R.D.C. MONTEIRO and P.R. ZANJ ACOMO. Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants. Technical report, Georgia Tech, Atlanta, GA, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....direction is guaranteed to be symmetric. It is the solution of (7) 8) and VZX ZV #ZX = X , 10) VX #X (11) S(n) is an auxiliary variable. They proved polynomial time convergence of the path following algorithm using this direction. Recently, Monteiro and Zanjacomo [25] discussed a computational aspects of this direction, and gave some numerical experiments. 1.3. The AHO Direction. Alizadeh, Haeberly, and Overton [2] proposed symmetrizing equation (6) by rewriting it as XZ ZX= 0, 12) and then applying Newton s method to (3) 4) and (12) The resulting ....

R. D. C. Monteiro and P. Zanjacomo. Implementation of primal-dual methods for semidefinite programming based on monteiro and tsuchiya newton directions and their variants. Technical report, Georgia Institute of Technology,Atlanta, Georgia, USA, 1997.

....the algorithm that we presented later. There exist various interior point algorithms that solve the semide nite program in polynomial time, based on either the primal scaling (using X) the dual scaling (using S) or the primal dual scaling (using both X and S) see, e.g. 1] 7] 10] 12] [11], 14] and [16] Other nonlinear programming based methods for semide nite programming include Burer, Monteiro and Zhang[3] Helmberg and Kiwiel [6] Vavasis [17] Vanderbei and Benson [18] etc. Recently, the dual scaling algorithm (see Benson et al. 2] has gained some attention, since it ....

R. D. C. Monteiro and P. R. Zanjacomo, \Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya directions and their variants," Technical Report, School of Ind. and Systems Engineering, Georgia Institute of Technology, Atlanta, 1997.

....1 2 #Z X 1 2 = X 1 2 Z X 1 2 , 10) V X 1 2 X 1 2 V = #X (11) 4 MASAKAZU MURAMATSU AND ROBERT J. VANDERBEI where V # S(n) is an auxiliary variable. They proved polynomial time convergence of the path following algorithm using this direction. Recently, Monteiro and Zanjacomo [25] discussed a computational aspects of this direction, and gave some numerical experiments. 1.3. The AHO Direction. Alizadeh, Haeberly, and Overton [2] proposed symmetrizing equation (6) by rewriting it as X Z Z X = 0, 12) and then applying Newton s method to (3) 4) and (12) The ....

R. D. C. Monteiro and P. Zanjacomo. Implementation of primal-dual methods for semidefinite programming based on monteiro and tsuchiya newton directions and their variants. Technical report, Georgia Institute of Technology, Atlanta, Georgia, USA, 1997.

....local convergence and prove convergence of Q order 1.5 or 2 for predictor corrector algorithms using certain search directions from the Monteiro Zhang family. Nor do we discuss the amount of computational work involved in computing our search directions in much detail; see Monteiro and Zanjacomo [24] and Toh [32] for flop counts for some of these directions. In Section 2 we discuss the central path and how directions might be defined to approximate a sequence of points on the path. We also introduce some notation and various notions of geometric mean. The following section describes the ....

.... , E#X F#S = REF , 7) where the operators E = E(X, S) and F = F(X,S) are the derivatives of # with respect to X and S respectively, evaluated at (X, S) and REF = REF (X, S) #(X,S) Such a general derivation of search directions was considered also in Monteiro and Zanjacomo [24]. For some directions it will be more convenient to introduce another variable Z # SIR n n and to consider the system A#X = r p , A # #y #S = R d , E#X 2GZ = RE , F#S 2HZ = RF , 8) for certain operators E, F , G, and H, again depending on X and S. The factor 2 and the ....

[Article contains additional citation context not shown here]

R. D. C. Monteiro and P. R. Zanjacomo, Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants, manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 1997.

.... [1, 2] are available in both of SDPA and SDPT3, we employed the HRVW KSH M direction in our numerical experiments because its computation is the cheapest among the three directions (particularly, for sparse data matrices) when we employ the method proposed by Fujisawa et al. 7] Monteiro et al. [17] recently showed that in theory, the NT direction requires less computation for dense matrices. However, their method needs large amount of computational memory and does not efficiently exploit the sparse data structures. Actually, according to their numerical results, the computation of the ....

....direction is almost the same as when we employ the NT direction. Furthermore, the HRVW KSH M direction is faster than the NT direction on all problems. We can also observe the same tendency for the HRVW KSH M direction using SDPT3 [21, 22] excepting the ETP Problem. Monteiro and Zanj acomo [17] showed similar numerical results. Therefore, we mainly focus our attention on the HRVW KSH M direction in this paper. As we have already seen in Section 1, there exists some MATLAB implementations besides SDPT3. Also, there are some other full implementations, for example, SDPSOL and CSDP, which ....

R.D.C. Monteiro, P.R. Zanj'acomo, "Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya directions and thier variants, " Technical Report, School Industrial and Systems Engineering, Georgia Tech., Atlanta, GA 30332, July 1997, Revised August 1997.

.... and the AHO direction [1] are available in SDPA, we employed the HRVW KSH M direction in our numerical experiments because its computation is the cheapest among the three directions (particularly, for sparse data matrices) when we employ the method proposed by Fujisawa et al. 5] Monteiro et al. [12] recently showed that in theory, the NT direction requires less computation for dense matrices. However, their method needs large amount of computational memory and does not efficiently exploit the sparse data structures. Actually, according to their numerical results, the computation of the ....

R.D.C. Monteiro, P.R. Zanj'acomo, "Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya directions and thier variants," Technical Report, School Industrial and Systems Engineering, Georgia Tech., Atlanta, GA 30332, July 1997, Revised August 1997.

....= ff p Gammar T (y k ; z k )d( z k )y : 8) Unlike linear programming, positive semidefinite programming requires a significant amount of the time to compute the system of equations that determines the step direction. For arbitrary symmetric matrices A i , Monteiro and Zanj acomo [20] demonstrated an efficient implementation of several primal dual step directions. The AHO direction [3] can be computed in 5nm 3 n 2 m 2 O(maxfm;ng 3 ) operations. The HRVW KSH M direction [13] 16] 21] uses 2nm 3 n 2 m 2 O(maxfm;ng 3 ) operations, and the NT direction [26] ....

R. D. C. Monteiro and P. R. Zanj'acomo, "Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya directions and their variants," Technical Report, School of Ind. and Systems Engineering, Georgia Institute of Technology, Atlanta, 1997.

.... [1, 2] are available in both of SDPA and SDPT3, we employed the HRVW KSH M direction in our numerical experiments because its computation is the cheapest among the three directions (particularly, for sparse data matrices) when we employ the method proposed by Fujisawa et al. 7] Monteiro et al. [18] recently showed that in theory, the NT direction requires less computation for dense matrices. However, their method needs large amount of memory and does not efficiently exploit the sparse data structures. Actually, according to their numerical results, the computation of the HRVW KSH M ....

....1.37e 09 Control and system theory 60 231 1.91e 07 1.18e 05 Maximum cut 100 100 1.55e 09 3.08e 06 5.45e 09 Graph equipartition 100 101 5.33e 07 1.65e 03 Maximum clique 100 1024 2.50e 09 6.26e 09 4.47e 09 using SDPT3 [22, 23] excepting the ETP Problem. Monteiro and Zanj acomo [18] showed similar numerical results. Therefore, we mainly focus our attention on the HRVW KSH M direction in this paper. As we have already seen in Section 1, there exists some MATLAB implementations besides SDPT3. Also, there are some other full implementations, for example, SDPSOL and CSDP, which ....

R.D.C. Monteiro, P.R. Zanj'acomo, "Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya directions and their variants, " Technical Report, School Industrial and Systems Engineering, Georgia Tech., Atlanta, GA 30332, July 1997, Revised August 1997.

....4] and the NT direction [7] obtained from (1.6) by taking P equal to I, Z 1=2 , and [Z 1=2 (Z 1=2 XZ 1=2 ) Gamma1=2 Z 1=2 ] 1=2 respectively. Among these directions, the AHO direction has been observed to achieve the highest accuracy. We also mention that Monteiro and Zanj acomo [6], and Toh [9] recently reported other search directions that can attain high accuracy. All the above mentioned search directions involve the linearization of a specific symmetric central path equation. In this paper, we show that the nonsymmetric central path equation (1.4) can be directly used ....

....of computing the XZ direction by using formula (3.4) is 4mn 3 2m 2 n 2 O(maxfm;ng 3 ) Remark 3.3 The complexity of computing most commonly used search directions for SDP is of the form ffmn 3 fim 2 n 2 O(maxfm;ng 3 ) 3. 5) where ff and fi are two positive constants (see [6, 9]) We note that the third term in (3.5) cannot be neglected because sometimes it may contribute significantly to the complexity, especially when extra matrix factorizations are used. We also note that the computation of the XZ direction needs the least number of matrix factorizations. This feature ....

R. D. C. Monteiro and P. Zanj'acomo. Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, July 1997. Revised August 1997.

No context found.

Monteiro, R.D.C., P.R. Zanj'acomo (1997). Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants, manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta. To appear in Optimization Methods and Software.

....to rederive the polynomial convergence of these two algorithms in a unified way; more specifically, we obtain the same iteration complexity of O(n p L) for the first map and a slightly worse iteration complexity of O(n 2 L) for the second map. The third and fourth maps have been studied in [15], 16] and [23] yet no polynomial convergence analysis of long step algorithms based on these maps have been established. We show that the third and fourth maps also fit nicely into our general framework and hence obtain for the first time polynomially convergent feasible long step algorithms for ....

....based on the other two maps, namely the L T x SL x map and the V 2 map, where L x j chol (X) and V j W 1=2 XW 1=2 = W Gamma1=2 SW Gamma1=2 with W being the unique symmetric matrix such that S = WXW , are studied here for the first time. The two last maps have been introduced in [15], 16] and [23] and were studied there from different points of view. We now state a technical lemma which will be used in the upcoming subsections. Lemma 4.1 Let (X; S) 2 C, G j G (X;S) and R j R (X;S) Then, for all A 2 S n , kG(A)k 2 F k( Phi s ) Gamma1 (A)k 2 F kR 2 k ; ....

R. D. C. Monteiro and P. R. Zanj'acomo. Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants. Technical Report, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, July 1997. To appear in Optimization Methods and Software.

....so far. While for MZ family the iteration complexity bound depends on a certain condition number associated with the sequence fP k g of scaling matrices, the corresponding bound for the MT family does not depend on this sequence. After the release of this paper, Monteiro and Zanj acomo [20] have reported promising computational results for algorithms based on the pure Newton direction (13) and two other pure Newton directions based on the central path equations: S 1=2 XS 1=2 = I ; L T S XL S = I ; respectively, where L S denotes the Cholesky lower triangular factor of S, that ....

R. D. C. Monteiro and P. R. Zanj' acomo, Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants, Technical Report, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, July 1997.

.... and Overton [2] Freund [3] Helmberg, Rendl, Vanderbei and Wolkowicz [4] Jarre [5] Kojima, Shida and Shindoh [8] Kojima, Shindoh and Hara [10] Lin and Saigal [11] Luo, Sturm and Zhang [12] Monteiro [14, 15] Monteiro and Zhang [20] Monteiro and Tsuchiya [18] Monteiro and Zanj acomo [19], Nesterov and Nemirovskii [23] Nesterov and Todd [26, 25] Potra and Sheng [27] Sturm and Zhang [28] Tseng [30] Vandenberghe and Boyd [31] and Zhang [33] Most of these more recent works are concentrated on primal dual methods. The first algorithms for SDP and SDLCP that are extensions of ....

....path following algorithms, respectively [18] They also consider a subclass of the MT family which contains the HRVW KSH M and NT directions, and establish an O(n 3=2 L) iteration complexity bound for any long step path following algorithm based on this subclass. Monteiro and Zanj acomo [19] report promising computational results for algorithms based on some directions of the MT family. Unified analysis for the KSH family of search directions are provided in Kojima, Shindoh and Hara [10] This paper deals with primal dual path following algorithms for the semidefinite linear ....

R. D. C. Monteiro and P. R. Zanj'acomo. Implementation of primal-dual methods for semidefinite programming based on monteiro and tsuchiya newton directions and their variants. Technical Report, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, July 1997.

No context found.

R.D.C. MONTEIRO and P.R. ZANJ ACOMO. Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants. Technical report, Georgia Tech, Atlanta, GA, 1997.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC