K. Fujisawa, M. Fukuda, M. Kojima and K. Nakata, "Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm)," in: H. Frenk, K. Roos, T. Terlaky and S. Zhang eds., High Performance Optimization, (Kluwer Academic Press, 2000) pp.267-301.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and Ye [19] which is the analogue of the dual scaling algorithm for linear programming. The dual scaling algorithm uses only S to generate the iterate direction. The third is the primaldual scaling algorithm which uses both X and S to generate iterate directions (see Todd [16] Fujisawa et al. [8], and references therein) All these algorithm possess O( p n log(1=ffl) iteration complexity to yield accuracy ffl. Although they can be solved in polynomial time, large scale SDP problems are difficult to solve. In practice, only problems with dimension (n) in hundreds are being solved and ....

K. Fujisawa, M. Fukuda, M. Kojima and K. Nakata, "Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm)," Research Report B-330, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, September 1997.

....and Zhao [32] Zhao, Karisch, Rendl, and Wolkowicz [35] 36] To the best of our knowledge, the largest problem that could be solved was at n = 900 from their reports. After the initial version of this paper was submitted, one more implementation came out: Fujisawa, Fukuda, Kojima and Nakata [10] reported that they could solve a maximum cut semidefinite program with n = 1250, using a powerful work station. The practical winner of solving semidefinite programs was Helmberg and Rendl [12] an implementation of a non interior point algorithm called the bundle method. They reported the ....

....n was only a few hundred or less, so that no available computation result could be compared to ours. After our results reported, a study of using a primal dual algorithm for solving relative larger problems, including the maximum cut problem, was conducted by Fujisawa, Fukuda, Kojima, and Nakata [10]. They tested solving sparse maximum cut semidefinite programs with dimension up to 1250. On a sparse problem with dimension of 1000, they required 63,130 seconds; on a problem of dimension 1250, they used 111,615 seconds. Their computations were performed on a DEC AlphaServer 8400 with a ....

K. Fujisawa, M. Fukuda, M. Kojima and K. Nakata, "Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm)," Research Report B-330, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, September 1997.

....SDPs e#ciently. The SDPA is written in C language, so that one can call its routines from his own C program. Some numerical results of the SDPA 4. 0, an old version of the SDPA with its comparison to some other software packages including the SDPT3 and the CSDP were reported in the paper [8]. Since then, the performance of the SDPA and those software packages has been considerably improved. In particular, the SDPA 6.0 (the latest version at the moment of writing) incorporated LAPACK [1] and adopted one dimensional array data structure to store dense matrices so as to fit to dense ....

....both F i and F j are dense, and the second the case where F i is dense and F j is sparse, and the third the case where both F i and F j are sparse. This significantly contributes to the computational e#ciency of the SDPA. This sparsity handling technique was employed also in the SDPT3 [20] See [7, 8] for more details. We summarize the entire algorithm of the SDPA below. The Primal Dual Interior Point Algorithm of the SDPA Step 0 (Initialization) Choose an initial point (x ) satisfying X O. Let k = 0. Step 1 (Checking Feasibility) If (x ) is an (#, # # ) approximate ....

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K. Fujisawa, M. Fukuda, M. Kojima and K. Nakata, "Numerical Evaluation of SDPA (SemiDefinite Programming Algorithm)," in: H. Frenk, K. Roos, T. Terlaky and S. Zhang eds., High Performance Optimization, (Kluwer Academic Press, 2000) pp.267-301.

....can be as large as n(n 1) 2 even if the constraint matrices A 1 ; A 2 ; Am are assumed to be linearly independent. As m becomes larger for a fixed n, more cpu time is spent for (a) the computation of the coefficient matrix B, and (b) the computation of the solution dz of Bdz = s. See [5, 23]. Fujisawa Kojima Nakata [7] proposed an efficient method for computing the coefficient matrix B when the data matrices A p 2 S n (p = 1; 2; m) are sparse. Also, the computation of the coefficient matrix B is carried out efficiently, when the data matrices A p 2 S n (p = 1; 2; ....

K. Fujisawa, M. Fukuda, M. Kojima and K. Nakata, "Numerical evaluation of SDPA (SemiDefinite Programming Algorithm)," in: H. Frenk, K. Roos, T. Terlaky and S. Zhang, eds., High Performance Optimization (Kluwer Academic Publishers, Dordrecht, 1999) 267--301.

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