Nesterov, Yu., Todd, M. J., Ye, Y. \Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems", Math. Prog. A v. 84 (1999), 227-267.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....search directions. 7.4. Infeasible start. As we already commented, the standard initialization techniques as given in [16] can be applied. We could also apply the surface following idea developed in [17] However, a particularly attractive choice would be an e ective analogue of the approach of [21]. Such analogues seem possible and the development of such techniques is left for future work. The idea to solve the problem by tracing the primal path is, of course, a common place. The idea to trace what we call here the dual path is not new either (it originates from Nesterov [14] for a more ....

Nesterov, Yu., Todd, M. J., Ye, Y. \Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems", Math. Prog. A v. 84 (1999), 227-267.

....theorem (Section 2.5 of [8] Theorem 2.2 Any closed convex cone admits a self concordant, logarithmically homogeneous barrier function. The following straightforward, but usefully, properties of the # logarithmically homogeneous function can be found in Nesterov and Nemirovski [8] see also [9]. Proposition 2.3 Suppose that F (x) is an # logarithmically homogeneous barrier function for K. Then the following identities hold where x # int K and t 0: #F (tx) 1 t #F (x) 3) # 2 F (tx) 1 t 2 # 2 F (x) 4) # 2 F (x)x = #F (x) 5) #F (x) T x = #. 6) 5 ....

Yu. Nesterov, M.J. Todd, and Y. Ye, Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems, Mathematical Programming 84, 227 -- 267, 1999.

....the center of recent research activities in the interior point community. Several groups of authors independently extended the self dual embedding technique to semidefinite programming, viz. Potra and Sheng [20] De Klerk, Roos and Terlaky [6] Luo, Sturm and Zhang [10] and Nesterov, Todd and Ye [19]. The latter two papers concern the more general case of conic convex programming, each with a different emphasis. Nesterov, Todd and Ye [19] analyzed the application of logarithmically homogeneous barrier techniques to self dual embeddings and considered the associated complexity issues. Luo, ....

....technique to semidefinite programming, viz. Potra and Sheng [20] De Klerk, Roos and Terlaky [6] Luo, Sturm and Zhang [10] and Nesterov, Todd and Ye [19] The latter two papers concern the more general case of conic convex programming, each with a different emphasis. Nesterov, Todd and Ye [19] analyzed the application of logarithmically homogeneous barrier techniques to self dual embeddings and considered the associated complexity issues. Luo, Sturm and Zhang [10] were concerned with a general duality theory for conic problems and with the question how to determine the status of the ....

[Article contains additional citation context not shown here]

Y. Nesterov, M.J. Todd, and Y. Ye. Infeasible--start primal--dual methods and infeasibility detectors for nonlinear programming problems. Technical Report 1156, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 1996.

....e x q s q Gamma 1 fl fl fl fl fl (29) and centering steps can be taken until the desired level of centering and feasibility is reached. Excellent references on convergence of this method as well as detailed descriptions of proximity measures are Anstreicher and Vial [1] Nesterov, Todd and Ye [30] and Ralph and Wright [31] In practice, we implemented a pure Newton step. 5 Jacobian matrix approximations Faced with an algorithm that uses derivative information (the Jacobian rF ) one asks if it is not possible to keep the spirit of the method while avoiding the derivatives evaluations. The ....

Yu. Nesterov, M.J. Todd and Y. Ye, "Infeasible Start PrimalDual Methods and Infeasibility Detectors for Nonlimear Programming Problems", Working Paper, Department of Management Sciences, The University of Iowa (April 1996), to appear in Mathematical Programming.

....c T s = 0 and s 2 (A K ) n Gamma K : Hence, CP( Gammac=kck 2 2 ; 0; A Ker c T ; K) has no one sided dual level directions if and only if the conic convex program CP(b; c; A; K) is dual infeasible and has no one sided dual directions. 2 We remark that Nesterov, Todd and Ye [33] called a closed conic convex program strictly infeasible if it has a dual improving interior direction. Corollary 3 shows that a program is strictly infeasible in the sense of [33] if and only if it is infeasible and has no one sided directions. We will see in Corollary 4 that strict ....

....CP(b; c; A; K) is dual infeasible and has no one sided dual directions. 2 We remark that Nesterov, Todd and Ye [33] called a closed conic convex program strictly infeasible if it has a dual improving interior direction. Corollary 3 shows that a program is strictly infeasible in the sense of [33] if and only if it is infeasible and has no one sided directions. We will see in Corollary 4 that strict infeasibility implies strong infeasibility. The relation between dual directions and primal strong feasibility has now been fully investigated. We now proceed to study the relationships ....

Yu. Nesterov, M.J. Todd, and Y. Ye. Infeasible--start primal--dual methods and infeasibility detectors for nonlinear programming problems. Technical Report 1156, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 1996.

No context found.

Yu. E. Nesterov, M. J. Todd, and Y. Ye. Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math. Progr., 84:227-- 267, 1999.

....of all ones, and X Z means that X Gamma Z is positive semidefinite. Obviously, SDP) is feasible if (QP) is feasible, since xx 0 2 n Thetan is feasible for (SDP) if x 2 n is feasible for (QP) Thus, we assume that (SDP) is feasible throughout this paper (see Nesterov, Todd and Ye [14] for detecting the feasibility of (SDP) The dual of the (SDP) problem is Z SD = Minimize P m i=1 b 2 i Delta i e 0 y subject to P m i=1 i A i D(y) Q; 2) where D(y) is the diagonal matrix such that d(D(y) y 2 n . Note that the dual is always feasible and it has an ....

Yu. E. Nesterov, M. J. Todd, and Y. Ye, "Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems," Technical Report No. 1156, School of Operations Research and and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, 1996.

....of all ones, and X Z means that X Gamma Z is positive semidefinite. Obviously, SDP) is feasible if (QP) is feasible, since xx 0 2 n Thetan is feasible for (SDP) if x 2 n is feasible for (QP) Thus, we assume that (SDP) is feasible throughout this paper (see Nesterov, Todd and Ye [14] for detecting the feasibility of (SDP) The dual of the (SDP) problem is Z SD = Minimize P m i=1 b 2 i Delta i e 0 y subject to P m i=1 i A i D(y) Q; 2) where D(y) is the diagonal matrix such that d(D(y) y 2 n . Note that the dual is always feasible and it has an ....

Yu. E. Nesterov, M. J. Todd, and Y. Ye, "Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems," Technical Report No. 1156, School of Operations Research and and Industrial Engineering, Cornell University, Ithaca, NY 14853-3801, 1996.

....and self dual interior point algorithm for solving linear programming (LP) problems. The algorithm can start from arbitrary (infeasible) interior points and achieves the best known complexity in term of the number of iterations without using a big initial constant. Recently, Nesterov, Todd, and Ye [17] proposed another type of homogeneous and self dual interior point algorithm for solving nonlinear conic problems. Although the homogeneous and self dual system treated in [17] resembles that in [23] the algorithm seems rather di#erent from [23] because it generates a sequence of points along ....

....complexity in term of the number of iterations without using a big initial constant. Recently, Nesterov, Todd, and Ye [17] proposed another type of homogeneous and self dual interior point algorithm for solving nonlinear conic problems. Although the homogeneous and self dual system treated in [17] resembles that in [23] the algorithm seems rather di#erent from [23] because it generates a sequence of points along the central path such that the parameter of the duality gap diverges. Indeed, Nesterov, Todd, and Ye seek a recession direction of a convex set, as in the primal method of de ....

[Article contains additional citation context not shown here]

Yu. E. Nesterov, M. J.Todd and Y. Ye, Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems, to appear in Mathematical Programming.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC