B-469 Simplified Copositive and Lagrangian Relaxations for Linearly Constrained Quadratic Optimization Problems in Continuous and Binary Variables

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1 B-469 Simplified Copositive and Lagrangian Relaxations for Linearly Constrained Qadratic Optimization Problems in Continos and Binary Variables Naohiko Arima, Snyong Kim and Masakaz Kojima October 2012, Revised November 2012 Abstract. For a qadratic optimization problem QOP with linear eqality constraints in continos nonnegative variables and binary variables, we propose three relaxations in simplified forms with a parameter λ: Lagrangian, completely positive, and copositive relaxations. hese relaxations are obtained by redcing the QOP to an eqivalent QOP with a single qadratic eqality constraint in nonnegative variables, and applying the Lagrangian relaxation to the reslting QOP. As a reslt, an nconstrained QOP with a Lagrangian mltiplier λ in nonnegative variables is obtained. he other two relaxations are a primal-dal pair of a completely positive programming CPP relaxation in a variable matrix with the pper-left element set to 1 and a copositive programming CP relaxation in a single variable. he CPP relaxation is derived from the nconstrained QOP with the parameter λ, based on the recent reslt by Arima, Kim and Kojima. he three relaxations with a same parameter vale λ > 0 work as relaxations of the original QOP. he optimal vales ζλ of the three relaxations coincide, and monotonically converge to the optimal vale of the original QOP as λ tends to infinity nder a moderate assmption. he parameter λ serves as a penalty parameter when it is chosen to be positive. hs, the standard theory on the penalty fnction method can be applied to establish the convergence. Key words. Nonconvex qadratic optimization, 0-1 mixed integer program, Lagrangian relaxation, copositve programming relaxation, completely positive programming relaxation. AMS Classification. 90C20, 90C25, 90C26 Department of Mathematical and Compting Sciences, okyo Institte of echnology, Oh-Okayama, Megro-k, okyo Japan. arima.n.ab@m.titech.ac.jp Department of Mathematics, Ewha W. University, 11-1 Dahyn-dong, Sdaemoon-g, Seol Korea. he research was spported by NRF 2012-R1A1A and NRF skim@ewha.ac.kr Department of Mathematical and Compting Sciences, okyo Institte of echnology, Oh-Okayama, Megro-k, okyo Japan, and Research and Development Initiative & JS CRES, Cho University, , Kasga, Bnkyo-k, okyo Japan. his research was spported by Grant-in-Aid for Scientific Research B and the Japan Science and echnology Agency JS, the Core Research of Evoltionary Science and echnology CRES research project. kojima@is.titech.ac.jp 1

2 1 Introdction We consider a class of linearly constrained qadratic optimization problems QOPs in continos nonnegative and binary variables. he standard 0-1 mixed integer linear optimization problem is inclded in this class as a special case. his class of QOPs has long been stdied for varios soltion methods, inclding semidefinite programming SDP relaxation. Recently, completely positive programming CPP relaxation for this class of QOPs was proposed by Brer [3]. he class was extended to a more general class of QOPs by Eichfelder and Povh [6, 7] and by Arima, Kim and Kojima [1]. heoretically strong reslts were presented in their papers [3, 6, 7, 1] showing that the exact optimal vales of QOPs in their classes coincide with the optimal vales of their CPP relaxation problems. A general CPP problem is characterized as linear optimization problems over closed convex cones as a general SDP problem. It is well-known, however, that solving a general CPP problem, or even a simple CPP problem derived as the CPP relaxation problem of the QOP over the simplex [2] is mch more difficlt than solving a general SDP problem. Efficient nmerical methods for solving CPPs have not been developed whereas the primal-dal interior-point methods have been effective for solving SDPs. In fact, the fndamental problem of determining whether a given variable matrix is completely positive or copositive still remains a very challenging problem. If developing efficient nmerical methods for CPP relaxations from QOPs in the classes is an important goal to achieve in the ftre, a first step toward that goal may be representing the QOPs in a simplified form. We say that the QOPs are in a simplified form if the nmbers of constrains and variables are redced. hen, CPP relaxations derived from the simplified QOPs have a redced nmber of constraints and variables, alleviating some of difficlties of handling a large nmber of constrains and variables. his will decrease the difficlty of solving CPP relaxations. For this prpose, three types of extremely simple relaxations are proposed for a class of linearly constrained QOPs in continos and nonnegative variables and binary variables, the class stdied in Brer [3]. he first relaxation is an nconstrained QOP in nonnegative variables. he other two relaxations are a primal-dal pair of an nconstrained CPP problem in a variable matrix whose pper-left element is fixed to 1 and a copositive programming CP problem in a single variable. We may regard that sch a CPP is one of the simplest CPPs, except trivial ones over the completely positive cone with no constraint, and that sch a CP problem is one of the simplest CPs, except trivial ones with no variable. If the problem of determining whether a given variable matrix is completely positive is resolved in the ftre [15], the proposed relaxations can be sed for designing efficient nmerical methods for solving the class of QOPs. A techniqe to redce a linearly constrained QOP in continos nonnegative variables and binary variables to a QOP with a single qadratic eqality constraint in nonnegative variables was introdced by Arima, Kim and Kojima in [1]. he reslting QOP was relaxed to a CPP problem with a single linear eqality constraint in a variable matrix with pper-left element fixed to 1. hey showed that the optimal vale of the CPP relaxation problem coincides with the optimal vale of the original QOP. aking the dal of the CPP problem leads to a CP problem in two variables. As will be shown in the sbseqent section, the dal CP has no optimal soltion in general. his is the second motivation of this paper. he first proposed relaxation is obtained by applying the Lagrangian relaxation to the QOP with a single qadratic eqality constraint in nonnegative variables, which has been redced from the given QOPs sing the techniqe in [1]. he application of the Lagrangian relaxation reslts in an nconstrained QOP with the Lagrangian mltiplier parameter λ in nonnegative variables. For 2

3 any fixed λ, the optimal vale of this nconstrained QOP in nonnegative variables, denoted by ζλ, bonds the optimal vale of the original linearly constrained QOP in continos nonnegative variables and binary variables, denoted by ζ, from below. If the Lagrangian mltiplier parameter λ is chosen to be positive, λ works as a penalty parameter. hs, the standard theory on the penalty fnction method [8] can be tilized to prove that the optimal vale ζλ of the nconstrained QOP with the parameter vale λ > 0 monotonically converges to ζ as λ tends to nder a moderate assmption. In addition, the nconstrained QOP may play an important role for developing nmerical methods. More precisely, if the feasible region of the original QOP can be scaled into the nit box [0, 1] n, then the box constraints can be added to the nconstrained QOP. Many global optimization techniqe developed for this type of QOPs can be sed. See, for example, [4, 5]. he application of CPP relaxation in [1] to the QOP with the parameter λ > 0 provides a primal-dal pair of an nconstrained CPP problem with the parameter λ > 0 in a variable matrix with the pper-left element fixed to 1 and a CP problem with the parameter λ in a single variable. We show nder the same moderate assmption that the primal-dal pair of problems with the parameter vale λ > 0 both have optimal soltions with no dality gap, and share the optimal vale ζλ with the QOP for the parameter vale λ > 0. After introdcing notation and symbols in Section 2, we state or main reslts, heorems 3.1 and 3.3 in Section 3. We give a proof of heorem 3.1 in Section 4, and some remarks in Section 5. 2 Notation and symbols We se the following notation and symbols throghot the paper. R n = the space of n-dimensional colmn vectors, R n + = the nonnegative orthant of R n, S n = the space of n n symmetric matrices, S n + = the cone of n n symmetric positive semidefinite matrices, C = { A S n : x Ax 0 for all x R n +} the copositive cone, { } C = x j x j : x j R n + j = 1, 2,..., r for some r 1 the completely positive cone, Y Z = trace of Y Z for every Y, Z S n the inner prodct, cl conv G = the closre of the convex hll of G S n. 3 Main reslts 3.1 Linearly constrained QOPs in continos and binary variables Let A be a q m matrix, b R q, c R m and r m a positive integer. We consider a QOP of the form minimize sbject to Q 0 + 2c R m +, A + b = 0, i 1 i = 0 i = 1, 2,..., r. Brer [3] stdied this type of QOP, and proposed a completely positive cone programming CPP relaxation whose objective vale is the same as the QOP 1. When Q 0 = O is taken, the problem 3 1

4 becomes a standard 0-1 mixed integer linear optimization problem. We assme throghot the paper that QOP 1 has an optimal soltion with the optimal vale ζ. We first convert the QOP 1 into a QOP with a single eqality constraint according to the discssions in Section 5.1 and 5.2 of [1]. Note that the constraint i 1 i = 0 implies 0 i 1 i = 1, 2,..., r, ths, we may assme withot loss of generality that q r, m 2r, and that the eqalities i + i+r 1 = 0 i = 1, 2,..., r are inclded in the eqality A + b = 0. More precisely, we assme that A and b are of the forms A = A11 A 12 A 13 I I O b1, b = e, 2 where I denotes an r r matrix and e the r-dimensional colmn vector of ones. If the 0-1 constraint i 1 i = 0 is replaced by the complementarity constraint i i+r = 0 i = 1, 2,..., r and the linear eqality constraint A + b = 0 by a qadratic eqality constraint A + b A + b = 0, QOP 1 is rewritten by minimize sbject to Q 0 + 2c R m +, A + b A + b = 0, i i+r = 0 i = 1, 2,..., r. 3 Notice that the left hand sides of all qadratic eqality constraints A + b A + b = 0 and i i+r = 0 i = 1, 2,..., r are nonnegative for every R m +. hs, we can nify the constraints into a single eqality constraint to redce the QOP 3 to minimize Q 0 + 2c sbject to R m +, g = 0, 4 where g = A + b A + b + i i+r. In the main reslts presented in heorems 3.1 and 3.3, one of the following conditions will be imposed on 4. a he feasible region of QOP 4 is bonded. b Q 0 is copositive-pls and the set of optimal soltions of QOP 4 is bonded. Here A C is called copositive-pls if 0 and A = 0 imply A = A parametric nconstrained QOP over the nonnegative orthant We now introdce a Lagrangian relaxation of QOP 4 by defining a Lagrangian fnction f : R m + R R by f, λ = Q 0 + 2c + gλ, minimize f, λ sbject to R m +. 5 Notice that 5 is an nconstrained QOP over the nonnegative orthant with the Lagrangian mltiplier parameter λ R for the eqality constraint of QOP 4. Let s choose a positive nmber for λ so that g λ : R m + R serves as a penalty fnction for QOP 4. In fact, we see that g 0 for every R m +, g = 0 if and only if R m + satisfies the eqality constraint of 4, gλ as 0 λ otherwise. 4

5 For each λ > 0, define a level set Lλ = { R m + : ζ f, λ } and the optimal objective vale ζλ = inf { f, λ : R m + } of QOP 5. hen, for 0 < λ < µ, Lλ Lµ the set of optimal soltions of 4, ζλ = inf {f, λ : Lλ} ζµ ζ. 6 Hence, if L λ is bonded for some λ > 0 and λ λ, then QOP 5 has an optimal soltion with the finite objective vale ζλ, and all optimal soltions of QOP 5 are contained in the bonded set L λ. he next theorem ensres that QOP 5 with the parameter λ > 0 serves as seqential nconstrained QOPs over R m + for solving QOP 4 nder condition a or b. heorem 3.1. i If condition a is satisfied, then Lλ is bonded for every sfficiently large λ > 0. ii If condition b is satisfied, then Lλ is bonded for any λ > 0. iii Assme that L λ is bonded for some λ > 0. Let { λ k λ : k = 1, 2,..., } be a seqence diverging monotonically to as k, and { k R m + : k = 1, 2,..., } a seqence of { optimal soltions of QOP 5 with λ = λ k. hen, any accmlation point of the seqence k R m } + is an optimal soltion of QOP 4, and ζλ k converges monotonically to ζ as k. Althogh some of the assertions in heorem 3.1 can be proved easily by applying the standard argments on the penalty fnction method for example, see [8], we present complete proofs of all assertions for completeness in Section 4. It wold be desirable if any optimal soltion of the problem 5 were an optimal soltion of 4 for every sfficiently large λ. However, this is not tre in general, as shown in the following illstrative example: minimize 2 1 sbject to 1 0, 1 1 = 0, 7 which has the niqe optimal soltion 1 = 1 with the optimal vale ζ = 2. In this case, the Lagrangian relaxation problem is described as minimize λ sbject to 1 0. For every λ > 1, this problem has the niqe minimizer 1 = 1 1/λ with the optimal vale ζλ = 2 1/λ. 3.3 Completely positive cone programming and copositive cone programming relaxations of QOP 5 with the parameter λ > 0 We now relate the Lagrangian relaxation 5 of QOP 4 to the CPP relaxation of QOP 4, which was discssed in Sections 5.1 and 5.2 of [1]. Let n = m, x = R n 0 c, Q = S n, c Q 0 C i = the m m matrix with the i, i + rth component 1/2 and 0 elsewhere H 0 = i = 1, 2,..., r, 1 0 S n b b b A, H 0 O 1 = A b A A C i + C i.

6 hen, Q xx 0 = Q 0 = Q 0 + 2c 0, H 0 xx 0 = H 0 H 1 xx 0 = H 1 0 for every x = rewrite QOP4 as and QOP 5 as Here, 0 0 = 2 0, = b 0 + A b 0 + A + i i+r R n. It shold be noted that H 0, H 1 C. Using these identities, we minimize Q X sbject to X G 1, 8 minimize Q + H 1 λ X sbject to X G 0. 9 G 0 = { xx S n : x R n +, H 0 xx = 1 } = { xx S n : x R n +, x 2 1 = 1 }, { G 1 = xx G } 0 : H 1 xx = 0 = { xx S n : x R n +, H 0 xx = 1, H 1 xx = 0 }. It was shown in [1] that the constraint sets cl conv G 0 and cl conv G 1 coincide with their CPP relaxations Ĝ0 and Ĝ1, respectively, where Ĝ 0 = {X C : H 0 X = 1} = {X C : X 11 = 1}, { } Ĝ 1 = X Ĝ0 : H 1 X = 0 = {X C : H 0 X = 1, H 1 X = 0}. See heorem 3.5 of [1]. Since the objective fnctions of the problems 8 and 9 are linear with respect to X S n, 8 has the same optimal objective vale as minimize Q X sbject to X Ĝ1 = cl conv G 1, 10 and 9 has the same optimal objective vale as minimize Q + H 1 λ X sbject to X Ĝ0 = cl conv G We note that CPP 10 and QOP 4 have the eqivalent optimal vale ζ, and that both CPP 11 and QOP 5 have the optimal vale ζλ ζ. and As dal problems of 10 and 11, we have maximize y 0 sbject to Q H 0 y 0 + H 1 y 1 C, 12 maximize y 0 sbject to Q H 0 y 0 + H 1 λ C, 13 6

7 respectively. Either 12 or 13 forms a simple copositive cone programming CP. he difference is that y 1 is a variable in 12 whereas λ > 0 is a parameter to be fixed in advance in 13. As a reslt, 13 involves jst one variable. Obviosly, CPP 11 has an interior feasible soltion. By the standard dality theorem see, for examples, heorem of [14], CP 13 and CPP 11 have an eqivalent optimal vale. It is already observed that CPP 11 and QOP 5 share the common optimal vale ζλ. hs, heorem 3.1 leads to the following theorem. It shows that, nder condition a or b, 11 and 13 serve as a parametric CPP relaxation and a parametric CP relaxation, respectively, and these parametric CP and CPP relaxations bond the optimal vale ζ of QOP 4 from below by ζλ converging monotonically to ζ as λ. heorem 3.2. Assme that condition a or b holds. iv CPP 11 and QOP 5 have the eqivalent optimal vale ζλ ζ, which converges monotonically to ζ as λ. v CP 13 and QOP 5 have the eqivalent optimal vale ζλ ζ, which converges monotonically to ζ as λ. Now, we discss whether the dal 12 of CPP 10 has an optimal soltion with the same optimal objective vale ζ of CPP 10. heorem 3.3. vi rank X n rank A if X Ĝ1. vii Assme that condition a or b is satisfied. hen, the strong dality eqality between CPP 10 and CP 12 { } ζ = min Q X : X Ĝ1 = sp {y 0 : Q H 0 y 0 + H 1 y 1 C} 14 holds. Since C S n + and any X S n with rank X < n lies on the bondary of S n +, vi of heorem 3.3 implies that CPP 10 does not have an interior feasible soltion. Hence, the standard dality theorem can not be applied to the primal dal pair of 10 and 12. hs, the assertion vii is important. Proof of vi: Sppose that xx G 1. hen, x R n +, 0 = H 1 xx b b b A 0 0 = A b A + A 0 C i + C i b b b A A b A xx 0 0 0, A 0 C i + C xx 0. i It follows that xx, 0 = x b b b A A b A A x = x b A b A x. 7

8 hs, b A xx = O holds for every xx G 1. Conseqently, we have that b A X = O for every X cl conv G 1 = Ĝ1, which implies that rank X n rank b A = n rank A. Proof of vii: By assertion ii of heorem 3.2, we know that his implies the desired reslt. ζ = lim ζλ = sp max {y 0 : Q H 0 y 0 + H 1 λ C}. 0<λ λ>0 In general, 12 does not have an optimal soltion with the optimal objective vale ζ. o verify this, assme on the contrary that 12 has an optimal soltion y0, y 1 with the optimal vale y0 = ζ. Since H 1 C, y0, y 1 remains an optimal soltion with the optimal vale y0 = ζ for every y 1 y1. hs, 13 attains the optimal vale ζ for every λ > max{0, y1 }. By the weak dality relation, 11 hence 5 attains the optimal vale ζ of QOP 4 for every λ > max{0, y1 }. his contradicts what we have observed in the simple example 7. Frthermore, if we reformlate the simple problem 7 as 8, the constraint of 12 becomes y0 + y 1 1 y 1 C y 1 y 1 We can verify nmerically that if y 0 = ζ = 2, then, for any finite y 1 R, the matrix on the left side has a negative eigenvale λ with a eigenvector v > 0. hs, v 2 + y1 1 y 1 v = λv v < 0. 1 y 1 y 1 As a reslt, the matrix on the left side of 15 with y 0 = ζ = 2 can not be in C for any y 1 R. his is a direct proof for the assertion that 12 does not have an optimal soltion with the optimal vale y 0 = ζ. If y 0 < ζ = 2, then the matrix on the left side of 15 becomes positive definite for every sfficiently large y 1 > 0. hs, the strong dality eqality 14 holds. Remark 3.4. Assme that QOP 1 satisfies condition a and we know a positive constant α sch that e α for any feasible soltion of QOP 1 and the m-dimensional vector of ones e. hen, QOP 1 can be transformed into a QOP in a strictly positive qadratic form, Q 0 with all [Q 0 ] ij > 0, to be minimized over a bonded feasible region. hs, the reslting QOP satisfies condition b. We show this nder the assmption that the last component m of R m serves as a slack variable for the ineqality constraint that bonds the sm of the other components j j = 1, 2,..., m 1 by α, and that the last row of the eqality constraint A + b = 0 is of the form e α = 0. hen, for any β R and every feasible soltion of 1, Q 0 + ce α + ec α + βee = Q 0 + 2c + βα 2. hs, the objective fnction Q 0 +2c can be replaced by Q 0 + ce α + ec α + βee. aking β > 0 sfficiently large, we have a QOP in a strictly positive qadratic form. herefore, in 0 0 the relaxation problems 10, 11, 12 and 13, we can assme that Q is of the form 0 Q 0 with all [Q 0 ] ij > 0. his conversion can be sed for developing nmerical methods for the QOP 1 with a bonded feasible region in the ftre. However, it may not be nmerically efficient becase the coefficient matrix Q 0 of the reslting QOP becomes flly dense. 8

9 4 Proof of heorem 3.1 A seqence { k, λ k R m+1 : k = 1, 2,..., } is sed in the proofs of assertions i, ii and iii below sch that We note that sch a seqence always satisfies k 0, λ k > 0, 16 ζ f k, λ k = k Q 0 k + 2c k k + g k λ k. 17 A k + b A k + b 0, k i k i+r 0, 18 ζ k Q 0 k + 2c k k Proof of assertion i Assme on the contrary that Lλ is nbonded for any λ > 0. hen, λ k and k as k for some seqence { λ k, k : k = 1, 2,..., } satisfying 16 throgh 19. We may assme withot loss of generality that k / k converges to some nonzero d 0. We divide the ineqality 17 by λ k k 2 and the ineqalities in 18 by k 2, respectively, and take the limit on the reslting ineqalities as k. hen, we observe that d 0 satisfies 0 Ad Ad + d k i d k i+r, Ad Ad 0, d k i d k i+r It follows that Ad = 0 and d i = 0 i = 1, 2,..., 2r recall that A is of the form 2. Hence, + νd remains a feasible soltion of 4 for every ν 0. his can not happen if condition a is satisfied. 4.2 Proof of assertion ii Let s fix λ > 0 arbitrarily. Assme that { 0 : ζ f, λ} is nbonded. We can take a seqence { k : k = 1, 2,..., } satisfying 16 throgh 19 with λ k fixed to λ and k as k. We may assme that k / k converges to some nonzero d 0. Since Q 0 is assmed to be copositive-pls, it follows from 17 that ζ 2c k k + A k + b A k + b + k i k i+r λ, 21 and from 19 that ζ 2c k. We first divide the ineqality 21 by λ k 2, the ineqalities in 18 by k 2, and the ineqalities 19 by k 2, respectively. Next, take the limit on the reslting ineqalities and the ineqality ζ / k 2c k / k as k. hen, we obtain 20, which implies Ad = 0 and d i = 0 i = 1, 2,..., 2r, and 0 2c d. In addition, 0 d Q 0 d, which implies d Q 0 d = and Q 0 d = 0 recall that Q 0 is copositive-pls. herefore, + νd remains a feasible soltion of QOP 4 for every ν 0 and that ζ + νd Q 0 + νd + 2c + νd for every ν 0. his contradicts condition b. 9

10 4.3 Proof of assertion iii We observe that the seqence { k, λ k } satisfies 16 throgh 19 with f k, λ k = ζλ k. Let { k j : j = 1, 2,..., } be a sbseqence that converges to ū 0. From 17, ζ λ k j kj Q 0 kj + 2c k kj λ k j + g k j, g k j 0, ζ ζλ k j k j Q 0 k j + 2c k k j j = 1, 2,..., hold. aking the limit as j in the ineqality above, we have gū = 0 and ζ lim j ζλ k j ū Q 0 ū+2c ū. hs, ū is an optimal soltion of 4 and lim j ζλ k j = ζ. By 6, we also know that ζλ ζµ ζ if λ µ. Conseqently, the seqence { ζλ k } itself converges monotonically to ζ as k. 5 Conclding remarks We have proposed the three relaxations, the nconstrained QOP 5 in nonnegative variables, the CPP problem 11 and the CP problem 13. Compting an optimal soltion of any of the proposed relaxations nmerically is difficlt. As mentioned in Section 1, to develop a nmerical method for the second and third relaxations, the problem of checking whether a given matrix lies in the completely positive cone and the copositive cone needs to be resolved, respectively, which was shown in [12] as a co-np-complete problem. One practical method to overcome this difficlty is to relax the completely positive cone C by the so-called dobly nonnegative cone S n + N in 11, where N denotes the cone of n n symmetric matrices with nonnegative components. his idea of replacing C by S n + N in CPP problems has been sed in [9, 16]. Even in their cases, the reslting SDPs can not be solved efficiently when the size of variable matrix X is large becase of the n 1n/2 ineqality constraints X ij 0 1 i < j n. In particlar, the ineqality constraints make it difficlt to exploit sparsity in the SDPs, which is an effective tool to solve large scale SDPs efficiently. If we replace C by S n + N in 11, we have the following problem rewritten as an SDP, which serves as a relaxation of 11. minimize Q + H 1 λ X sbject to X 11 = 1, X ij 0 1 i < j n, X S n Note that the nmber of ineqalities is increased. As a reslt, sparsity can not be exploited efficiently, althogh it involves only a single eqality constraint X 11 = 1. o exploit sparsity in 22, we need to redce the nmber of ineqality constraints. For example, we can take K = {i, j : 1 i < j n, [Q + H 1 λ] ij 0}, expecting that K satisfies a strctred sparsity in practice, and replace the ineqality constraints by X ij 0 i, j K. Frther stdies and nmerical experiments along this direction are important sbjects of ftre research. We refer to [10, 11, 13] for exploiting sparsity in SDPs. References [1] N. Arima, S. Kim and M. Kojima. A qadratically constrained qadratic optimization model for completely positive cone programming. Research Report B-468, Department of Mathematical and Compting Sciences, okyo Institte of echnology, Oh-Okayama, Megro-k, okyo , September

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