data are deterministic: b = and (a t ; b t ) = ( 2 ; ) for t 2 ]; ] : This uncertain problem can be modeled as (USP ) inf x x 2 s.t. ha t (v t ); xi b t (w t ); 8t 2 T; with uncertain mapping U such that U = V f g and U t = f(; ; )g for all t 2 ]; ] ; and associated functions a (v ) = v ; a t (v t ) = v t = 2 ; for all t 2 ]; ] ; and b t (w t ) = w t =, for all t 2 [; ]: Observe that there exists a one-to-one correspondence between the selections of U and the elements of V because the unique uncertain constraint is ha ; xi b : So, the robust counterpart and the robust moment cone are and (RSP ) inf x x 2 s.t. hv ; xi ; 8v 2 V ; h 2 ; xi ; 8t 2 ]; ] ; M = S v 2V co conef( v ; ) ; ( 2 ; )g; respectively. Thus M is the union of countable many 2-dimensional convex cones having a common edge on the vertical axis. Obviously, pm is neither p convex nor closed. As F = cl B; the unique optimal solution is 2=2; 2=2 and min(rsp ) = p 2: Concerning the robust dual problem (ODP ); it is inconsistent because (; ) =2 co conefv ; 2 g = R + fv g whichever v 2 V we consider (cos 6= sin for any 2 [; 2]\Q); so that max(odp ) = by convention, i.e., there is an in nite duality gap.

6 6 Goberna M.A. et al. Theorem If the robust moment cone M is closed and convex, then inf(rsp ) = max(odp ): Proof. Let := inf(rsp ) 2 R. It follows that [ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph U] ) hc; xi ; (3) where u t = (v t ; w t ): De ne g : R n! R by g(x) = sup f ha t (v t ); xi + b t (w t )g: As g is supremum of a collection of a ne functions, g is convex, F = fx : g(x) g; and (3) is equivalent to g(x) ) hc; xi : Let f(x) := hc; xi + F (x). This implies that f(x) for all x 2 R n. So, we have (see, e.g., [9] and [23]) (; ) 2 epif = cl(epi(hc; :i) + epi F ) Note that [g] = sup (g) and = epi(hc; :i) + epi F = fcg R + + epi F : epi sup f i = cl co i2i (see, e.g., [4, (2.3)]). It follows that! [ epifi i2i (4) epi F = epi sup (g) = cl [ epi(g) A : As g = sup (t;ut)2gph Ufh a t (v t ); xi+b t (w t )g, we have by applying again (4) epi(g) = cl [ [( a t (v t ); b t (w t )) + f n g R + ] A :

7 Robust linear semi-in nite programming duality under uncertainty 7 This gives us that epi F = cl [ epi(g) A 8 9 = cl [ < cl [ = [ (a t (v t ); b t (w t )) + f n g R + ] A A : ; = cl [ [ [ (a t (v t ); b t (w t )) + f n g R + ] A (5) = cl [ [ [(a t (v t ); b t (w t )) + f n g R ] A : On the other hand, Lemma 5. of Appendix shows that [ [ [(a t (v t ); b t (w t )) + f n g R ] A = co M; and hence epi F = cl co M: (6) As M is closed and convex by our assumption, we have epi F = so, (; ) 2 (fcg R + ) M: M, and This implies that there exists bu = (bv t ; bw t ) 2 U and b 2 R (T ) + such that c = b t a t (bv t ) and b t b t ( bw t ): So, we see that P tb t (w t ) max(odp ). Thus the conclusion follows from the weak duality, and we also conclude that (bu; ) b is optimal for (ODP ). Observe that (ODP ) is more easily manageable than the ordinary (or Haar) dual problem of (RSP ): P (DRSP ) sup (gph U) 2R (t;u + t)2gph U (t;u t)b (t;ut)(w (t;ut)) s.t. c + P (7) (t;u t)a (t;ut)(v (t;ut)) = : Now consider the so-called characteristic cone of (RSP ) K = co conef(a t (v t ); b t (w t )); (t; u t ) 2 gph U; (; )g: (8) Then (DRSP ) is equivalent to sup f : (c; ) 2 Kg in the sense that both problems have the same optimal value and are simultaneously solvable or not. It is worth observing that K = co M: In fact, co M K because

8 8 Goberna M.A. et al. M K trivially and K is convex. To show the reverse inclusion, take an arbitrary generator of K di erent of (; ) 2 M; say (a s (v s ); b s (w s )); with (s; u s ) 2 gph U: As (a s (v s ); b s (w s )) 2 co conef(a t (v t ); b t (w t )); t 2 T ; (; )g; we have (a s (v s ); b s (w s )) 2 M: Thus, K co M and so K = co M: From the linear SIP strong duality theorem, inf(rsp ) = max(drsp ) whenever K is closed (see, e.g., [5, Chapter 8]). If M is closed and convex, then K = co M = M is closed and so inf(rsp ) = max(drsp ) = max f : (c; ) 2 Kg = max f : (c; ) 2 Mg = max(odp ): Thus, we have obtained an alternative proof of Theorem appealing to linear SIP machinery. In Example, the characteristic cone K = co n conef( v ; ) ; v 2 V ; (; )g = x 2 R 3 : x 3 < p o x 2 + x2 2 [ cone fv f gg is not closed, (DRSP ) is not solvable and inf(rsp ) = p 2 = sup(drsp ) > sup(odp ) = : (The equality inf(rsp ) = sup(drsp ) comes from [5, Theorem 8.(v)].) 3 Robust Moment Cones: Convexity and Closure We say that (RSP ) satis es the convexity condition if for every t 2 T; a t () is a ne, i.e., a t (v t ) = (a t (v t ); : : : ; a n t (v t )) and each a j t() is an a ne function, j = ; : : : ; n, and b t () is concave. Proposition Let U be convex-valued. Suppose that (RSP ) satis es the convexity condition. Then the robust moment cone M is convex. Proof. Let a ; a 2 2 M and 2 [; ]. Let a := a +( )a 2 and denote a = (z; ) 2 R n R, with a = (z ; ) 2 R n R and a 2 = (z 2 ; 2 ) 2 R n R. Then, there exist u = (vt ; wt ) 2 U; = ( t ) 2 R (T ) +, and such that (z ; ) = t (a t (vt ); b t (wt )) + (; ): Similarly, there exist u 2 = (v 2 t ; w 2 t ) 2 U; 2 = ( 2 t ) 2 R (T ) + and 2 such that (z 2 ; 2 ) = 2 t (a t (v 2 t ); b t (w 2 t )) + (; 2 ):

9 Robust linear semi-in nite programming duality under uncertainty 9 Then, we have z + ( )z 2 = t a t (v t ) + ( ) 2 t a t (v 2 t ) and + ( ) 2 = t b t (w t ) + ( ) 2 t b t (w 2 t ) + + ( ) 2 : We associate with t 2 T the scalar t := t + ( ) 2 t and the vector (v t ; w t ) := Then (v t ; w t ) 2 U t and ( (v t ; w t ); if t = ; t t (v t ; w t ) + ( )2 t t (v 2 t ; w 2 t ); if t > : t (v t ; w t ) + ( ) 2 t (v 2 t ; w 2 t ) = t (v t ; w t ): By our convexity assumption, we see that, for each t 2 T, and t a t (v t ) + ( ) 2 t a t (v 2 t ) = t a t (v t ); (9) t b t (w t ) + ( ) 2 t b t (w 2 t ) t b t (w t ): () Then, there will exist such that a = (z; ) = (z ; ) + ( )(z 2 ; 2 ) = (z + ( )z 2 ; + ( ) 2 )! = t a t (v t ); t b t (w t ) + ( n ; + ( ) 2 + ) = t (a t (v t ); b t (w t )) + ( ( ) 2 )( n ; ); and this implies that a = (z; ) 2 M: In particular, the robust moment cone M is convex in the important a nely data parametrization case [2], i.e. U t = f(a t ; b t ) = (a t ; b t ) + q u j t(a j t; b j t) : (u t ; : : : ; u q t ) 2 Z t g; j= where Z t is closed and convex for each t 2 T. In this case, U is convex-valued and the convexity condition holds with b t () being also a ne. It is easy to see from Example that M may not be convex when U is not convex-valued. In fact, in this example, U = V fg is not even connected.

10 Goberna M.A. et al. The set-valued mapping U : T R q ; with (T; d) being a metric space, is said to be (Hausdor ) upper semicontinuous at t 2 T if for any > there exists > such that U s U t + B q 8 s 2 T with d(s; t) : In particular, U is uniformly upper semicontinuous on T if for any > there exists > such that U s U t + B q 8 s; t 2 T with d(s; t) : If, additionally, T is compact, then there exists a nite set ft ; :::; t m g T such that d (t; ft ; :::; t m g) < for all t 2 T: Then, U s S i=;:::;m (U t i + B q ) 8 s 2 T; so that gph U is bounded whenever U is compact-valued, i.e., for each t 2 T, U t is a compact set. We say that (RSP ) satis es the Slater condition when there exists x 2 R n (called Slater point) such that ha t (v t ); x i > b t (w t ) for all (t; u t ) 2 gph U; i.e. when there exists a strict solution of the constraint system of (RSP ): Proposition 2 Suppose that the following three assumptions hold: (i) T is a compact metric space; (ii) U is compact-valued and uniformly upper semicontinuous on T ; (iii) the mapping gph U 3 (t; u t ) 7! (a t (v t ) ; b t (w t )) 2 R n+ () is continuous on gph U; (iv) (RSP ) satis es the Slater condition. Then, the robust moment cone M is closed. Remark (before the proof) (ii) and (iii) conjointly imply that a t (V t ) and b t (W t ) are compact. From now on we shall call (iii) continuity condition. Proof. Let (z k ; r k ) 2 M = [ co conef(a t (v t ); b t (w t )) [ ( n ; ) : t 2 T g; k = ; 2; :::; u2u such that (z k ; r k )! (z; r). Then, for each k, there exists u k u k t = (vt k ; wt k ) 2 U t for all t 2 T, such that 2 U; with (z k ; r k ) 2 co conef(a t (v k t ); b t (w k t )) [ ( n ; ) : t 2 T g: From the Carathéodory theorem, we can nd k i, i = ; : : : ; n + 2, k, ft k ; : : : ; t k n+2g T and (vi k; wk i ) 2 U t, i = ; : : : ; n + 2, such that k i n+2 (z k ; r k ) = k i (a t k i (vi k ); b t k i (wi k )) + k ( n ; ): (2) i= As T is compact, we may assume that t k i! t i 2 T, i = ; : : : ; n + 2.

11 Robust linear semi-in nite programming duality under uncertainty Fix i = ; : : : ; n + 2. Since we are assuming that U is uniformly upper semicontinuous and so, for any >, there exists > such that U t U ti + B q ; for all t such that d(t; t i ) : It follows that d((v k i ; w k i ); U ti )! as k! : Since U ti is compact, we may assume the existence of (v i ; w i ) 2 U ti such that As (RSP ) satis es (iii) (v k i ; w k i )! (v i ; w i ) as k! : (3) (a t k i (v k i ); b t k i (w k i ))! (a ti (v i ); b ti (w i )) as k! : Now, we show that l k := P n+2 i= k i + k is bounded. Granting this, by passing to subsequence if necessary, we may assume that k i! i 2 R + and k! 2 R + ; as each k i and k are non-negative. Then, passing to the limit in (2) we have n+2 (z; r) = i (a ti (v i ); b ti (w i )) + ( n ; ) 2 M: i= Therefore, the robust moment cone M is closed. To show the boundedness of l k, we proceed by the method of contradiction and assume without loss of generality that l k := P n+2 i= k i + k! +: By passing to subsequence if necessary, we may assume that k i l k! i 2 R +, k l k! 2 R + and n+2 i + = : (4) i= Dividing by l k both members of (2) and passing to the limit, we obtain that n+2 ( n ; ) = i (a ti (v i ); b ti (w i )) + ( n ; ): i= So, we have P n+2 i= ia ti (v i ) = n and P n+2 i= ib ti (w i ) = and so, taking a Slater point x ; we have n+2 i (ha ti (v i ); x i b ti (w i )) = : i=

12 2 Goberna M.A. et al. On the other hand, since (v i ; w i ) 2 U ti, assumption (iii) implies that ha ti (v i ); x i b ti (w i ) > for all i = ; : : : ; n + 2. Note that ( ; : : : ; n+2 ) 6= n+2 (otherwise, = P n+2 i= ir i = and so, ( ; : : : ; n+2 ; ) = n+3 which contradicts (4)). This implies that n+2 i (ha ti (v i ); x i b ti (w i )) > : i= This is a contradiction and so, fl k g is a bounded sequence. Example violates assumptions (ii) and (iii) in Proposition 2 because U is neither compact nor convex, U is not upper semicontinuous at ; and the function in () is not continuous. In fact, if we consider the sequence (t k ; u tk ) = (=k; ( 2 ; )) which converges to (; ( 2 ; )); we have (a tk (v tk ) ; b tk (w tk )) = ( 2 ; ) which does not converge to (a (v ) ; b (w )) as v = 2 =2 V : As an immediate consequence of the previous results, we obtain the following su cient condition for robust duality. In the special case when jt j < +, this result collapses to the robust strong duality result for linear programming problems in []. Corollary Suppose that the following assumptions hold: (i) T is a compact metric space; (ii) U is compact-convex-valued and uniformly upper semicontinuous on T ; (iii) (RSP ) satis es the convexity, the continuity and the Slater conditions. Then, the robust duality holds, i.e. inf(rsp ) = max(odp ). Proof. The conclusion follows from Theorem, Proposition, and Proposition 2. We now present an example verifying Corollary. Example 2 Let T = [; ] and consider the following uncertain linear SIP problem: inf x s.t. a t x b t ; t 2 T; where the data a t ; b t are uncertain, a t 2 [ 2t; + 2t] and b t. This uncertain problem can be captured by our model as inf x s.t. a t (v t )x b t (w t ); t 2 T; where (v t ; w t ) are the uncertain parameter, a t (v t ) := + 2v t, v t 2 V t := [ t; t] and b t (w t ) :=. Let U t := V t fg: Then the robust counterpart is inf x s.t. a t (v t )x b t (w t ); 8 (t; u t ) 2 gph U:

13 Robust linear semi-in nite programming duality under uncertainty 3 t : It can be veri ed that the feasible set is [ ; =3] : So, the optimal value of the robust counterpart is. The optimistic counterpart of the dual problem is (ODP ) sup f 2R (T ) + ; u2u + t ( + 2v t ) = g: Let 2 R (T ) + such that = and t = for all t 2 T nfg. Let u = (v; ) 2 U such that v = : Then + P t( + 2v t ) = + ( + 2v ) = P t = : So, max(odp ) = and the robust strong dual- and ity holds. In fact, M = co cone f( 3; ) ; ( ; )g is closed and convex. Finally, one can see that all the conditions in the preceding corollary are satis ed. The following theorem shows that our assumption is indeed a characterization for robust strong duality in the sense that the convexity and closedness of the robust moment cone hold if and only if the robust strong duality holds for each linear objective function of (RSP ). Theorem 2 The following statements are equivalent to each other: (i) For all c 2 R n, ( ) inffhc; xi : x 2 F g = t a t (v t ) = : max 2R (T ) + ; u2u t b t (w t ) : c + (ii) The robust moment cone M is closed and convex. Proof. [(ii) ) (i)] It follows by Theorem. [(i) ) (ii)] We proceed by contradiction and let (c ; r ) 2 (cl co M ) n M: So, (6) implies that ( c ; r ) 2 epi F where F := fx : g(x) g and Thus, we have So, for every x 2 F, It follows that g(x) = sup f F ( c ) r : hc ; xi r : ha t (v t ); xi + b t (w t )g: r inffhc ; xi : ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph Ug: Thus, the statement (i) gives that ( r max 2R (T ) + ;u2u t b t (w t ) : c + t a t (v t ) = ) :

14 4 Goberna M.A. et al. and so, there exists 2 R (T ) + and (v t ; w t ) 2 U t for all t 2 T; with c + P ta t (v t ) = and such that r t b t (w t ): This shows that (z ; r ) 2 co conef(a t (v t ); b t (w t )); t 2 T ; (; )g Mg; which constitutes a contradiction. 4 Robust Semi-in nite Farkas Lemma In this Section, as consequences of robust duality results of previous Sections, we derive two forms of robust Farkas lemma for a system of uncertain semi-in nite linear inequalities. Related results may be found in [8, 8 2]. Corollary 2 (Robust Farkas Lemma: Characterization I) The following statements are equivalent to each other: (i) For all c 2 R n, the following statements are equivalent: ) [ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph U] ) hc; xi r 2) 9 = ( t ) 2 R (T ) + and (v t ; w t ) 2 U t ; t 2 T; P c + such that ta t (v t ) = and P tb t (w t ) r (ii) The robust moment cone M is closed and convex. Proof. The conclusion follows immediately from Theorem 2. Next we compare the previous results with similar ones involving the characteristic cone K de ned in (8) and the standard dual (DRSP ) introduced in (7) instead of M and (ODP ), respectively. Recall that, the assumptions in Proposition and 2 guarantee that the robust moment cone M is convex and closed (and so, the characteristic cone K is also closed). In the following, we show that the assumptions in Proposition 2 alone ensure that the characteristic cone K is closed. Proposition 3 Under the same assumptions as in Proposition 2, the characteristic cone K is closed and inf(rsp ) = max(drsp ) Proof. We rst prove that the characteristic cone K of the robust linear SIP problem is closed. Condition (ii) in Proposition 2 guarantees the closedness of the index set gph U: In fact, let f(t r ; v r ; w r )g gph U be a sequence such that (t r ; v r ; w r )! (t; v; w) 2 T R q+q2 : Then (v r ; w r ) 2 U tr for all r 2 N; t r! t; and (v r ; w r )! (v; w) : Assume that (t; v; w) =2 gph U; i.e. that (v; w) =2 U t : Since U t is closed, there exists > such that (v; w) =2 U t +B q :

15 Robust linear semi-in nite programming duality under uncertainty 5 Let > be such that U s U t + 2 B q for any s 2 T with d(s; t) : We have d(t r ; t) and d ((v r ; w r ) ; (v; w)) < 2 for su ciently large r; so that (v r ; w r ) 2 U tr U t + 2 B q and (v; w) 2 U t + B q (contradiction). We have also seen that condition (ii) implies the boundedness of gph U; which turns out to be compact. On the other hand, condition (iii) states that the mapping from gph U to R n+ such that (t; u t ) 7! (a t (v t ) ; b t (w t )) is continuous and so, the set f(a t (v t ); b t (w t )); (t; u t ) 2 gph Ug is compact. As we are assuming that the Slater condition holds, K is closed by [5, Theorem 5.3 (ii)]. Finally, the closedness of K and [5, (8.5)-(8.6)] imply inf(rsp ) = max(drsp ): We have seen that, under the assumptions of Proposition 2, the characteristic cone K is closed, and this entails that any Slater point is an interior point of F ([5, Theorem 5.9 (iv)]). Thus, F is full dimensional. Proposition 4 The following statements are equivalent to each other: (i) For all c 2 R n, n P inffhc; xi : x 2 F g = max (gph U) (t;ut)2gph U (t;u t)b (t;ut)(w (t;ut)) : 2R + c + P o (t;u t)a (t;ut)(v (t;ut)) = : (ii) The characteristic cone K is closed. Proof. It is straightforward consequence of [5, Theorem 8.4]. Corollary 3 (Robust Farkas Lemma: Characterization II) The following statements are equivalent to each other: (i) For all c 2 R n, the following statements are equivalent: ) [ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph U] ) hc; xi r: (gph U) 2 ) 9 = ( t ) 82 R + >< c + such that >: and P P (t;ut)a (t;ut)(v (t;ut)) = ; (t;ut)b (t;ut)(w (t;ut)) r: (ii) The characteristic cone K is closed. Proof. It is straightforward consequence of Proposition 4. 5 Uncertainty in all the data In this section, we consider linear SIP problems where uncertainty occurs in both objective function and in the constraints. This situation can be modeled as the parameterized linear SIP problem ( ]USP ) inf x2r n hc (v s ) ; xi s.t. ha t (v t ); xi b t (w t ); 8t 2 T; (5)

16 6 Goberna M.A. et al. where the uncertain parameter v s ranges on some set Z R q3 and c : Z! R n. For convenience, we always assume that s =2 T. The problem ( ]USP ) can be equivalently rewritten as inf (x;y)2rn R y s.t. ha t (v t ); xi b t (w t ); 8t 2 T; y hc (v s ) ; xi So, the robust (or pessimistic) counterpart of ( ]USP ) can be formulated as ( ]RSP ) inf (x;y)2rn R y s.t. ha t (v t ); xi b t (w t ); 8 (t; u t ) 2 gph U; y hc (v s ) ; xi ; 8v s 2 Z; (6) (7) whose decision space is R R n : In other words, the robust counterpart ( ]RSP ) is indeed a linear SIP problem with n+ decision variables y; x ; :::; x n and deterministic objective function as follows: ( ]RSP ) inf (x;y)2rn R h(; n ) ; (y; x)i s.t. hea t (v t ); (y; x)i e b t (w t ); 8 (t; u t ) 2 gph e U; (8) where x = (x ; :::; x n ) ; T e := T [fsg (as s =2 T ), q := q +q 2 +q 3 ; U e : T e R q is the extension of U : T R q to T e which results of de ning U e s := Z fg ; and the functions ea t : V t! R n+ and e b t : W t! R are (; c (vs )) ; if t = s; ea t (v t ) := (; a t (v t )) ; otherwise, and e bt (w t ) := ; if t = s; b t (w t ); otherwise. The constraint system of the optimistic counterpart for ( ]RSP ), say (^ODP ), is t + = ; a t (v t ) c (v s ) where v t 2 V t ; t 2 T; v s 2 Z; 2 R (T ) + ; and 2 R + : Eliminating = we get ( ) (^ODP ) sup u2u;v s2z;2r (T ) + t b t (w t ) : c (v s ) + Finally, the robust moment cone of ( ]RSP ) is fm = S co conef(ea t (v t ); e b t (w t )); t 2 T e ; (; )g = u2 e U S u=(v t;w t) 2U v s2z t a t (v t ) = co conef(; a t (v t ); b t (w t )); t 2 T ; (; c (v s ) ; ) ; (; ; )g: :

17 Robust linear semi-in nite programming duality under uncertainty 7 Proposition 5 Suppose that U is convex-valued, that Z is a convex set, and that all the components of c () and of a t (), t 2 T; are a ne whereas the functions b t (); t 2 T; are concave. Then the robust moment cone f M is convex. Proof. By assumption, U e t = U t is convex for all t 2 T and U e s = Z fg is a convex too, so that U e is convex-valued. Moreover, ( ]RSP ) satis es the convexity condition. The conclusion follows from Proposition. It easily follows from Proposition 5 that the robust moment cone f M is convex in the case where the data (a t ; b t ; c) are a ne. Proposition 6 Suppose that the following assumptions hold: (i) T is a compact metric space; (ii) U is compact-valued and uniformly upper semicontinuous on T, and Z is compact; (iii) c : Z! R n is continuous and the mapping from gph U in R n+ such that (t; u t ) 7! (a t (v t ) ; b t (w t )) 2 R n+ is continuous on gph U; (iv) there exists x 2 R n such that ha t (v t ); x i > b t (w t ) for all (t; u t ) 2 gph U: Then, the robust moment cone f M is closed. Proof. By assumption (i), we can take a scalar greater than the diameter of T: We extend the distance d to T e by de ning d (s; t) = d(t; s) = for all t 2 T and d (s; s) = : Obviously, ( T e ; d) is a compact metric space. By assumption (ii), U e is compact-valued since U is compact-valued and eu s = Z fg. Given > ; there exists > such that U t U t + B q for all t ; t 2 T with d(t ; t) < : So, if t ; t 2 T e are such that t 6= s 6= t; we have U e t U e t + B q : Otherwise, d(t ; t) < implies that t = t = s; so that eu t U e t + B q trivially holds. So, U e is uniformly upper semicontinuous on et : By (iii), and by the isolation of s in T e ; the mapping gph e U 3 (t; u t ) 7! (ea t (v t ) ; e b t (w t )) 2 R n+2 is continuous on gph e U: Moreover, (y ; x ) is a Slater point for ( ]RSP ) whenever y > max vs2z hc (v s ) ; x i : The conclusion follows from Proposition 2 applied to ( ]RSP ) and f M: Theorem 3 Suppose that the following assumptions hold: (i) T is a compact metric space; (ii) U is compact-convex-valued and uniformly upper semicontinuous on T; and Z is a compact and convex subset of R q3 ; (iii) c : Z! R n is continuous and the mapping from gph U in R n+ such that (t; u t ) 7! (a t (v t ) ; b t (w t )) 2 R n+ is continuous; (iv) The components of c () and of a t (), t 2 T; are a ne whereas the functions b t (); t 2 T; are concave;

18 8 Goberna M.A. et al. (v) There exists x 2 R n such that ha t (v t ); x i > b t (w t ) for all (t; u t ) 2 gph U: Then, the robust duality holds, i.e. inf( ]RSP ) = max(^odp ): Proof From Proposition 5 and Proposition 6, we see that the robust moment cone ~ M is closed convex. Thus the conclusion follows from Theorem. Uncertainty in the objective function In the particular case of uncertainty in the objective and deterministic constraints, the parameterized associated problem is ( [USP ) inf hc(v s ); xi s.t. ha t ; xi b t ; 8t 2 T; where c : Z! R n ; Z R q3 ; and a t 2 R n ; b t 2 R for all t 2 T, s =2 T: Its robust counterpart is the particular case of ( ]RSP ) in (8) in which eu t = Z fg ; if t = s; ftg ftg ; otherwise, (so that V t = ftg and W t = ftg), (; c (vs )) ; if t = s; ea t (v t ) := (; a t ) ; otherwise, e bt (w t ) := ; if t = s; b t ; otherwise, the robust moment cone is fm = S co conef(; a t ; b t ); t 2 T ; (; c (v s ) ; ) ; (; ; )g; v s2z and nally, the optimistic counterpart is ( (^ODP ) sup v s2z;2r (T ) + t b t : c (v s ) + t a t = As a consequence of Proposition 5 and 6, we obtain robust duality for ( [USP ). Corollary 4 Suppose that the following assumptions hold: (i) T is a compact metric space and the functions t 7! a t and t 7! b t are continuous on T ; (ii) Z is compact and convex, and c : Z! R n is continuous; (iii) all the components of c () are a ne; (iv) there exists x 2 R n such that ha t ; x i > b t for all t 2 T: Then, the robust duality holds, i.e. inf( ]RSP ) = max(^odp ): Proof The conclusion follows from Proposition 4., Proposition 4.2 and Theorem 3. ) :

19 Robust linear semi-in nite programming duality under uncertainty 9 6 Appendix In this Section, we provide a technical lemma which is used in the proof of Theorem. This lemma provides a characterization for the convex hull of the robust moment cone. Lemma Let T be an arbitrary index set, a t : V t! R n ; b t : W t! R; V t R q ; W t R q2, t 2 T and q ; q 2 2 N: Let u t = (v t ; w t ), t 2 T, and let the robust moment cone M be de ned as in (2). Then, we have co M = [ [ (a t (v t ); b t (w t )) + fg ( ; ] A : Proof. [] Let (z; r) 2 [ [ (a t (v t ); b t (w t ))+fg( ; ]. Then, there exists, s 2 T and (v s ; w s ) 2 U s such that (z; r) 2 (a s (v s ); b s (w s )) + fg ( ; ]: So, Letting = b s (w s ) z = a s (v s ) and r b s (w s ): r, this implies that (z; r) = (a s (v s ); b s (w s )) + (; ) [ 2 co conef(a t (v t ); b t (w t )); t 2 T ; (; u=(v t;w t) 2U )g: Thus, the de nition of M gives us that [ [ (a t (v t ); b t (w t )) + fg ( ; ] M; and so, [ [ (a t (v t ); b t (w t )) + fg ( ; ] A co M: [] Let (z; r) 2 M. Then, there exists u = (v t ; w t ) 2 gph U such that (z; r) 2 co conef(a t (v t ); b t (w t )); t 2 T ; (; )g: Note that conef(a t (v t ); b t (w t )) [ (; ) : t 2 T g [ [ (a t (v t ); b t (w t )) + fg ( ; ] :

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