M. Muramatsu and R.J. Vanderbei. Primal-dual affine-scaling algorithm fails for semidefinite programming. Technical Report SOR 97-04, Princeton University, NJ 08544, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by De Klerk et al. in [15] from LP to SDP. These algorithms minimize the duality gap over ellipsoids in the scaled primal dual space, where the matrix L = D 1 2 is used for the scaling. The primal dual method fails if either of the scalings L = X 1 2 or L = S 1 2 from Table 1 is used [40]. 5.4 Infeasible start methods Several infeasible start algorithms have been suggested. A review of traditional big M initialization strategies may be found in [58] One of the first infeasible start predictor corrector algorithms was by Potra and Sheng [46] Other references include [31, 37] ....

M. Muramatsu and R.J. Vanderbei. Primal-dual affine-scaling algorithm fails for semidefinite programming. Technical Report SOR 97-04, Princeton University, NJ 08544, 1997.

....scaling direction is the so called (primal) HKM affinescaling direction, where RHY = 1 2 Gamma XY S Gamma1 S Gamma1 Y X Delta . As mentioned, this search direction is not globally convergent for any choice of step length. In particular, it can converge to a non optimal point [35]. Moreover, it cannot be used for copositive programming because a copositive matrix S can be singular despite hX; Si 0 for all X 2 K n fOg, e.g. S = 4 2 2 1 , which is linearly independent from E. Finally, a primal dual affine scaling direction for SDP which is also defined for ....

.... of hE; dXi = 0 ; hA i ; dXi = 0 ; i 2 f1; mg ; Edy 0 P m i=1 A i dy i Gamma dS = O ; dS 1 2 Gamma X Gamma1 (dX)S S(dX)X Gamma1 Delta = GammaS: As with the (primal) HKM direction, no primal dual SDP algorithm using this search direction is globally convergent [35]. The preceding observations prove two things: Gamma Using only primal dual affine scaling directions in interior point methods for conic programming does not necessarily lead to a globally convergent algorithm; Gamma One cannot guarantee a fixed feasible step length for all primal dual affine ....

M. Muramatsu, R.J. Vanderbei (1997), Primal-dual affine-scaling algorithm fails for semidefinite programming. Technical Report SOR 97-04, Princeton University, NJ 08544.

....which yields the same search direction as in [21] for the usual path following primal dual methods. However, the so called V space interpretation presented in [26] can be used for other non path following schemes as well. For primal dual affine scaling method, Muramatsu and Vanderbei [20] investigated the performance of various search directions. For several of the known search directions they showed that the convergence fails, even for a simple example, except for the direction based on Nesterov and Todd [21] This gives an indication that the symmetrization based on Nesterov and ....

M. Muramatsu and R.J. Vanderbei. Primal--dual affine--scaling algorithms fail for semidefinite programming. Technical Report, Department of Mechanical Engineering, Sophia University, Japan, 1997.

....iteration complexity to yield the duality gap accuracy ffl. Other scaling algorithms have been proposed in the past. For example, an SDP equivalent of Dikin s affine scaling algorithm could be very fast. However this algorithm may not even converge. Muramatsu [22] and Muramatsu and Vanderbei [23] showed an example in which these affine scaling algorithms will not converge to an optimal answer. There are also quite a few computational results and implementations of these interior algorithms, see Anstreicher and Fampa [4] Alizadeh, Haeberly, and Overton [3] Fujisawa, Kojima and Nakata ....

M. Muramatsu and R. Vanderbei, "Primal-dual affine scaling algorithms fail for semidefinite programming, " Technical Report, SOR, Princeton University, Princeton, NJ, 1997.

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