R.D.C. MONTEIRO. Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim., 7(3):663--678, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....clique problems of graphs, and selected SDPLIB problems [4] to moderately high accuracies, but at reasonable costs. Numerical experiments indicate that our method is promising in solving large SDPs. But there is a slight limitation in that our method cannot be adapted for the HRVW KSH M direction [15, 16, 19], for reasons that we will explain later. The paper is organized as follows. In Section 2, the derivation of the reduced augmented system is presented. The implementation of the PCR method for solving the SCE is given in Section 3. The implementation of the PSQMR method for solving the reduced ....

....for is likely to be easier than that for the highly ill conditioned matrix M . Remark. Notice that the derivation of the reduced augmented system (15) depends on our ability to find the eigenvalue decomposition of W 1 W 1 . For the HRVW KSH M direction direction described in [15, 16, 19], W 1 W 1 is replaced by X 1 Z. Unfortunately, unlike the former, the eigenvalue decomposition of the latter is not readily available even if those of X and Z are known. Because of this reason, the augmented system (12) cannot be reduced to the form in (15) for the HRVW KSH M direction. ....

R. D. C. Monteiro, Primal-dual path following algorithms for semidefinite programming, SIAM J. Optimization, 7 (1997), pp. 663--678.

....linear programming have been successfully extended to solve the SDP problems (1. 3) For a survey of results obtained before 1993 in this field see the paper of Alizadeh [1] More recent results can be found in [4, 2, 3, 7, 9, 15] Kojima, Shindoh and Hara [9] Nesterov and Todd [13] and Monteiro [12] extended some interior point methods for LP to SDP. In the latter paper Monteiro developed a new formulation of the primal dual search direction originally introduced in [9] All above mentioned methods, with the exception of the infeasible interior point potential reduction method of ....

....Clearly i = 1 Gamma )R i ; i = 1; m; R d = 1 Gamma )R d : 2.13) Correspondingly, we define = 1 Gamma ) 2.14) Le us note that in setting up the linear systems (2.4) and (2.11) it is not necessary to compute square roots of matrices. Indeed, as pointed out by Monteiro [12], it is easily seen that an equation of the form = 2(oeI Gamma X 6 can be written equivalently under the form 2XV X USX SXU = 2(oeX Gamma XSX) Summarizing, we can formally define our algorithm as follows: Algorithm 2.1 Choose (X ) 2 N (ff; 0 ) with 0 = 0 = X ) n ....

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R. D. C. Monteiro. Primal-dual path following algorithms for semidefinite programming. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA, September 1995.

....of method here and below, to stress that we are concerned here with the Newton system, which defines the direction; many di#erent methods can use this direction, depending on their choices of the centering parameter and the step sizes. As a second example, the HRVW KSH M direction for SDP (see [8, 11, 17]) has the identity, xvs 1 s 1 vx) 2, g x# n times s 1 , and h = x. It is easily seen that these choices satisfy the conditions of case (b) as well as the extra condition. Another instance of case (b) is the Nesterov Todd (NT) direction for SDP see [20, 21] Here is the ....

....the operator v wvw, where w : x [x sx ] 1 2 x is the unique positive definite matrix with wsw = x, and , g, and h are as above. Then, if ws w = x, it is easy to see that w = # #) w, so again the conditions are simple to check. The dual HRVW KSH M direction for SDP (see [11, 17]) is an instance of case (c) Here takes v to (svx 1 x 1 vs) 2, is the identity, g x# n times x 1 , and h = s. We presented the NT direction above in the form that is most useful for computing the directions, and only for SDP. But it is applicable in more general self scaled conic ....

R. D. C. Monteiro. Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim., 7:663--678, 1997.

....The HRVW KSH M Direction. This direction is derived by using (7) 9) as is, and then taking the symmetric part of the resulting #X . This method to make a symmetric direction was independently proposed by Helmberg, Rendl, Vanderbei and Wolkowicz [13] Kojima, Shindoh and Hara [18] and Monteiro [21]. Polynomial time convergence was proved for the path following algorithms using this direction. For related work, see also the papers of Lin and Saigal [19] Potra and Sheng [32] and Zhang [40] The HRVW KSH M direction is currently very popular for practical implementation because of its ....

R. D. C. Monteiro. Primal-dual path-following algorithms for semidefinite programming. Technical report, School of Industrial and System Engineering, Georgia Tech, Atlanta, GA30332, 1995. (To appear in SIAM Journal on Optimization).

....relaxation by #SOCP2 MC#. 4. SOCP3: SOCP2 with the triangle inequalities (27) SDPA 5.01 ( 9] an implementation of primal dual interior point method, was used to solve SDP. SOCP was solved by our own implementation of the primal dual interior point method. In both solvers, the HKM direction ([14, 17, 18]) was used and the Mehrotra type predictor corrector method was adopted. All computations were performed on an Intel Pentium based computer (CPU: Intel Celeron 733 MHz, Memory: 512 MB, OS: VINE Linux 2.1, C and C compilers: egcs 2.91.66) Our SOCP solver, which we implemented from scratch to ....

Monteiro, R.D.C. (1995). Primal-dual path-following algorithms for semidefinite programming. SIAM Journal on Optimization 7, 663--678.

....[3, 21] for more applications. The Primal Dual Interior Point Method (PDIPM) is known as one of the most powerful numerical methods for general SDPs. The PDIPM not only has excellent features such as polynomial time convergence in theory, but also solves various SDPs e#ciently in practice. See [6, 11, 12, 13, 17, 18, 22]. The SDPA (SemiDefinite Programming Algorithm) is computer software for general SDPs based on the PDIPM. In their paper [23] Yamashita Fujisawa Kojima reported high performance of the SDPA 6.0 to various problems including SDPLIB [7] benchmark problems. The high performance of the SDPA 6.0 is ....

....1, 2, m) dXY X# dY = R, dY = 2) where P = F 0 F i x i X, d i = c i . Y (i = 1, 2, m) and R = # I XY . Note that the symmetrization of the matrix dY is needed for Y dY being a symmetric matrix. This search direction is called the HRVW KSH M direction [11, 12, 13]. There have been proposed various search directions as systems of di#erent modified Newton equations. The HRVW KSH M direction is employed in the SDPT3 [18] and the CSDP [6] besides the SDPA. We update the current point (dx, dX, dY ) to the new iterate (x # p dx, X # p dX,Y # d dY ) where ....

R.D.C. Monteiro, "Primal-dual path following algorithms for semidefinite programming," SIAM Journal on Optimization 7 (1997) 663--678.

....problems. It has a lot of applications in various fields such as combinatorial optimization [9] control theory [4] robust optimization [3, 23] and chemical physics [15] See [19, 22, 23] for a survey on SDPs and the papers in their references. The PDIPA (primal dual interior point algorithm) [10, 12, 14, 16] is known as the most powerful and practical numerical method for solving general SDPs. The method is an extension of the PDIPA [11, 18] developed for LPs. The SDPA (SemiDefinete Programming Algorithm) presented in this paper is a PDIPA software package for general SDPs based on the paper [7, 12] ....

....# # ) approximate optimal solution of the SDP (1) if it is an # approximate feasible solution and the relative duality gap is less than # # . To compute the search direction (dx, dX, dy) the SDPA employs Mehrotra type predictor corrector procedure [13] with use of the HRVW KSH M search direction [10, 12, 14]. Let (x, X,Y ) x ) be a current point satisfying X O. In Mehrotra type predictor procedure, we first choose a centering parameter # p for the predictor step. If (x, X,Y ) is an # approximate feasible solution of the SDP (1) then let # p = 0, else let # p = #, where # is a ....

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R.D.C. Monteiro, "Primal-dual path following algorithms for semidefinite programming," SIAM Journal on Optimization 7 (1997) 663--678.

....given by 1 [pMp i p TMTpT] Hp(M) for any matrix M, and where the scaling matrix P determines the symmetrization strategy. Some popular choices for P are listed in Table 1. The resulting linear systems are now P Reference [X21 (X21 SX21 ) 21 X21 ] Nesterov and Todd (NT) 8] X Monteiro [7], Kojima et al. 5] S Monteiro [7] Helmberg et al. 3] Kojima et al. 5] I Alizadeh, Haeberley and Overton (AHO) 1] Table 1: Choices for the scaling matrix P. solvable (for the AHO direction (P = I) solvability is only guaranteed if (X, S) lie in a 1 certain neighbourhood of the central ....

....: for any matrix M, and where the scaling matrix P determines the symmetrization strategy. Some popular choices for P are listed in Table 1. The resulting linear systems are now P Reference [X21 (X21 SX21 ) 21 X21 ] Nesterov and Todd (NT) 8] X Monteiro [7] Kojima et al. 5] S Monteiro [7], Helmberg et al. 3] Kojima et al. 5] I Alizadeh, Haeberley and Overton (AHO) 1] Table 1: Choices for the scaling matrix P. solvable (for the AHO direction (P = I) solvability is only guaranteed if (X, S) lie in a 1 certain neighbourhood of the central path) The choice P = X in Table 1 ....

R.D.C. Monteiro. Primal-dual path-following algorithms for semidefinite programming. SIAM Journal on Optimization, 7(3):663-678, 1997.

....On the other hand the computation of a single step is computationally rather expensive and this limits these methods to problems of moderate size, say around 3000 constraints. The important references on interior point methods for SDP include Alizadeh et al. [2] Helmberg et al. [23] Monteiro [44], Todd et al. [55] These are primal dual algorithms. Benson et al. [7] propose a dual scaling algorithm. Finally Zhang [63] develops a framework to extend some primal dual interior point algorithms from linear programming to semidefinite programming. In this section we discuss interior point ....

R.D.C Monteiro, Primal Dual path following algorithms for semidefinite programming, SIAM Journal on Optimisation, 7(1997), pp. 663-678.

....example where the Newton direction is not well defined at a pair of strictly feasible solutions. This does not seem to cause di#culties in practice. A more general approach is to apply a similarity to XS before symmetrizing it. This was discussed for a specific pair of similarities by Monteiro [39], and then in general by Zhang [70] So let P be nonsingular, and let us replace the last part of # P by 1 2 (PXSP 1 P T SXP T ) #I. 14) Zhang showed that this is zero exactly when XS = #I as long as X and S are symmetric. An alternative way to view this is to scale (P) so that the ....

....search directions are known as the HRVW KSH M, dual HRVW KSH M, and NT directions. The first was introduced by Helmberg, Rendl, Vanderbei, and Wolkowicz [28] and independently Kojima, Shindoh, and Hara [33] using di#erent motivations, and then rediscovered from the perspective above by Monteiro [39]. The second was also introduced by Kojima, Shindoh, and Hara [33] and rediscovered by Monteiro; since it arises by switching the roles of X and S, it is called the dual of the first direction. The last was introduced by Nesterov and Todd [46, 47] from yet another motivation, and shown to be ....

R. D. C. Monteiro. Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim., 7:663--678, 1997.

.... Gamma1 1 L Gamma1 2 Delta Delta Delta L Gamma1 Gamma1 is a lower triangular matrix with possible nonzero elements specified in F . 3 Primal dual interior point method In this section, we describe a generic framework for primal dual interior point methods applied to SDPs (1) and (2) [10, 13, 16, 18, 21, 24]. Various search directions have been proposed so far for primal dual interior point methods [21] Among others, we restrict ourselves to the HRVW KSH M search direction [10, 13, 16] in this article. In addition, we only consider a simple primal dual path following interior point method which is ....

.... we describe a generic framework for primal dual interior point methods applied to SDPs (1) and (2) 10, 13, 16, 18, 21, 24] Various search directions have been proposed so far for primal dual interior point methods [21] Among others, we restrict ourselves to the HRVW KSH M search direction [10, 13, 16] in this article. In addition, we only consider a simple primal dual path following interior point method which is not a Mehrotra type since we implement this simple method in the completion method described in section 5. On the other hand, we can solve the standard equality form SDP resulting ....

R. D. C. Monteiro, Primal-dual path-following algorithms for semidefinite programming, SIAM Journal on Optimization 7 (1997) 663--678.

....X 1 = WD 1 W T (4) where W = L 1 1 L 1 2 L 1 # 1 is a lower triangular matrix with possible nonzero elements specified in F . 3 Primal dual interior point method In this section, we describe a generic framework for primal dual interior point methods applied to SDPs (1) and (2) [10, 13, 16, 18, 21, 24]. Various search directions have been proposed so far for primal dual interior point methods [21] We restrict ourselves to the HRVW KSH M search direction [10, 13, 16] in this article. In addition, we only consider a simple primal dual path following interior point method which is not a Mehrotra ....

.... In this section, we describe a generic framework for primal dual interior point methods applied to SDPs (1) and (2) 10, 13, 16, 18, 21, 24] Various search directions have been proposed so far for primal dual interior point methods [21] We restrict ourselves to the HRVW KSH M search direction [10, 13, 16] in this article. In addition, we only consider a simple primal dual path following interior point method which is not a Mehrotra type since we implement this simple method in the completion method described in section 5. On the other hand, we can solve the standard equality form SDP resulting ....

R. D. C. Monteiro, Primal-dual path-following algorithms for semidefinite programming, SIAM Journal on Optimization 7 (1997) 663--678.

....# R , the minimization of g( over R m and the minimization of g p ( over R m are equivalent. 8 Suppose that X # S n and S # S n . Then we can describe the coe#cient matrix # M (or M) of the so called Schur complement equation for the HRVW KSH M search direction [14, 17, 19] applied to the primal dual pair of SDPs (5) and (6) or applied to the primal dual pair of SDPs (1) and (2) as follows M # # # # # # M 11 M 12 M 1m M 21 M 22 M 2m . Mm1 Mm2 Mmm # # # # # # S m , # M # # # M h h T hm 1 # # # S m 1 # # # # # # # # # # # # ....

....condition X # S # = # I for some # # R . Therefore one additional iteration of a primal dual interior point method is expected to work very e#ectively. Now we will show how we perform one iteration of the primal dual interior point method using the HRVW KSH M search direction [14, 17, 19] to get approximate optimal solutions with a higher accuracy. We first compute a search direction (dX , dy, dw, dS) by solving A p . X # dX) a p (p = 1, 2, m) I . X # dX) b, m # p=1 A p (y # p dy p ) I(w # dw) S # dS) C, X # dS dXS # = # ....

R. D. C. Monteiro, "Primal-dual path-following algorithms for semidefinite programming, " SIAM J. Optim. 7 (1997) 663--678.

....solve mixed semidefinite linear programming problems. The algorithm used by CSDP is a predictor corrector variant of the primal dual interior point method of Helmberg, Rendl, Vanderbei, and Wolkowicz [6] This method was also independently discovered by Kojima, Shindo and Hara [7] and Monteiro [8]. Thus the method is known as the HRVW KSH M or HKM method. This method is also known as the XZ method because the complementarity condition is expressed in the form XZ = I. The problem data consist of the matrices A i , the objective matrix C, and the right hand side vector a. In addition, an ....

Renato D. C. Monteiro. Primal--dual path--following algorithms for semidefinite programming. SIAM Journal on Optimization, 7(3):663--678, 1997.

....versions of the semidefinite complementarity condition in (3.1) which result in di#erent search directions. The form we use, 1 2 (x S z S z S xS ) 0, results in the AHO [7] also known as the XZ ZX search direction) whereas using xS z S = 0 (see Remark 1.2. 1) results in the KSH HRVW M [49, 59, 75] (also known as the XZ direction) of the SDP literature. Yet another direction is the NT direction [108] which is obtained by using the condition w 1 2 (x S z S )w 1 2 = 0, with w = x 1 2 S (x 1 2 S z S x 1 2 S ) 1 2 x 1 2 S . More generally, using the Monteiro Zhang symmetrization ....

R. D. C. Monteiro. Primal--dual path--following algorithms for semidefinite programming. SIAM Journal on Optimization, 7:663--678, 1997.

....: d x ; d y ; d s ) resulting from this system of equations the Nesterov Todd system or N T system is called the Nesterov Todd direction or N T direction. It was later recognized by Todd Toh Tutuncu [42] that in the SDP case 20 the N T direction is a member of the Monteiro Zhang family [27, 45, 28]. In the linear programming case the third N T equation becomes 0 B B B B s1 x 1 . sn xn 1 C C C C A d x d s = 0 B B B B x 1 Gamma s 1 . xn Gamma s n 1 C C C C A : 1.9) If we multiply this equation from the left with the operator X = Diag(x) r 2 F (x) Gamma1=2 ....

R.D.C. Monteiro. Primal-dual path following algorithms for semidefinite programming. SIAM Journal on Optimization, 7:663--678, 1997.

....and Computer Engineering, McMaster University, Hamilton, Ontario, L8S 4L7, Canada. 3 Econometric Institute, Erasmus University Rotterdam, The Netherlands. 1 Introduction Recently, there have been many interior point algorithms developed for semidefinite programming (SDP) see for example [1, 2, 3, 7, 11, 13, 14, 16, 20]. These algorithms differ in their choices of scaling matrix, the size of the central path neighborhoods, and stepsize rules, among others. In particular, the algorithms of Kojima Shindo Hara [7] and Nesterov Todd [13, 14] are based on the primal dual scaling and they both can be viewed as ....

Monteiro, R.D.C., "Primal-dual path following algorithms for semidefinite programming," Technical Report, School of Industrial and Systems Engineering, Georgia Tech, Atlanta, Georgia, U.S.A., 1995. To appear in SIAM Journal on Optimization. 25

.... Delta Bmm 1 C C C A 2 S m ; B j 0 C c c T Bm 1;m 1 1 A 2 S 1 m : 9 = 9) The matrix B above corresponds to the coefficient matrix of the so called Schur complement equation for the HRVW KSH M search direction [15, 18, 20]. In particular, the matrix B is known to be symmetric and positive definite. It should be also noted that when the primal dual pair of matrix variables X 2 S n and S 2 S n satisfies XS = I as in the succeeding discussions, the matrix B is also identical to the coefficient matrix of the ....

R. D. C. Monteiro, "Primal-dual path-following algorithms for semidefinite programming, " SIAM J. Optim. 7 (1997) 663--678.

....(1.1) and (1.2) if and only if they are solutions of the following nonlinear system: A i ffl X = b i ; i = 1; m; 1.3a) m X i=1 y i A i S = C; 1.3b) XS = 0; X 0; S 0: 1.3c) Over the last couple of years many interior point methods for solving (1. 3) have been investigated (cf. [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]) In the present paper we consider a class of primal dual interior point algorithms for SDP that generalize the Mizuno Todd Ye predictor corrector method [9] The latter method was originally introduced for linear programming and was later generalized by Nesterov and Todd [12] for a more general ....

....1=2 s S 1=2 P Gamma1 ] T [J 1=2 s S 1=2 P Gamma1 ] I: 6 The lemma is proved by taking Q T x = J 1=2 x X Gamma1=2 P Gamma1 and Q T s = J 1=2 s S 1=2 P Gamma1 . Note that J x = J s = I and P = X Gamma1=2 or S 1=2 define the directions formulated by Monteiro [10] which are particular cases of the direction originally proposed by Kojima, Shindoh and Hara [6] The direction defined by J x = J s = I and P = S 1=2 was derived independently by Helmberg, Rendl, Vanderbei and Wolkowicz [2] Finally, the case J x = X 1=2 SX 1=2 ] 1=2 ; or J s = S 1=2 ....

R. D. C. Monteiro. Primal-dual path following algorithms for semidefinite programming. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA, September 1995.

No context found.

R.D.C. MONTEIRO. Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim., 7(3):663--678, 1997.

No context found.

R. D. C. Monteiro, "Primal-dual path-following algorithms for semidefinite programming, " SIAM J. Optim. 7 (1997) 663--678.

No context found.

R.D.C. Monteiro, "Primal-dual path following algorithms for semidefinite programming," Working Paper, School Industrial and Systems Engineering, Georgia Tech., Atlanta, GA 30332, September 1995. To appear in SIAM Journal on Optimization.

No context found.

R.D.C. MONTEIRO. Primal-dual path-following algorithms for semidefinite programming. SIAM J. Optim., 7(3):663--678, 1997.

No context found.

R. D. C. Monteiro. Primal-dual path following algorithms for semidefinite programming. Working paper, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA, September 1995.

No context found.

R.D.C. Monteiro. Primal-dual path following algorithms for semidefinite programming. SIAM Journal on Optimization, 7:663--678, 1997.

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