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...ent interior-point methods has turned these techniques into powerful practical tools; see Alizadeh [Ali92b, Ali91, Ali92a], Kamath and Karmarkar [KK92, KK93], Helmberg, Rendl, Vanderbei and Wolkowicz =-=[HRVW94]-=-. For a more detailed survey of semidefinite programming in combinatorial optimization, we refer the reader to the recent paper by Alizadeh [Ali95]. Control and system theory Semidefinite programming ...

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...irections In this package, the user has four choices of symmetrizations resulting in four different search directions, namely, (1) the AHO direction [3], corresponding to P = I; (2) the HKM direction =-=[14]-=-,[16],[21], corresponding to P = Z 1=2 ; (3) the NT direction [26], corresponding to P = N \Gamma1 , where N is the unique matrix such that D := N ZN = N \Gamma1 XN \Gamma is a diagonal matrix (then W...

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...ix of order � ns j=1 sj. In the current release of this package, the user has two choices of scaling operators parametrized by P and G, resulting in two different search directions: the HKM direction =-=[10,12,16]-=-, and the NT direction [17]. See also Tsuchiya [26, 27] for the second-order case. (1) The HKM direction. This choice uses the scaling matrix P = Z1/2 for the semidefinite blocks and a symmetric block...

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...if the problem is defined over large matrix variables or a huge number of constraints interior point methods grow terribly slow and consume huge amounts of memory. The most efficient methods of today =-=[15, 23, 2, 32, 45, 29]-=- are primal-dual methods that require, in each iteration of the interior point method, the factorization of a dense matrix of order equal to the number of constraints and one to three factorizations o...

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...ansformed into an algorithm for SDP in a mechanical way. Since then several authors have proposed interior-point algorithms for solving SDP problems including Helmberg, Rendl, Vanderbei and Wolkowicz =-=[5]-=-, Jarre [8], Kojima, Shindoh and Hara [11], Nesterov and Nemirovskii [16], Nesterov and Todd [19, 18] and Vandenberghe and Boyd [20]. Among the above works, Kojima, Shindoh and Hara [11] and Nesterov ...

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...rix with WSW = X, so that X = S. The resulting 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, an...

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...one of the H..K..M directions; and the last on the NT direction. The second direction has a convoluted history: it apparently first appeared in a preprint of Helmberg, Rendl, Vanderbei, and Wolkowicz =-=[5] in late 1-=-993; was rediscovered independently as one member of a family of directions introduced in early 1994 by Kojima, Shindoh, and Hara [8]; and was most recently derived (as well as a "dual" dire...