Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often appear n varous felds such as network flows and computer tomography. In ths paper, we propose an algorthm for solvng those problems and prove the convergence of the proposed algorthm. 1. Introducton Consder the multcommodty transportaton problem wth convex quadratc cost functon 1.1) mnmze subject to 1 2 x x0 ) T Qx x 0 ) γ Ax δ where A = a j ) s a gven m n matrx whose th row s a T, x0 R n, r, δ R m are gven vectors, Q s a gven n n symmetrc postvedefnte matrx and the superscrpt T denotes transposton. We assume that matrx A does not contan any row of whch elements are all zero. The pars of nequalty constrants n problem 1.1) are referred to as nterval constrants. Interval constrants often appear n optmzaton problems that arse n varous felds such as network flows and computer tomography. Recently, varous row-acton methods [1,2,8], whch orgnate from the classcal Hldreth s method[4], have drawn much attenton. Those Receved October 13, 1995. 1991 Mathematcs Subject Classfcaton: 49M37. Key words and phrases: multcommodty transportaton problem, quadratc programmng problem, network flows. Ths paper was supported by research fund of Inha unversty, 1993.
8 Yong Joon Ryang methods are partcularly useful for large and sparse problems, because they act upon rows of the orgnal coeffcent matrx one at a tme. They are adaptatons of coordnate descent methods such as Gauss- Sedel method or ts varants, for solvng the dual of a gven quadratc programmng problem. To obtan the soluton of problem 1.1), t wll be helpful to consder the dual of problem 1.1). 1.2) mnmze φz) subject to z 0 where φ : R 2m R s a convex quadratc functon defned by 1.3) φz) = 1 2 zt ÂQ 1 Â T z + z T b Âx0 ), Â s the 2m n matrx 1.4) Â = a 1, a 1, a 2, a 2,, a m, a m ) T, b s the 2m-vector 1.5) b = δ 1, γ 1, δ 2, γ 2,, δ m, γ m ) T, and z s the 2m-vector 1.6) z = z + 1, z 1, z+ 2, z 2,, z+ m, z m) T. Note that z +, z ) s a par of dual varables assocated wth the th par of the nterval constrants of 1.1),.e., z + and z correspond to the constrants a T x δ and a T x γ, respectvely. By takng nto account the specal structure of problem 1.1), Herman and Lent[3] extended Hldreth s algorthm to deal wth nterval constrants drectly, thereby economzng the number of dual varables by half [2]. Ryang[7] have recently proposed a method that deal wth the nterval constrants n a drect manner. In ths paper we propose a method for solvng those problems, whch may be regard as the applcaton of the Jacob method to the dual of the orgnal problems. We prove the convergence of the proposed algorthm. In secton 2, a row-acton method s presented. In secton 3, the proposed algorthm s shown to converge to the soluton of 1.1).
An teratve row-acton method 9 2. Row-Acton Method In ths secton, we state an algorthm for solvng the nterval constrant problem 1.1). Algorthm 2.1. Intalzaton : Let x 0), x 0) ) := x 0, 0), k := 0 and choose a relaxaton parameter ω > 0. Iteraton k : ) For = 1,, m, where f z +k) else endf z k) c +k) c k) c k) c +k) z +k+1) z k+1) then := mn{z +k), ω k) }, := mn{z k), ωγ k) + c +k) } := mn{z k), ωγ k) }, := mn{z +k), ω k) + c k) } := z +k) := z k) c +k), c k), k) Γ k) := δ a T xk) α, := γ a T xk) α. ) Let where x k+1) := x k) + Q 1 m =1 c +k) c k) )a. 2.1) α = a T Q 1 a, = 1,, m.
10 Yong Joon Ryang Note that α are all postve, snce Q s postve defnte and a 0. Note also that, snce γ δ, the followng nequaltes are always satsfed : 2.2) Γ k) k), = 1,, m. Lemma 2.1. Let {x k) } and {z k) } be generated by Algorthm 2.1. Then for all k, we have 2.3) x k) = x 0 Q 1 Â T z k), 2.4) z k) 0, 2.5) z +k) z k) = 0, = 1,, m. Proof. 2.3) and 2.4) drectly follow from the manner n whch {x k) } and {z k) } are updated n the algorthm. We prove 2.5) by nducton. For k = 0, t trvally holds. For each, we assume z +k) z k) = 0 and show that t s also true for k + 1. Wthout loss of generalty, we may only consder the case where x +k) x k), because a parallel argument s vald for the opposte case. Frst note that, when z +k) Moreover, f z +k) c k) z k) = mn{0, ω k) ω k) Γ k) from 2.2). Therefore we must have z k+1) z +k) < ω k), 2.4) mples z +k) 0 and z k) = 0. holds, then we have c +k) = ω k) and )} = 0, where the last equalty follows = 0. On the other hand, f = z +k), whch n turn mples holds, then we have c +k) z +k+1) = 0. Thus 2.5) s satsfed for k + 1. For each, ether z +k) Moreover, we can deduce the followng relatons : If z +k) z k) 2.6) c +k), c k) ) =,.e., z +k) = 0 or z k) = 0 must always hold by 2.5). 0, z k) = 0, then ω k), 0), f z +k) ω k), z +k), 0), f ω k) z +k) ωγ k), + z +k) ), f ωγ k) z +k). z +k), ωγ k)
If z +k) 2.7) An teratve row-acton method 11 z k),.e., z +k) = 0, z k) 0, then c +k), c k) ) = 0, ωγ k) ), f z k) ωγ k), 0, z k) ), f ωγ k) z k) ω k), ω k) + z k) 3. Convergence of Algorthm 2.1, z k) ), f ω k) z k). In ths secton, we prove convergence of Algorthm 2.1. Frst, we consder an algorthm for solvng general lnear complementarty problems. Then we show that Algorthm 2.1 can be reduced to ths algorthm. Let us consder symmetrc lnear complementarty problem, whch s to fnd y R l such that 3.1) Mu + q 0, y 0, y T My + q) = 0, where M s an l l symmetrc matrx and q s a vector n R l. If M s postve semdefnte, then ths problem s equvalent to the problem 3.2) mnmze 1 2 yt My + q T y subject to y 0. Mangasaran [5] proposes the followng algorthm for problem 3.1). Algorthm 3.1. Intalzaton : Let y 0) := 0 and k := 0. Iteraton k : Choose an l l dagonal matrx E k) and an l l matrx K k), and let 3.3) y k+1) := y k) ωe k) My k) + q + K k) y k+1) y k) ))) +, where, for any vector y, y + denotes the vector wth elements y + ) = max{0, y }. Varous choces for {E k) } and {K k) } are possble and each partcular choce yelds a dfferent algorthm [5]. In the followng, we show that Algorthm 2.1 s a partcular realzaton of Algorthm 3.1.
12 Yong Joon Ryang Frst observe that problem 1.2) can be wrtten as problem 3.2) wth l = 2m by settng 3.4) M = ÂQ 1 Â T, 3.5) q = b Âz0, 3.6) y = z. Note that the postve defnteness of Q mples that the matrx M defned by 3.4) s postve semdefnte. We wll show that Algorthm 2.1 can be reduced to Algorthm 3.1 by choosng matrces E k) and K k) approprately. Specfcally, let E k) and K k) be 2m 2m matrces such that 3.7) E k) = 3.8) K k) = D1 1 0... 0 D 1 m K k) 1 0... 0 K k) m where 3.9) D 1 = 1 ) 1 0, α 0 1 3.10) K k) =,, ) 0 0 α, f z +k) 0 ω α ) 0 ω, otherwse, 0 0 z k), and α are defned by 2.1) for all = 1,, m. Snce matrx K k) gven by 3.8) are block dagonal, the par y k+1) 2 1, yk+1) 2 ) of varables n problem 3.1), whch corresponds to z +k+1), z k+1) ) n problem 1.2) by 1.6) and 3.6), can be updated separately from each other, that s, n parallel for = 1,, m [6].
An teratve row-acton method 13 Theorem 3.1. Let M, q and y n problem 3.1) be defned by 3.4)- 3.6). Then the sequence {z k) } generated by Algorthm 3.1 wth E k) and K k) gven by 3.7)-3.10) s dentcal wth the sequence {z k) } generated by Algorthm 3.1 for problem 1.2). Proof. The formula 3.3) may be wrtten componentwse as follows : For = 1,, m, f z +k) z k) then 3.11) z +k+1) := z +k) ω ) a T Q 1 Â T z k) + δ a T x 0 ), α + 3.12) otherwse z k+1) := 3.13) z k+1) := z k) ω α a T Q 1 Â T z k) γ + a T x 0 α ω z+k+1) z +k) ) )) z k) ω ) a T Q 1 Â T z k) γ + a T x 0 ), α + + 3.14) z +k+1) := For smplcty, let z +k) ω α a T Q 1 Â T z k) +δ a T x 0 α )) ω z k+1) z k) ) 3.15) x k) := x 0 Q 1 Â T z k), +. 3.16) k) := δ a T xk) α, 3.17) Γk) := γ a T xk) α.
14 Yong Joon Ryang Then 3.11)-3.14) are rewrtten as follows : z +k) z k) then For = 1,, m, f 3.18) z +k+1) = 3.19) z k+1) = otherwse 3.20) z k+1) = 3.21) z +k+1) = z +k) ω ) δ a T x k) ) α + = max{0, z +k) = z +k) mn{z +k) k) ω } ω k) }, z k) ω γ + a T x k) α z +k+1) α ω = max{0, z k) = z k) + ω Γ k) + z +k+1) z +k) )} )) ) z +k) + mn{z k), ω Γ k) z +k+1) z +k) )} z k) ω ) γ a T x k) ) α + = max{0, z k) = z k) ω Γ k) } mn{z k) ω Γ k) }, z +k) ω δ a T x k) α z k+1) α ω = max{0, z +k) = z +k) k) ω + z k+1) z k) )} mn{z +k), ω k) z k+1) )) ) z k) + z k) )}. Besdes, defne c +k) then z k) and c k) as follows : For = 1,, m, f z +k) 3.22) c +k) := mn{z +k) k), ω },
3.23) c k) otherwse 3.24) c k) An teratve row-acton method 15 := mn{z k), ω Γ k) + c +k) } := mn{z k), ω Γ k) }, 3.25) c +k) := mn{z +k) k), ω + c k) }. It then follows from 3.18), 3.19), 3.22) and 3.23) that, f z +k) z k), we have 3.26) z +k+1) = z +k) z +k), 3.27) z k+1) = z k) z k). On the other hand, f z +k) < z k), then 3.20), 3.21), 3.24) and 3.25) mply that the same relatons 3.26) and 3.27) also hold. Moreover note that x k+1) = x 0 Q 1 m =1 = x 0 Q 1 m z +k+1) z +k) =1 = x k) + Q 1 m c +k) =1 z k+1) )a z k) )a + Q 1 c k) )a, m c +k) j=1 c k) )a where the frst and the thrd equaltes follow from 3.24), whle the second follows from 3.26) and 3.27). Snce both Algorthms 2.1 and 3.1 start wth z 0) = 0, we can nductvely show that 3.28) x k) = x k), 3.29) k) 3.30) c +k) for all k, where {x k) }, { k) = k), Γk) = Γ k), = 1,, m, = c +k), c k) = c k), = 1,, m, }, {Γ k) }, {c +k) } and {c k) } are the sequences generated by Algorthm 2.1. Thus the sequence {z k) } generated by Algorthms 2.1 and 3.1 are dentcal.
16 Yong Joon Ryang References [1] Y. Censor, Row-Acton Methods for Huge and Sparse Systems and Ther Applcatons, SIAM Revew 23 1981), 444 466. [2] Y. Censor and A. Lent, An Iteratve Row-Acton Method for Interval Convex Programmng, Journal of Optmzaton Theory and Applcatons 34 1981), 321 353. [3] G.T. Herman and A. Lent, A Famly of Iteratve Quadratc Optmzaton Algorthms for Pars of Inequaltes wth Applcaton n Dagnostc Radology, Mathematcal Programmng Study 9 1979), 15 29. [4] C. Hldreth, A Quadratc Programmng Procedure, Naval Research Logstc Quarterly 4 1957), 79 85. [5] O.L. Mangasaran, Soluton of Symmetrc Lnear Complementarty Problems by Iteratve Methods, Journal of Optmzaton Theory and Applcatons 22 1977), 465 485. [6] O.L. Mangasaran and R. De Leone, Parallel Successve Overrelaxaton Methods for Symmetrc Lnear Complementarty Problems and Lnear Programs, Journal of Optmzaton Theory and Applcatons 54 1987), 437 446. [7] Y.J. Ryang, A Method for Solvng Nonlnear Programmng Problems, Inha Unversty R.I.S.T. 21 1993), 277 285. [8] S.A. Zenos and Y. Censor, Massvely Parallel Row-Acton Algorthms for Some Nonlnear Transportaton Problems, SIAM Journal on Optmzaton 1 1991), 373 400. [9] S.A. Zenos, On the Fn-Gran Decomposton on Multcommodty Transportaton Problems, SIAM Journal on Optmzaton, Forth comng. Dept. of Computer Scence and Engneerng Inha Unversty Incheon, 402-751, Korea