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IMPLEMENTATION OF PRIMAL-DUAL METHODS FOR SEMIDEFINITE PROGRAMMING BASED ON MONTEIRO AND TSUCHIYA NEWTON DIRECTIONS AND THEIR VARIANTS RENATO D.C. MONTEIRO and PAULO ZANJA COMO
School of Industrial and Systems Engineering, Georgia Tech, Atlanta, 30332, USA
(
[email protected],
[email protected]) (Received date to be inserted)
Monteiro and Tsuchiya [22] have proposed two primal-dual Newton directions for semide nite programming, referred to as the MT directions, and established polynomial convergence of pathfollowing methods based on them. This paper reports some computational results on the performance of interior-point predictor-corrector methods based on the MT directions and a variant of these directions, called the S-Ch-MT direction. We discuss how to compute these directions eciently and derive their corresponding computational complexities. A main feature of our analysis is that computational formulae for these directions are derived from a uni ed point of view which entirely avoids the use of Kronecker product. Using this uni ed approach, we also present schemes to compute the Alizadeh-Haeberly-Overton (AHO) direction, the Nesterov-Todd direction and the HRVW/KSH/M direction with computational complexities (for dense problems) better than previously reported in the literature. We present some computational results for small dense problems, which are quite promising. We have obtained better performance for the methods based on the AHO, NT and HRVW/KSH/M directions. We have also observed that the method based on the S-Ch-MT direction compares favorably with the new implementation of the methods based on the NT direction and the HRVW/KSH/M direction. KEY WORDS: Semide nite programming, interior-point methods, path-following methods, predictor-corrector methods, higher-order methods, Newton directions, central path, numerical implementation
1 Introduction Many authors have proposed interior-point algorithms for solving semide nite programming (SDP) problems (see [1, 2, 5, 9, 12, 14, 15, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34]). Most of these more recent works are concentrated on primal-dual methods. One of the main goals of this paper is the implementation of primal-dual path-following and predictor-corrector algorithms based on two pure This work was partly supported by the NSF grants CCR-9700448, CCR-9902010 and INT9600343.
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R.D.C. MONTEIRO AND P. ZANJA COMO
Newton directions proposed in Monteiro and Tsuchiya [22], which we refer to as the X-MT and S-MT directions (or simply MT directions). In addition to these two directions, we also consider a variant of the S-MT direction which can be vaguely described as follows. By de nition, the S-MT direction is the Newton step with respect to the central path equation S 1=2 XS 1=2 ? I = 0, where X is the primal matrix variable and S is the dual slack matrix variable. The variant of the S-MT direction that we consider, referred to as the S-Ch-MT direction, is the Newton step with respect to the alternative central path equation LTs XLs ? I = 0, where Ls is the Cholesky factor of S. The S-Ch-MT direction has similar theoretical properties as the MT directions (see Section 2), but its computational features are quite distinct from the latter ones. For example, the S-Ch-MT direction is considerably cheaper to compute than the MT directions. In a similar manner, it is possible to derive a variant of the X-MT direction, called the X-Ch-MT direction; however, according to our numerical experiments, the X-Ch-MT direction is not as ecient as the S-Ch-MT direction, and for this reason we omit the description of the X-Ch-MT direction in this paper. Our paper is organized as follows. In Section 2, we derive the S-Ch-MT direction and study the complexity of primal-dual path-following algorithms based on this direction. We argue that all the results obtained for the MT directions can be generalized in a similar way to the S-Ch-MT direction. In Section 3, we derive the S-MT, X-MT and S-Ch-MT directions from a computational point of view. Our approach is a uni ed one in the sense that we develop general formulae for the Newton path-following and the predictor-corrector steps associated with a general central path equation of the form (X; S) ? I = 0, where is an appropriate \central path map". Our formulas are based on certain operators determined by the Newton system and completely avoid the use of Kronecker products. Using this general framework, we give in Subsections 3.1 to 3.3 the derivation of the S-MT, X-MT and S-Ch-MT directions, respectively. In addition to these three directions, we also use our framework to give in Subsections 3.4, 3.5 and 3.6 alternative derivations for the Alizadeh-Haeberly-Overton (AHO) direction introduced in [2], the HRVW/KSH/M direction introduced independently by Helmberg, Rendl, Vanderbei and Wolkowicz [9] and Kojima, Shindoh and Hara [15], and later rediscovered by Monteiro [19] using a di erent insight, and the Nesterov and Todd (NT) direction introduced in [27]. We also obtain computational complexities for these three directions which are better than those previously reported in the literature. In Subsection 3.7, we brie y summarize the computational complexities to obtain the various Newton directions mentioned above. In Section 4, we provide computational results of our implementation of a Mehrotra predictor-corrector algorithm based on the X-MT, S-MT and S-Ch-MT directions. We also consider a hybrid version, referred to as the S-MT-Hyb variant, which uses either the S-MT direction or the S-Ch-MT direction, depending on the proximity of the iterate to the central path. We compare these methods against similar methods based on the AHO direction, the HRVW/KSH/M direction and the Nesterov and Todd (NT) direction. Results of numerical experiments of primal-dual interior-point algorithms based on the AHO, HRVW/KSH/M and NT directions
IMPLEMENTATION OF PRIMAL-DUAL SDP METHODS
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are given in several papers including Alizadeh, Haeberly and Overton [2], Helmberg, Rendl, Vanderbei and Wolkowicz [9], Brixius, Potra and Sheng [4], Fujisawa, Kojima and Nakata [6], Kojima [13], Todd, Toh and Tutuncu [30]. Our code is based on an earlier version of Todd, Toh and Tutuncu's code, which in turn is based on one of Alizadeh, Haeberly and Overton. The methods based on the AHO and the NT directions of this earlier version of Todd, Toh and Tutuncu's code have been slightly modi ed to compute symmetric matrix products more eciently (see A.10 of the Appendix). We run a total of 50 random problems which were chosen over two di erent classes of problems. The problems we tested were not large in size, and involved only dense matrices. An implementation for large scale problems with sparse matrices should take advantage of the presence of sparsity. However, since the main objective of this paper is to test the robustness of the new MT directions and compare them to other known SDP directions, we have not tried to solve SDP problems with sparse data. Section 5 is devoted to nal remarks. We now give a brief summary of the conclusions of our numerical experiments. First, the S-Ch-MT method is as fast as the HRVW/KSH/M and the NT methods. Second, the S-MT and X-MT method is as robust as the AHO method in terms of nding highly accurate solutions (that is, solutions with small duality gap and feasibility violation). Third, the S-Ch-MT and S-MT-Hyb method in addition to being fast are also robust in nding highly accurate solutions. 1.1 Notation and terminology
The following notation is used throughout the paper. The superscript T denotes transpose.
