F. ALIZADEH. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....been extended to problems involving matrix inequalities. Nesterov and Nernirovsky [30, Ch.4] describe three potential reduc tion algorithms for problems involving matrix inequalities: a generalization of Karmarkar s method, a projective method, and a generalization of the method of Ye [36] In [3, 2],Alizadeh describes several potential reduction methods for problems involving matrix inequalities, emphasizing their similarity to the analogous methods for linear programming. These potential reduction methods all share an important advantage over the earlier path following meth ods: they allow ....

F. Alizadeh, "Combinatorial optimization with interior point methods and semi-definite matrices", PhD thesis, University of Minnesota (October 1991).

....a saddle point method for eigenvalue mimimization due to Pyatnitski and Skorodinsky [PS83] Interior point methods for eigenvalue minimization have recently been developed by several researchers. The first were Nesterov and Nemirovsky [NN88, NN90b, NN90a, NN91a, NN93] others include Alizadeh [Ali92b, Ali91, Ali92a], Jarre [Jar91a] and Vandenberghe and Boyd [VB93] Of course, general interior point methods (and the method of centers in particular) have a long history. Early work includes the SUMT book by Fiacco and McCormick [FM68] the method of centers described by Huard et al. LH65, Hua67] and Dikin s ....

....form of an affine matrix inequality C(x) 0. The idea that affine matrix inequalities can be used to represent a wide variety of convex constraints can be found in Nesterov and Nemirovsky [NN90b, NN90a, NN93] who formalize the idea of a positive definite representable function) and Alizadeh [Ali92b, Ali91]. 2.1 Multiple constraints We first note that multiple constraints on x, expressed as the affine matrix inequalities C i (x) 0, i = 1; l, are equivalent to the single affine matrix inequality C 1 (x) Phi Delta Delta Delta Phi C l (x) 0. 2.2 Linear constraints The constraint ....

F. Alizadeh. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, Univ. of Minnesota, October 1991.

....convexity can be enforced via semide nite programming formulations. More precisely, the problem of nding the best convex quadratic approximation in the least squares sense, may be formulated as a semide nite programming problem. Semide nite programming problems can be solved eciently nowadays ([1, 10, 15, 16, 22]) We note that especially in the eld of Computer Aided Design much attention has been given to convexity preserving properties for several interpolation and approximation techniques ( 11, 12] However, this research is mostly restricted to splines and to the univariate and bivariate cases. ....

....(f(z) g(z) 2 ; 10) and then g can be found by solving min 0 t : s X z2Z s(z) 2 t; Q 0 1 A : 11) For the rst two cases the resulting problems (7) and (9) have linear constraints and a semide nite constraint Q 0. Such a semide nite programming problem can eciently be solved ([1, 10, 15, 16, 22, 26]) The third resulting problem (11) again can be eciently solved, since the new constraint is a second order cone (Lorentz cone) constraint ( 22] In practice one sometimes want to add the condition that the approximation is exact or an upper or underestimate in several points in Z . Observe ....

Alizadeh, F. (1991), Combinatorial optimization with interior point methods and semide nite matrices, Ph.D. Thesis, University of Minnesota, Minneapolis.

....rather than satisfiability [31] 2.2 Semidefinite programming Recently much attention has been devoted to the field of semidefinite programming. It was shown that efficient approximation algorithms for hard combinatorial optimization problems can be obtained using semidefinite relaxations [12, 1], while there are also applications in control theory [28] Using interior point methods, semidefinite programs can be solved (to a given accuracy) in polynomial time. For the reader that is unfamiliar with semidefinite programming, we review some of the basic concepts. The standard primal (P) ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi--definite matrices. PhD thesis, University of Minnesota, Minneapolis, USA, 1991.

....(IPMs) provide a powerful approach for solving SDO problems, and a lot of results have been reported. A comprehensive list of the publications on this topic can 1 be found on the SDO homepages maintained by Alizadeh [2] and Helmberg [6] Pioneering work in this direction is due to Alizadeh [1] and Nesterov and Nemirovskii [12] Most IPMs for SDO can be viewed as natural extensions of existing central path following IPMs for linear optimization (LO) and have similar polynomial complexity results. However, di erent from the LO case, there are many search directions in so called ....

F. Alizadeh, Combinatorial Optimization with Interior Point Methods and Semidenite Matrices, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 1991.

....alternative to the usual convex logarithmic barrier potential. 6 Further work It has already been mentioned that the new potential function (1) has an extension to the semi definite programming (SDP) case. The recent revival of interest in SDP started 9 more or less with the work of Alizadeh [1], and an excellent review on developments and applications up to 1995 is given by Vanderberghe in [13] One reason for the recent interest in SDP is that most interior point methods for LP can be extended to SDP. This is presently an active research area, as can be seen by the number of recent ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi-- definite matrices. PhD thesis, University of Minnesota, Minneapolis, USA, 1991.

....from linear programming (LP) to the more general semidefinite programming (SDP) problem. In their seminal work [15] Nesterov and Nemirovskii present three methods, namely a generalization of Karmarkar s method [10] a projective method, and an extension of an LP algorithm of Ye [25] Alizadeh [1] independently analysed several methods which have analogies in the LP literature. A primal dual potential reduction method suited for the structure of linear matrix inequalities arising in control theory applications was analysed by Vandenberghe and Boyd in [23] A general potential reduction ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi--definite matrices. PhD thesis, University of Minnesota, Minneapolis, USA, 1991.

....has found a couple of impressive applications in various fields. To mention a few, we refer to Vandenberghe and Boyd [19] which gives a survey of the theory and applications, to Boyd et al. [2] for a discussion of a wide range of powerful 1 applications in system theory, and to Alizadeh [1] and Goemans and Williamson [5] for some applications in combinatorial optimization. Now let us turn to the smallest eigenvalue problem for a symmetric n Theta n matrix A. This problem can be formulated as a semidefinite optimization problem. Denoting the eigenvalues of A by 1 2 Delta Delta ....

....Since both A Gamma I and X are SPSD, one has Tr ( A Gamma I) X) 0, proving the first statement in the lemma. It follows that the pair ( X) is certainly optimal if = 1 and Tr ( A Gamma 1 I) X) 0: 1) The latter equality implies (A Gamma 1 I) X = 0, as has been shown in, e.g. Alizadeh [1]. To make the paper self supporting we have included a proof of this property in Appendix A. Since the last 2 equality can be written as AX = 1 X, it is clear that (1) simply means that the column space of X must be a subspace of the eigenspace of A with respect to 1 . Finally, for proving ....

F. Alizadeh. Combinatorial Optimization with Interior Point Methods and Semi--Definite Matrices. PhD thesis, University of Minnesota, Minneapolis, Minnesota, USA, 1991.

....centers for semi definite programming. iii 1 Introduction In recent years, a great revival of interest in semi definite programming (SDP) has taken place. The reason is basically twofold: Important applications in control theory [4] structural optimization [3] and combinatorial optimization [1], to name but a few, have been formulated as SDP problems. A review of applications may be found in [15] Secondly, it has become clear that most interior point algorithms for linear programming (LP) may be extended to semi definite programming. The polynomial complexity of these methods and ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi-- definite matrices. PhD thesis, University of Minnesota, Minneapolis, USA, 1991.

....papers appeared on this topic. Some recent review papers include [20] and [26] Several new books on the subject have also appeared recently, including [50] and [59] The first extension of interior point algorithms from LP to SDP was by Nesterov and Nemirovskii [43] and independently by Alizadeh [1] in 1991. Nesterov and Nemirovskii actually considered a more general class of convex optimization problems, where the nonlinearity is banished to a convex cone, like X 0. They show that such conic optimization problems can be solved by sequential minimization techniques, where the conic ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi--definite matrices. PhD thesis, University of Minnesota, Minneapolis, USA, 1991.

....to include information inherent in the positive semidefinite program. 3 POSITIVE SEMIDEFINITE PROGRAMMING ALGORITHMS There are actually several polynomial algorithms that can solve positive semidefinite programs. One is the primal scaling algorithm (Nesterov and Nemirovskii [33] Alizadeh [1], Vandenberghe and Boyd [42] and Ye [44] which is the analogue of the primal potential reduction algorithm for linear programming. This algorithm uses X to generate 8 APPLICATIONS AND ALGORITHMS OF COMPLEMENTARITY the iterate direction. Another is the dual scaling algorithm (Vandenberghe and ....

F. Alizadeh. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, Minneapolis, MN, 1991.

....[11, 12, 21, 28, 34] discuss extensions of IPMs for classes of nonlinear problems. In recent years the majority of research is devoted to IPMs for Semidefinite Optimization (SDO) SDO has a wide range of interesting applications not only in such traditional areas as combinatorial optimization [1], but also in control, and different areas of engineering, more specifically structural [8] and electrical engineering [30] For surveys on algorithmic and complexity issues the reader may consult [5, 6, 7, 4, 21, 22, 24, 27] Teaching Interior Point Methods After years of intensive research a ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi--definite matrices. Ph.D. thesis, University of Minnesota, Minneapolis, USA, 1991.

....9] LS91] who have demonstrated the power of semidefinite relaxations on some very hard combinatorial problems. The recent development of efficient interior point methods has turned these techniques into Semidefinite programming relaxations 5 powerful practical tools; see Alizadeh [Ali92b, Ali91, Ali92a] Kamath and Karmarkar [KK92, KK93] Helmberg, Rendl, Vanderbei and Wolkowicz [HRVW94] For a more detailed survey of semidefinite programming in combinatorial optimization, we refer the reader to the recent paper by Alizadeh [Ali95] 3 SEMIDEFINITE PROGRAMMING AND CONTROL THEORY ....

F. Alizadeh. Combinatorial optimization with interior point methods and semi-definite matrices. PhD thesis, Univ. of Minnesota, October 1991.

....a proof, see Nesterov and Nemirovsky [NN94, p.238] or Alizadeh [Ali91, x2.2] The semidefinite program (8) has dimensions m = 1 k p(p 1) 2 and n = 2p. These results can also be extended to problems involving absolute values of the eigenvalues, or weighted sums of eigenvalues (see Alizadeh [Ali91, Chapter 2]) Another related problem is to minimize the (spectral, or maximum singular value) norm kA(x)k of a matrix A(x) A 0 x 1 A 1 Delta Delta Delta x k A k 2 R p Thetaq . Here the A i need not be symmetric. This can be cast as the semidefinite program minimize t subject to tI A(x) ....

....by Grotschel, Lov asz, and Schrijver [GLS88, Chapter 9] LS91] who have demonstrated the power of semidefinite relaxations on some very hard combinatorial problems. The recent development of efficient interior point methods has turned these techniques into powerful practical tools; see Alizadeh [Ali92b, Ali91, Ali92a], Kamath and Karmarkar [KK92, KK93] Helmberg, Rendl, Vanderbei and Wolkowicz [HRVW94] For a more detailed survey of semidefinite programming in combinatorial optimization, we refer the reader to the recent paper by Alizadeh [Ali95] Control and system theory Semidefinite programming problems ....

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F. Alizadeh. Combinatorial optimization with interior point methods and semi-definite matrices. PhD thesis, Univ. of Minnesota, October 1991.

.... work of Nesterov and Nemirovskii much of which is summarized in [92] They also implemented the rst interior point method for SDP in [91] Independently, Alizadeh extended interior point polynomial time algorithms from linear programming to SDP and studied applications to discrete optimization [3, 4]. Non smooth optimization methods for solving semide nite programming problems have also been proposed (see e.g. 46] Before we proceed let us observe that X = I is a strictly positive de nite feasible point for SDP1 (usually refereed to as a Slater point) and therefore strong duality holds, ....

F. ALIZADEH. Combinatorial optimization with interior point methods and semidenite matrices. PhD thesis, University of Minnesota, 1991.

....condition # i # i =0,i=1, n. Observe that # i and # i are the limits of # i as 0, and Q may be taken to be a limit point (not necessarily unique) of the set Q : 0 . We have n and # 1 Interior point methods for semidefinite programming were originally introduced by [11, 4]. Early papers on primal dual methods include [17] and [6] A preliminary version of the present work appeared as [2] Convergence analysis of primal dual pathfollowing methods for SDP appeared first in [7, 13, 12] We are primarily concerned with four methods, which we call the XZ, XZ ZX, ....

F. Alizadeh, Combinatorial Optimization with Interior Point Methods and Semidefinite Matrices, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 1991.

....p Theta p matrices all admit p selfconcordant barriers. Therefore, the authors extend the revolutionary result of Karmarkar [Kar84] to a rather general class of convex programs. In this article we study interior point methods for semidefinite programs from an alternative point of view. Our work [Ali91] started 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 ....

....[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 ....

F. Alizadeh. Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices. PhD thesis, University of Minnesota, Minneapolis, Minnesota, 1991.

....by National Science Foundation Grant CCR 9101649. 1 A constraint qualification is required to establish these properties in general. It is sufficient to assume that the primal feasible region has nonempty relative interior. See [2] for details. 1 2 Alizadeh, Haeberly, Overton Several authors [1,2,6,8,9,10] have observed that the interior point methods which have been so successful for LP may also be applied to solve SDP and related problems. Nesterov and Nemirovskii [6] is the primary reference for theoretical properties of interior point algorithms for general convex programs. Alizadeh [1,2] ....

....have observed that the interior point methods which have been so successful for LP may also be applied to solve SDP and related problems. Nesterov and Nemirovskii [6] is the primary reference for theoretical properties of interior point algorithms for general convex programs. Alizadeh [1,2] argues that the various algorithmic techniques and mathematical proofs which have been developed for LP may be generalized in a systematic way to apply to SDP. Instead of the barrier term Gamma P n j=1 log x j used in LP, one introduces the barrier term Gamma log det X to correspond to the ....

F. Alizadeh. Combinatorial Optimization with Interior Point Methods and Semidefinite Matrices. PhD thesis, University of Minnesota, 1991.

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F. ALIZADEH. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, 1991.

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F. ALIZADEH. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, 1991.

No context found.

F. ALIZADEH. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, 1991.

No context found.

F. Alizadeh. Combinatorial optimization with interior point methods and semi-definite matrices. PhD thesis, Univ. of Minnesota, October 1991.

No context found.

F. ALIZADEH. Combinatorial optimization with interior point methods and semidefinite matrices. PhD thesis, University of Minnesota, 1991.

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F. Alizadeh, Combinatorial Optimization With Interior Point Methods and Semi-Definite Matrices, Ph.D. thesis, University of Minnesota, Minneapolis, MN, Oct. 1991.

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F. ALIZADEH. Combinatorial optimization with interior point methods and semidenite matrices. PhD thesis, University of Minnesota, 1991.

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