F. ALIZADEH. Optimization over positive semi-definite cone; interior-point methods and combinatorial applications. In P.M. Pardalos, editor, Advances in Optimization and Parallel Computing, pages 1--25. North--Holland, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are the new interior point and new estimate, respectively. This proves polynomial time complexity for the algorithm. 77 9 Semidefinite Programming Relaxations There has been a great deal of interest recently in solving semidefinite programming relaxations of combinatorial optimization problems [5, 6, 40, 144, 145, 152, 54, 53, 57, 112, 123, 111]. The semidefinite relaxations are solved by an interior point approach. These papers have shown the strength of the relaxations, and some of these papers have discussed cutting plane and branch and cut approaches using these relaxations. The bounds obtained from semidefinite relaxations are often ....

F. Alizadeh. Optimization over positive semi-definite cone: Interiorpoint methods and combinatorial applications. In P.M. Pardalos, editor, 82 Advances in Optimization and Parallel Computing, pages 1--25. North-- Holland, Amsterdam, 1992.

....ellipsoid method. However, for practical experiments we have used the Bundle Trust algorithm ( 18] in combination with a Lanczos routine for computing the maximum eigenvalue. Recently, several other methods have been proposed for minimization of the maximum eigenvalue of a parametrized matrix ([1, 13]) but their practical efficiency has not yet been investigated thoroughly. 5.3 Semi infinite programs Our main theorem opens new ways to derive even tighter relaxations of the graph bisection problem, by combining the polyhedral approach relying on a (partial) description of the cut polytope ....

F. Alizadeh, Optimization over the positive semi-definite cone: Interior point methods and combinatorial applications, Proc. IPCO 1992, pp. 385--405.

....the ellipsoid algorithm (see e.g. 24, 25] The ellipsoid method has polynomial time complexity, and works in practice for smaller problems, but can be slow for larger problems. Other algorithms specifically for LMI based problems are discussed in, e.g. 26, 27] Recently, various researchers [28, 1, 29, 30] have developed interior point methods for solving LMIbased problems, based on the work of Nesterov and Nemirovsky [31] Numerical experience shows that these algorithms solve LMI problems with extreme efficiency. In some specific cases (one is discussed below) these methods can solve LMI based ....

F. Alizadeh. Optimization over the positive semi-definite cone : Interior--point methods and combinatorial applications. In P. M. Pardalos, editor, Advances in Optimization and Parallel Computing, pages 1--25. North Holland, Amsterdam, The Netherlands, 1992.

....to linear programs, interior point methods for nonlinear programs converge in polynomial time only in special cases. For example, polynomial time methods exist for classes of convex programming problems [22, 17, 10, 11, 4] and for optimization problems over cones of positive semidefinite matrices [1, 12]. Recently, Nesterov and Nemirovsky [18] developed an elegant general framework for polynomial time interior point methods, based on the concept of self concordant barrier functions. 1.1. The problem In this paper, we extend the approach of [18] and we present a polynomial time interiorpoint ....

F. Alizadeh, "Optimization over the positive semi-definite cone: interior-point methods and combinatorial applications," in: P. Pardalos, ed., Advances in Optimization and Parallel Computing (NorthHolland, Amsterdam, 1992) pp. 1--25.

....optimization problems, and since 1984, numerous new algorithms have been proposed. However, in contrast to linear programs, interior point methods for nonlinear programs converge in polynomial time only in special cases, such as classes of convex programming problems; we refer the reader to [1, 10, 17, 18, 22, 23, 28] and the references given there. In [15] we developed a polynomial time interior point method for a class of fractional programs, namely the minimization of a single fraction of linear functions subject to convex constraints. We stress that such programs are not convex, but only pseudo convex. In ....

....boundary point of C 1;k 1 , and let F k denote the triangle with vertices 0, p k , and p k 1 . Note that (F k ) 1 2 ae k ae k 1 sin Deltat k ; 7. 21) where, by (2:19) 2:18) and the choice of j = 1 (and oe k = 1=18) t k Gamma t k 1 = Deltat k = ju 1 (x (k) t k )j 18 kx (k) [1] k : 7.22) By construction of the triangle F k , we have F k ae C 1;k and C 1;k 1 ae C 1;k n F ffi k ; and thus (C 1;k 1 ) C 1;k ) Gamma (F k ) 1 Gamma (F k ) C 1;k ) C 1;k ) 7.23) Next, recall the definition of d k : d 1;k in (7:3) and set q k : d k sin t ....

[Article contains additional citation context not shown here]

Alizadeh, F., Optimization Over the Positive Semi-Definite Cone: Interior-Point Methods and Combinatorial Applications, Advances in Optimization and Parallel Computing, Edited by P. Pardalos, North-Holland, Amsterdam, pp. 1--25, 1992.

....classes of global optimization problems. In this chapter, we restricted ourselves to applications of interior point methods for quadratic and combinatorial optimization problems, as well as nonconvex potential functions. Recently, a great amount of research activity on semidefinite programming [2, 28] has produced some very interesting results. The significance of semidefinite programming is that it provides tighter relaxations to many combinatorial and nonconvex optimization problems and, in theory, semidefinite programming can be solved in polynomial time. Preliminary implementations of ....

F. Alizadeh. Optimization over positive semi-definite cone: Interior-point methods and combinatorial applications. In P.M. Pardalos, editor, Advances in Optimization and Parallel Computing, pages 1--25. North--Holland, Amsterdam, 1992.

....ellipsoid method. However, for practical experiments we have used the Bundle Trust algorithm ( 18] in combination with a Lanczos routine for computing the maximum eigenvalue. Recently, several other methods have been proposed for minimization of the maximum eigenvalue of a parametrized matrix ([1, 13]) but their practical efficiency has not yet been investigated thoroughly. 6.3. Semi infinite programs. Our main theorem opens new ways to derive even tighter relaxations of the graph bisection problem, by combining the polyhedral approach relying on a (partial) description of the cut polytope ....

F. Alizadeh, Optimization over the positive semi-definite cone: Interior point methods and combinatorial applications, Proc. IPCO 1992, pp. 385--405.

....This proves polynomial time complexity for the algorithm. 60 J. E. MITCHELL, P. M. PARDALOS, AND M. G. C. RESENDE 9. SEMIDEFINITE PROGRAMMING RELAXATIONS There has been a great deal of interest recently in solving semidefinite programming relaxations of combinatorial optimization problems [5, 6, 40, 144, 145, 152, 54, 53, 57, 112, 123, 111]. The semidefinite relaxations are solved by an interior point approach. These papers have shown the strength of the relaxations, and some of these papers have discussed cutting plane and branch and cut approaches using these relaxations. The bounds obtained from semidefinite relaxations are often ....

F. Alizadeh. Optimization over positive semi-definite cone: Interior-point methods and combinatorial applications. In P.M. Pardalos, editor, Advances in Optimization and Parallel Computing, pages 1--25. North-- Holland, Amsterdam, 1992.

....[Meg89] 3 Since X, S and y are solution of the algebraic system of equations: XS = 0; AvecX = b and A T y S = C, there are algebraic solutions among all optimal solutions of an SDP problem with integral input. 4 I am indebted to Joshi Ramana for bringing to my attention an error in [Ali91, Ali92] where I had claimed that the norm of the solution to any SDP problem is bounded by 2 L . Joshi essentially provided this counter example. 17 Consider the following pair of primal and dual problems: min C ffl X Mx 1 s:t: Avec(X) b Gamma Avec(X 0 ) x 1 = b [Mat(A T y 0 ) S 0 Gamma ....

....any references to the entries under the LP column, with the corresponding entry under the SDP column. Proofs of convergence or polynomial time complexity may also be extended mechanically in the same manner. We have already verified this claim on the approaches of Gonzaga [Gon89] Ye [Ye91] see [Ali92]) and Monteiro and 18 LP SDP unknown vector: x unknown symmetric matrix: X inequality constraints: Lowner constraints: dual variable: y dual variable: y dual slack vector: s dual slack symmetric matrix: S 1 I linear scaling: linear scaling: x (x i = x 0 ) i ) n i=1 = Diag(x 0 ) ....

F. Alizadeh. Optimization Over Positive Semi-Definite Cone; Interior-Point Methods and Combinatorial Applications. In P. Pardalos, editor, Advances in Optimization and Parallel Computing. North--Holland, 1992.

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F. ALIZADEH. Optimization over positive semi-definite cone; interior-point methods and combinatorial applications. In P.M. Pardalos, editor, Advances in Optimization and Parallel Computing, pages 1--25. North--Holland, 1992.

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F. Alizadeh, Optimization over the positive semi-definite cone: Interior point methods and combinatorial applications, in Advances in Optimization and Parallel Computing, ed., P. Pardalos, North-Holland, Amsterdam, 1992, 1-25.

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F. ALIZADEH. Optimization over positive semi-definite cone; interior-point methods and combinatorial applications. In P.M. Pardalos, editor, Advances in Optimization and Parallel Computing, pages 1--25. North--Holland, 1992.

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