F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control and Opt., 31(5):1360--1377, September 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i.e. C . X : tr(CX) i, j C ij X ij ,and 0 means that X is positive semidefinite. SDP bears a remarkable resemblance to LP. In fact, it is known that several interiorpoint methods for LP and their polynomial convergence analysis can be naturally extended to SDP (see Alizadeh [1] Jarre [15], Nesterov and Nemirovskii [28, 29] Vandenberghe and Boyd [38] However, in extending primal dual interior point methods from LP to SDP, certain choices have to be made and the resulting search direction depends on these Date. March 4, 1998. 1991 Mathematics Subject Classification. Primary ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360--1377,1993.

....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 interior point ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. Technical Report 91-8, Stanford Optimization Laboratory, 1991. to appear in SIAM J. on Contr and Optimization.

.... (MSC 1991) 90C25, 90C09 1 Introduction Pioneered by the work of Lov asz on the Shannon capacity of graphs [27] the interest in semidefinite relaxations of combinatorial optimization problems has been steadily increasing [16, 28, 7, 34] With the development of interior point algorithms [23, 32, 1, 37, 21, 25, 2, 33] practical methods for computing these relaxations became available. This encouraged research in the field and, as a consequence, several new approximation results based on semidefinite relaxations appeared within short time [14, 29, 13, 5, 12] Most semidefinite relaxations can be improved in a ....

F. Jarre. An interior--point method for minimizing the maximum eigenvalue of a linear combination of matrices. Siam J. Control and Optimization, 31(5):1360--1377, Sept. 1993.

....roughly less than a hundred iterations, independently of the problem size. Each elementary iteration reduces to solving a least square problem which incurs the main computational overhead. Recent and thorough studies of interior point techniques for semi denite programming are, among others, Jarre [23], Vandenberghe and Boyd [43] Rendl, Vanderbei and Wolkowicz [34] and the master book by Nesterov and Nemirovski [28] Basically, the simple feasibility problem of semidenite programming consists in seeking a solution to the LMI F 0 z 1 F 1 : z r F r 0 ; 1) where the F i s are given ....

F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control and Opt., 31 (1993), pp. 13601377.

....the quality of the starting iterate, ranges from 1 Gamma O( p n) Gamma1 to 1 Gamma O(n) Gamma1 . Our analysis is fairly simple and parallels that for the LP and LCP cases. 1 Introduction Since the original work of Nesterov and Nemirovskii [26] followed by that of Alizadeh [1] and Jarre [11], there has been very active research on interior point methods for the semi definite linear programming problem (SDLP) and the semidefinite linear complementarity problem (SDLCP) In particular, various search directions have been proposed [2, 3, 9, 11, 13, 24, 27, 28] and properties of these ....

.... [26] followed by that of Alizadeh [1] and Jarre [11] there has been very active research on interior point methods for the semi definite linear programming problem (SDLP) and the semidefinite linear complementarity problem (SDLCP) In particular, various search directions have been proposed [2, 3, 9, 11, 13, 24, 27, 28] and properties of these search directions, such as their existence and their computation, have been analyzed [3, 13, 24, 28, 32] For SDLP, global linear convergence results have been obtained for feasible primal primal dual path following methods [5, 6, 17, 24, 27, 30, 34, 40] and primal ....

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F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control Optim., 31 (1993), 1360-1377.

....yet powerful unifying framework in which to study a wide variety of important results. Examples include Schur convexity (see for example [22] the convexity of eigenvalue functions ( 10, 6, 11, 3, 13, 19] calculations of Fenchel conjugates and subdifferentials of convex eigenvalue functions [24, 5, 12, 30, 28, 25, 26, 27, 15, 16, 1, 17, 19], von Neumann s original result [33] and generalizations (for example [4, 20] subdifferentials of unitarily invariant norms [34, 35, 36, 37, 38, 8, 7, 9, 20] and characterizations of extreme, exposed and smooth points of unit balls [2, 37, 38, 8, 7, 9, 20] This paper concentrates on convexity ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360--1377, 1993.

....to use the interior point methods to solve semidefinite programs. Alizadeh [5] showed how an interior point algorithm for linear programming could be directly generalized to handle semidefinite programming. Since the work of Alizadeh, there has been a great deal of research into such algorithms [83, 149, 131, 157, 75, 101, 53]. A simplex type 4 method was also discovered by Pataki [119] Among these varieties, the performance of interior point methods are the best in both theory and practice. Suppose there are constraints in a semidefinite program, whose variable is an n Theta n matrix. A near optimal solution for ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31(5):1360--1377, 1993.

....examples. Key words. eigenvalue optimization, convex optimization, semidefinite programming, proximal bundle method, large scale problems. AMS subject classifications. 65F15, 90C25; Secondary 52A41, 90C06. 1. Introduction. The development of interior point methods for semidefinite programming [19, 31, 1, 46] has increased interest in semidefinite modeling techniques in several fields such as control theory, eigenvalue optimization, and combinatorial optimization. In fact, interior point methods proved to be very useful and reliable solution methods for semidefinite programs of moderate size. However, ....

F. Jarre, An interior--point method for minimizing the maximum eigenvalue of a linear combination of matrices, Siam J. Control and Optimization, 31 (1993), pp. 1360--1377.

....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 Vandenberghe and Boyd in [VB93] later developed similar interior point algorithms for special forms of SDP. Polynomial time interior point methods for SDP have some interesting consequences for combinatorial optimization problems. In order to solve such a problem by the ellipsoid method, an ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. Technical Report SOL-91-8, Department of Operations Research, Stanford Univeristy, June 1991. To appear in SIAM J. Control Optim.

....optimization problems. Independently of Nesterov and Nemirovsky, Alizadeh [Ali92b] and Kamath and Karmarkar [KK92, KK93] generalized interior point methods from linear programming to semidefinite programming. Other recent articles on interior point methods for semidefinite programming are Jarre [Jar93], Vandenberghe and Boyd [VB95] Rendl, Vanderbei and Wolkowicz [RVW93] Alizadeh, Haeberly and Overton [AHO94] Kojima, Shindoh and Hara [KSH94] Faybusovich [Fay94] Gahinet and Nemirovsky [GN93] and Freund [Fre94] An excellent reference on interior point methods for general convex problems is ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control and Opt., 31(5):1360--1377, September 1993.

....one which can be solved easily thanks to an addition of structure: self concordance is used to obtain the polynomiality of interior point schemes. This is proved in a general framework in [32, 33] and more speci cally in the framework of semide nite programming (which includes (P ) in ([2, 3, 20, 30, 45], 9, Chap. II, Notes and References] Yet, it seems that the initial transformation has a price: some local information is lost, intuitively we understand that the smoothing e ect of interior point methods slightly modi es the local (second order) information of the problem. In order to speed ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control Optim., 31(5):1360-1377, September 1993.

....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] Helmberg, Rendl, Vanderbei and Wolkowicz [9] Jarre [12], Kojima, Shida and Shindoh [15] Kojima, Shindoh and Hara [16] Lin and Saigal [18] Lou, Sturm and Zhang [19] Monteiro [20, 21] Monteiro and Tsuchiya [23, 24] Monteiro and Zhang [25] Nesterov and Nemirovskii [28] Nesterov and Todd [31, 30] Potra and Sheng [32] Sturm and Zhang [33] Tseng ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360--1377, 1993.

....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] Freund [3] Helmberg, Rendl, Vanderbei and Wolkowicz [4] Jarre [5], Kojima, Shida and Shindoh [8] Kojima, Shindoh and Hara [10] Lin and Saigal [11] Luo, Sturm and Zhang [12] Monteiro [14, 15] Monteiro and Zhang [18] Nesterov and Nemirovskii [21] Nesterov and Todd [24, 23] Potra and Sheng [25] Sturm and Zhang [26] Tseng [28] Vandenberghe and Boyd [29] ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360--1377, 1993.

....to the constraints (3.6) The minimization problem (3.7) together with the constraint (3.6) has the standard form of a constrained nondifferentiable eigenvalue minimization problem. Such a problem has been well studied and can be solved using e.g. the interior point method [4] 23] 21] [16]. Alternatively, the following necessary and sufficient condition is of independent interest. This condition is equivalent to that in Theorem 3.1 and also supports the claim that the consistency problem can be solved as a constrained nondifferentiable convex programming problem. Theorem 3.2 Let ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SOL report 91-8, 1991.

....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 Wolkowicz [5] Jarre [8], Kojima, Shindoh and Hara [11] Nesterov and Nemirovskii [16] Nesterov and Todd [19, 18] and Vandenberghe and Boyd [20] Among the above works, Kojima, Shindoh and Hara [11] and Nesterov and Todd [18] present some algorithms which extend the primal dual methods for linear programming based on ....

F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM Journal on Control and Optimization, 31 (1993), pp. 1360--1377.

..... X : tr(CX ) P i, j C i j X i j , and X # 0 means that X is positive semidefinite. SDP bears a remarkable resemblance to LP. In fact, it is known that several interiorpoint methods for LP and their polynomial convergence analysis can be naturally extended to SDP (see Alizadeh [1] Jarre [15], Nesterov and Nemirovskii [28, 29] Vandenberghe and Boyd [38] However, in extending primal dual interior point methods from LP to SDP, certain choices have to be made and the resulting search direction depends on these Date. March 4, 1998. 1991 Mathematics Subject Classification. Primary ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360--1377, 1993.

....how to use the interiorpoint methods to solve semidefinite programs. Alizadeh [1] showed how an interiorpoint algorithm for linear programming could be directly generalized to handle semidefinite programming. Since the work of Alizadeh, there has been a great deal of research into such algorithms [18, 36, 31, 37, 15, 24, 10]. A simplex type method was also discovered by Pataki [29] Among these varieties, the performance of interior point methods are the best in both theory and practice. Suppose there are constraints in a semidefinite program, whose variable is an n Theta n matrix. A near optimal solution for the ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31(5):1360--1377, 1993.

....for the application of modern interior point methods. Numerous problems in control and systems theory [9] combinatorics [1] structural design, robust optimization and many others (see for example [4] can be expressed in terms of SDP. Various extensions to semide nite programming can be found in [1, 8, 16, 22, 24] and references therein. Methods of multipliers, involving nonquadratic augmented Lagrangians [25, 29, 6, 10, 7] successfully compete with the interior point methods in non linear programming. Especially ecient they are when a very high accuracy of solution is required. This success is partially ....

F. Jarre, An Interior-Point Method for Minimizing the Maximum Eigenvalue of a Linear Combination of Matrices, SIAM J. Control Optim. 31 (1993) 1360-1377.

....the application of modern interior point methods. Numerous problems in control and systems theory [9] combinatorics [1] structural design, robust optimization and many others (see for example [4] can be expressed in terms of SDP. Various extensions to semidefinite programming can be found in [1, 8, 15, 19, 21] and references therein. Methods of multipliers, involving nonquadratic augmented Lagrangians [6, 7, 10, 22, 26, 29] successfully compete with the interior point methods in non linear programming. Especially efficient they are when a very high accuracy of solution is required. This success is ....

F. Jarre, An Interior-Point Method for Minimizing the Maximum Eigenvalue of a Linear Combination of Matrices, SIAM J. Control Optim. 31 (1993) 1360-1377.

....be solved in polynomial time using a sequence of barrier subproblems. Alizadeh [1] presented a transparent approach to extend potential function methods from LP to SDP. Simultaneously, strong numerical results for SDP and the special case of the min max eigenvalue problem appeared in e.g. 28] and [29]. This started a flood of results in SDP from researchers in LP. The most popular methods are the p d i p methods. These are based on applying Newton s method to solving the optimality conditions from the barrier problems, i.e. the perturbed optimality conditions Z C Gamma A y = 0; b ....

F. JARRE. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control and Optimization, 31:1360--1377, 1993.

....how to use the interior point method to solve semidefinite programs. Alizadeh [1] showed how an interior point algorithm for linear programming could be directly generalized to handle semidefinite programming. Since the work of Alizadeh, there has been a great deal of research into such algorithms [8, 24, 20, 25, 6, 12, 3]. Essentially, the number of iterations is roughly O( p n) and each iteration involves either factoring a matrix or solving a least squares problem. The size of either problem is s Theta n 2 where s is the number of constraints defining the semidefinite program. Goemans and Williamson s ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31(5):1360--1377, 1993.

....time. In the past several years, a major part of the research into SDP has focused on both the theoretical and practical solution of SDP problems using extensions of interior point methods for LP. Many authors have proposed interior point algorithms for solving SDP problems (see for example [1, 2, 4, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26]) Many of the recent works on interior point algorithms for SDP are concentrated on primal dual methods. Feasible primal dual path following algorithms for SDP simultaneously solve the primal and dual SDP problems by maintaining primal feasibility in X and dual feasibility in (S; y) while ....

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM Journal on Control and Optimization, 31:1360--1377, 1993.

No context found.

F. Jarre. An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control and Opt., 31(5):1360--1377, September 1993.

No context found.

F. JARRE. An interior point method for minimizing the maximum eigenvalue of a linear combination of matrices. SIAM J. Control and Optimization, 31:1360--1377, 1993.

No context found.

F. Jarre, "An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices", SIAM J. Control and Opt., vol. 31, pp. 1360--1377, Sep. 1993.

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