T. TSUCHIYA. A polynomial primal-dual path-following algorithm for second-order cone programming. Technical report, The Institute of Statistical Mathematics, Tokyo, Japan, 1997. 64  Home/Search   Document Details and Download   Summary   Related Articles   Check

....positive semide nite. This can directly be applied on the dual problem of SOCP to obtain the dual problem of the corresponding SDP, as will be shown below. The relationship between the Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543. Note that [2, 3, 6, 9, 10] are related to second order cone programming, hence they are also included as references. two dual problems is clear. However, when we consider the primal SOCP and try to relate it to its corresponding primal SDP, the situation is not as simple for example, suppose x is an optimal solution of ....

Takashi Tsuchiya, A Polynomial Primal-Dual Path-Following Algorithm for Second-Order Cone Programming, Research Memorandum No. 649, The Institute of Statistical Mathematics, Tokyo, Japan, October (Revised: December 1997).

....cone programming (SOCP) is an optimization problem having linear constraints and second order cone constraints. SOCP is a special case of symmetric cone programming ( 7] which also includes SDP and LP as special cases. Recently, primaldual interior point algorithms were developed for both SOCP ([19, 28, 29]) and symmetric cone programming ( 22, 26, 20] Several programs have been implemented to solve SOCP (e.g. 1] and symmetric cone programming (e.g. 27] Numerical experiments show that the computational cost of solving SOCP is much less than that of SDP, and similar to LP. It is natural to ....

Tsuchiya, T. (1997) A polynomial primal-dual path-following algorithm for secondorder cone programming. Research Memorandum No. 649, The Institute of Statistical Mathematics, Tokyo, Japan.

....class is also proposed by Gu [7] from a di#erent point of view with the name TTT family. In this paper, we consider Commutative Class of search directions for linear programming over symmetric cones. Based on the techniques developed in Monteiro and Zhang [13] Monteiro and Tsuchiya [12] Tsuchiya [19, 20], and Sturm [18] we analyze complexity of short, semi long, and long step algorithms using search directions of this class. For short step algorithm, we prove O( # n log 1 #) complexity of the algorithm. For long step, the same complexity result as in Monteiro and Zhang [13] for semidefinite ....

Tsuchiya, T. (1997) A polynomial primal-dual path-following algorithm for second-order cone programming. Research Memorandum No. 649, The Institute of Statistical Mathematics, Tokyo, Japan.

....its similarities with LP and SDP. The work Nesterov and Todd [NT94, NT95] though not specifically on QCQP, presents algorithms that, when specialized to QCQP, are extensions of Monteiro Adler algorithms in LP, and have polynomial time complexity of p p for a p block problems. Recently Tsuchiya [Tsu97] studied classes of primal dual algorithms with polynomial time complexity. Several papers have studied applications of the QCQP problem in various areas. Andersen, Christiansen and Overton study an application where the maximum load on solid material is examined [ACO94] Conn and Overton [CO94] ....

Takashi Tsuchiya. A Polynomial Primal-Dual Path-Following Algorithm for Second-Order Cone Programming. Technical Report Research Memorandum No. 649, The Institute of Statistical Mathematics, Tokyo, Japan, 1997.

....proof also works for quadratic cones. More generally we present a framework that proofs and algorithms dealing with SDP can be extended directly to all symmetric cones. A related approach based on properties of Jordan algebras has been proposed by Faybusovich in [Fay97a, Fay97b] and Tsuchiya [Tsu97, Tsu98] (the latter concentrates on applying Jordan algebra methods to QCQP problems. Faybusovich [Fay97a] works exclusively with Jordan algebra and its multiplication and develops his complexity analysis within this confine. Tsuchiya uses a similar approach but focuses on the QCQP part. He manages to ....

....= S 1=2 and P = X 1=2 . And the Nesterov Todd direction uses P = W 1=2 ; W = X 1=2 (X 1=2 SX 1=2 ) 1=2 X 1=2 : The convergence result for the short step algorithm over quadratic cones with P chosen as in the XS and Nesterov Todd method has previously been obtained by Tsuchiya [Tsu97], Tsu98] A further consideration for QCQP is the computational cost of the operations involving P in an implementation. To obtain iterations that are polynomial time computable, one has to avoid the matrix representation L(x) of an element and work directly with the vector. For the algorithms ....

Takashi Tsuchiya. A Polynomial Primal-Dual Path-Following Algorithm for Second-Order Cone Programming. Technical Report Research Memorandum No. 649, The Institute of Statistical Mathematics, Tokyo, Japan, 1997. Page 22 RRR 17-98

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T. TSUCHIYA. A polynomial primal-dual path-following algorithm for second-order cone programming. Technical report, The Institute of Statistical Mathematics, Tokyo, Japan, 1997. 64

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