### Table 3: Primal Dual algorithm

"... In PAGE 9: ... The running time increases with the accuracy needed. The next Theorem states the running time and the correctness of the algorithm shown in Table3 . The proof is omitted here due to lack of space, but is similar to the one in [31].... In PAGE 9: ... Theorem 4. The algorithm in Table3 computes a (1 ) 3 optimal solution to the ow scaling problem in time polynomial in Q; L; n and 1 , where Q is the number of com- modities, L is the number of constraining sets, and n is the number of nodes. 6.... ..."

### Table 1: Comparison of two implementations of primal-dual Newton interior point method. If 0:1 then = 7, otherwise = 40. Fixed value of q = 6; q1 = 3.

1999

"... In PAGE 21: ...19 C(1) = D for i = 1,: : : , m do p(i) = C(i)V T i ^ Dii = Dii + Vi p(i) u(i) = 1 ^ Dii p(i) C(i+1) = C(i) ? u(i)p(i)T Table1 : Computing ^ D and u(i) Set W(1) = LV for i = 1; : : : ; m do for j = i + 1; : : : ; m do W(i+1) j = W(i) j ? LjiVi Rji = Lji + W(i+1) j u Table 2: Computing R to obtain the factorization of D + V DV T . Given V; D and D we can show that the recurrence relations for computing ^ Dii and u(i) are as given in Table 1.... In PAGE 21: ...19 C(1) = D for i = 1,: : : , m do p(i) = C(i)V T i ^ Dii = Dii + Vi p(i) u(i) = 1 ^ Dii p(i) C(i+1) = C(i) ? u(i)p(i)T Table 1: Computing ^ D and u(i) Set W(1) = LV for i = 1; : : : ; m do for j = i + 1; : : : ; m do W(i+1) j = W(i) j ? LjiVi Rji = Lji + W(i+1) j u Table 2: Computing R to obtain the factorization of D + V DV T . Given V; D and D we can show that the recurrence relations for computing ^ Dii and u(i) are as given in Table1 . We can compute R = L^ L in terms of the u(i), for i = 1,: : : , m by forward recurrence using Lemma V in [8].... In PAGE 21: ... This is re ected in the sparse algorithm by setting Rji = 0 whenever Lji = 0. For Vi = 0 we have from Table1 that u(i) = 0 and ^ Lri = 0 for r = i + 1; : : : ; m in (11). Thus Rri = Lri for r = i + 1; : : : ; m.... In PAGE 28: ... The test code is implemented in MATLAB The variable is a proximity measure of the interior point iterations to a solution of the linear programming problem and is the number of iterations (or corrections) allowed for the preconditioned conjugate gradient method. The percentages in Table1 are based on that approximately half of the direct solves are replaced by an iterative solution. The gain is therefor approximately the ratio of the di erence between the two methods and half of the total time.... In PAGE 28: ... The gain is therefor approximately the ratio of the di erence between the two methods and half of the total time. The results in Table1 show that the mixed primal-dual Newton (mixed PDN) interior-point method, which alternatively uses a direct (Cholesky factorization) method and a preconditioned (with the preconditioner described in Section 3) conjugate gradient method to solve (2), competes favourably with the primal-dual Newton (PDN) interior-point method on large-scale problems. The numerical results show that the mixed PDN method is promising and merits further study.... In PAGE 34: ...1 Dense algorithm De ne V = A; C = D for i = 1; : : : ; m do p = (Vi C)T Tii = Dii + Vi p u = (1=Tii) p C C ? u pT for j = i + 1; : : : ; m do Vj Vj ? LjiVi Rji = Lji + Vj u Algorithm C.2 Sparse algorithm De ne V = A; C = D; R = L; T = D for i = 1; : : : ; m do if Vi 6 = 0 then p = (Vi C)T Tii = Dii + Vi p u = (1=Tii) p C C ? u pT for j = i + 1; : : : ; m do if Lji 6 = 0 then Vj Vj ? LjiVi Rji = Lji + Vj u Table1 : Updating the triangular factors for matrices of the form: RTRT = LDLT + A D AT Algorithm D.1 PCGLS y0 is the initial starting vector r0 = h ? AT y0, v0 = AGr0, p0 = s0 = Mv0, 0 = sT 0 v0 k 0 While not converged do qk = AT pk k = k qT k Gqk yk+1 = yk + kpk rk+1 = rk ? kqk (1.... ..."

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### Table 3 Primal Dual

"... In PAGE 10: ... If the ith primal variable yi is urs, the ith dual constraint is an equality constraint. Table3 gives a more complete relationship between nonnormal primal and dual problems. Table 3 Primal Dual... ..."

### Table 2. The norms of the computed primal and dual directions in various stages of the SUMP in test case A, indicating the relation between the primal and dual directions.

### Table 2: Number of function evaluations, number of CG iterations and CPU time for the new primal-dual interior point method (NITRO) and LANCELOT (LAN). A * indicates that the method did not meet the stopping test in 10,000 iterations.

1999

"... In PAGE 20: ... constraint types objective CORKSCRW 456 350 free, bounded, xed linear eq, nonlinear ineq nonlinear COSHFUN 61 20 free nonlinear ineq linear DIXCHLNV 100 50 bounded nonlinear eq nonlinear GAUSSELM 14 11 free, bounded, xed linear ineq, nonlinear eq linear HAGER4 2001 1000 free, bounded, xed linear eq nonlinear HIMMELBK 24 14 bounded linear eq, nonlinear eq linear NGONE 100 1273 bounded, xed linear ineq, nonlinear ineq nonlinear OBSTCLAE 1024 0 bounded, xed nonlinear OPTCNTRL 32 20 free, bounded, xed linear eq, nonlinear eq nonlinear OPTMASS 1210 1005 free, xed linear eq, nonlinear ineq nonlinear ORTHREGF 1205 400 free, bounded nonlinear eq nonlinear READING1 202 100 bounded, xed nonlinear eq nonlinear SVANBERG 500 500 bounded nonlinear ineq nonlinear TORSION1 484 0 bounded, xed nonlinear Table 1: The main test problem set. In Table2 we present the results for the primal-dual version of our new algorithm, NITRO. For comparison we also solved the problems with LANCELOT [16] using sec- ond derivatives and all its default settings.... In PAGE 20: ... The runs of NITRO were terminated when E(xk; sk; 0) 10?7, and LANCELOT was stopped when the projected gradient and con- straint violations were less than 10?7; the termination criteria for these two methods are therefore very similar. Since both algorithms use the conjugate gradient method to compute the step, we also report in Table2 the total number of CG iterations needed for conver- gence. All runs were performed on a Sparcstation 20 with 32 MG of main memory, using a FORTRAN 77 compiler and double precision; the CPU time reported is in seconds.... In PAGE 21: ...of NITRO reported in Table2 are highly encouraging, particularly the number of function... ..."

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