Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457--466, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....somewhat later than, and independent of, that of [NN90] Nesterov and Nemirovskii obtain their complexity theorems by specializing their general results to SDP. We, on the other hand, take a specific interior point algorithm for linear programming (i. e Ye s projective potential reduction method [Ye90]) and extend it to SDP. Furthermore, we argue that essentially any known interior point linear programming algorithm can also be transformed into an algorithm for SDP in a mechanical way; proofs of convergence and polynomial time computability extend in a similar fashion. Jarre in [Jar91] and ....

....we develop a potential reduction method for solving the primal problem so that, within O( p nj log fflj) iterations, we get an approximate solution with at least ffl relative accuracy, if ffl is sufficiently small. Our development closely follows Ye s projective technique for linear programming [Ye90]. Ye s complexity analysis is also extended to semidefinite programs. 9 MIN MAX matrix or scalar, 0 C matrix or scalar, V matrix or scalar, 0 O matrix or scalar, A matrix, 0 N matrix R matrix, 0 S matrix, matrix or scalar, unrestricted T matrix or scalar, C matrix ....

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Y. Ye. A class of projective transformations for linear programming. SIAM J. Comput., 19(3), 1990.

....convex quadratic programs with convex quadratic constraints, and semidefinite programs all have explicit and easily computable self concordant barrier functions, and hence can be solved in polynomial time . On the other hand, Alizadeh [1] extends Ye s projective potential reduction algorithm [37] for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior point algorithms for solving SDP problems, including Alizadeh, Haeberly and Overton [2] Freund [5] ....

Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457--466, 1990.

....convex quadratic programs with convex quadratic constraints, and semidefinite programs all have explicit and easily computable self concordant barrier functions, and hence can be solved in polynomial time . On the other hand, Alizadeh [1] extends Ye s projective potential reduction algorithm [30] for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior point algorithms for solving the SDP problem and SDLCP, including Alizadeh, Haeberly and Overton [2] ....

Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457--466, 1990.

....quadratic programs with convex quadratic constraints, and semidefinite programs all have explicit and easily computable selfconcordant functions, and hence can be solved in polynomial time . Subsequently, Alizadeh [2] extends in a direct way Ye s projective potential reduction algorithm (see [21]) for LP to the context of SDP and argues that many known interior point LP algorithms can also be transformed 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 ....

Y. Ye, A class of projective transformations for linear programming, SIAM Journal on Computing, 19 (1990), pp. 457--466. 18

....convex quadratic programs with convex quadratic constraints, and semidefinite programs all have explicit and easily computable self concordant barrier functions, and hence can be solved in polynomial time . On the other hand, Alizadeh [1] extends Ye s projective potential reduction algorithm [37] for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior point algorithms for solving SDP problems, including Alizadeh, Haeberly and Overton [2] Freund [3] ....

Y. Ye, A class of projective transformations for linear programming, SIAM Journal on Computing, 19 (1990), pp. 457--466.

....convex quadratic programs with convex quadratic constraints, and semidefinite programs all have explicit and easily computable self concordant barrier functions, and hence can be solved in polynomial time . On the other hand, Alizadeh [1] extends Ye s projective potential reduction algorithm [32] for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior point algorithms for solving the SDP problem and SDLCP, including Alizadeh, Haeberly and Overton [2] ....

Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457--466, 1990.

....Systems Engineering, Georgia Tech, Atlanta, GA 30332. monteiro isye.gatech.edu) programs all have explicit and easily computable self concordant barrier functions, and hence can be solved in polynomial time . On the other hand, Alizadeh [1] extends Ye s projective potential reduction algorithm [33] for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior point algorithms for solving SDP problems, including Alizadeh, Haeberly and Overton [2, 3] Freund [4] ....

Y. Ye, A class of projective transformations for linear programming, SIAM Journal on Computing, 19 (1990), pp. 457--466.

....convex quadratic programs with convex quadratic constraints, and semidefinite programs all have explicit and easily computable self concordant barrier functions, and hence can be solved in polynomial time . On the other hand, Alizadeh [1] extends Ye s projective potential reduction algorithm [26] for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior point algorithms for solving SDP problems, including Alizadeh, Haeberly and Overton [2] Helmberg, Rendl, ....

Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457--466, 1990.

....[1] started somewhat later than, and independent of [48] Nesterov and Nemirovskii obtain their complexity theorems by specializing their general results to SDP. We, on the other hand, take a specific interior point algorithm for linear programming (i. e Ye s projective potential reduction method [66]) and extend it to SDP. Furthermore, we argue that many known interior point linear programming algorithms can also be transformed into an algorithm for SDP in a mechanical way; proofs of convergence and polynomial time computability extend in a similar fashion. Later Jarre in [35] and ....

....we develop a potential reduction method for solving the primal problem so that, within O( p nj log fflj) iterations, we get an approximate solution with at least ffl relative accuracy, if ffl is sufficiently small. Our development closely follows Ye s projective algorithm for linear programming [66]. Ye s complexity analysis is also extended to semidefinite programs. 3.1. Potential functions and projective transformations. First, recall that the interior of the cone of positive semidefinite matrices is the set of positive definite 3 Actually one can reduce the number of unknowns by ....

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Y. Ye, A class of projective transformations for linear programming, SIAM J. Comput., 19 (1990), pp. 457--466.

....(see below for definitions) On the other hand Alizadeh [Ali95] argued that, at least as far as certain classes of interior point methods are concerned, there is a close connection between linear and semidefinite programming. He made this point by showing that Ye s potential reduction method [Ye90] can be, in a sense word by word, extended to semidefinite programming. Similar extension can be made to many other LP algorithms. An analogous result later was derived for optimization over the quadratic cone by Nemirovskii and Scheinberg [NS96] Next, two sets of papers appeared that showed the ....

Y. Ye. A class of projective transformations for linear programming. SIAM J. Comput., 19(3):457--466, 1990.

.... Adler [20] Kojima, Mizuno and Yoshise [18] and Roos and Vial [23] Apart from these short step methods so called long step path following methods have been proposed in e.g. Gonzaga [10] Gonzaga [11] Roos and Vial [24] 4) Potential based affine scaling methods, due to Gonzaga [9] Ye [32] (see also Todd and Ye [26] Freund [6] Anstreicher and Bosch [2] and Kojima, Mizuno and Yoshise [19] Algorithms in the first category, as well as the long step algorithms in the third category, are polynomial and require O(nL) iterations. The overall complexity is O(n 4 L) operations. ....

Ye, Y. (1990) A Class of Projective Transformations for Linear Programming, SIAM Journal on Computing 19, 457--466.

.... proposed his projective algorithm [5] various primal dual potential reduction algorithms for linear programming have been developed by Anstreicher and Bosch [1] Freund [2] Gonzaga and Todd [4] Kojima, Mizuno and Yoshise [6] Liu and Goldfarb [7] McShane, Monma and Shanno [8] and Ye [10][11] among others. All of these algorithms are based on reducing a primal dual potential function that is first appeared in Todd and Ye [9] They showed that a Newton type step can reduce the function by a constant from an interior point close to the central path. Soon after, Ye [10] showed that it ....

....typical conflict between theoretical and practical best. In fact, recent discussions on large steps are actually on large ae s. First, let us distinct ae in the potential function and ae in the linear system, say, the former is ae p and the latter is ae s . Second, an interesting result of Ye [11] indicates that ae p and ae s may not necessarily be the same. In other words, we can still use ae p = n p n in the potential function to measure the iterative process, but use ae s n p n in the linear system (0) to obtain ffix and ffis. For example, if the ratio of (min(X k s k ....

Y. Ye, "A class of projective transformations for linear programming," manus-cript (1988), to appear in SIAM J. on Computing.

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Y. Ye. A class of projective transformations for linear programming. SIAM Journal on Computing, 19:457--466, 1990.

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Y. YE. A class of projective transformations for linear programming. SIAM J. Comput., 19(3):457-- 466, 1990.

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