## Semidefinite optimization (2001)

Venue: | Acta Numerica |

Citations: | 121 - 3 self |

### BibTeX

@ARTICLE{Todd01semidefiniteoptimization,

author = {M. J. Todd},

title = {Semidefinite optimization},

journal = {Acta Numerica},

year = {2001},

volume = {10},

pages = {515--560}

}

### Years of Citing Articles

### OpenURL

### Abstract

Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.

### Citations

3777 | Convex Analysis
- Rockafellar
- 1970
(Show Context)
Citation Context ...any such would be a nonzero X # 0 satisfying C . X = 0, AX = 0, showing that the set of optimal solutions of (P) is unbounded, a contradiction). Hence, by a separating hyperplane theorem (Rockafellar =-=[56]-=-, Corollary 11.4.1), there exist S # SIR nn and # # IR with S . X > # for any X # G 1 , S . Xsfor any X # G 2 . Since 0 # G 1 , # is negative. Since #uu T # G 1 for any positive # and any u # IR n , i... |

1194 |
Geometric Algorithms and Combinatorial Optimization
- Grotschel, Lov'asz, et al.
- 1988
(Show Context)
Citation Context ... thereby found the capacity of the pentagon, solving a longopen conjecture. At that time, the most e#cient method known for SDP problems was the ellipsoid method, and Grotschel, Lovasz, and Schrijver =-=[24]-=- investigated in detail its application to combinatorial optimization problems by using it to approximate the solution of both LP and SDP relaxations. Lovasz and Schrijver [36] later showed how SDP pr... |

974 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ... the powerful theory of self-concordant barrier functions. These works led to the huge recent interest in semidefinite programming, which was further increased by the result of Goemans and Williamson =-=[22]-=- which showed that an SDP relaxation could provide a provably good approximation to the max-cut problem in combinatorial optimization. Our coverage will necessarily be incomplete and biased. Let us th... |

845 | Semidefinite programming
- Vandenberghe, Boyd
- 1996
(Show Context)
Citation Context ...pproximation to the max-cut problem in combinatorial optimization. Our coverage will necessarily be incomplete and biased. Let us therefore refer the reader to a survey paper by Vandenberghe and Boyd =-=[63]-=- which discusses in particular a number of applications, especially in control theory; the book of Boyd et al. which describes the latter in much further detail and gives the history of SDP in control... |

687 | A new polynomial-time algorithm for linear programming
- Karmarkar
- 1984
(Show Context)
Citation Context ...ersley; see [50] and the references therein. The key contributions of Nesterov and Nemirovski [44, 45] and Alizadeh [1] showed that the new generation of interior-point methods pioneered by Karmarkar =-=[30]-=- for LP could be extended to SDP. In particular, Nesterov and Nemirovski established a general framework for solving nonlinear convex optimization problems in a theoretically e#cient way using interio... |

499 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(Show Context)
Citation Context ...mmers in the '80s, and this led to a series of papers by Overton and Overton and Womersley; see [50] and the references therein. The key contributions of Nesterov and Nemirovski [44, 45] and Alizadeh =-=[1]-=- showed that the new generation of interior-point methods pioneered by Karmarkar [30] for LP could be extended to SDP. In particular, Nesterov and Nemirovski established a general framework for solvin... |

345 |
On the Shannon capacity of a graph
- Lovasz
- 1979
(Show Context)
Citation Context ...ed that some hard graph-partitioning problems could be attacked by considering a related eigenvalue optimization problem -- as we shall see, these are closely connected with SDP. Then in 1979, Lovasz =-=[35]-=- formulated an SDP problem that provided a bound on the Shannon capacity of a graph and thereby found the capacity of the pentagon, solving a longopen conjecture. At that time, the most e#cient method... |

295 | Robust convex optimization
- Ben-Tal, Nemirovski
- 1998
(Show Context)
Citation Context ...uncertainty in the data of an optimization problem (or in the implementation of a solution) by requiring that the solution be feasible whatever the realization of the data (see Ben-Tal and Nemirovski =-=[7]-=-). Without loss of generality we can assume that the objective function is deterministic. Let us consider robust LP with ellipsoidal uncertainty. The problem max b T y, a T j y # c j for all (a j ; c ... |

278 | Cones of matrices and setfunctions and 0-1 optimization
- Lovász, Schrijver
- 1991
(Show Context)
Citation Context ..., Lovasz, and Schrijver [24] investigated in detail its application to combinatorial optimization problems by using it to approximate the solution of both LP and SDP relaxations. Lovasz and Schrijver =-=[36]-=- later showed how SDP problems can provide tighter relaxations of (0, 1)-programming problems than can LP. Fletcher [17, 18] revived interest in SDP among nonlinear programmers in the '80s, and this l... |

219 | An interior-point method for semidefinite programming
- Helmberg, Rendl, et al.
- 1996
(Show Context)
Citation Context ...rix with WSW = X, so that X = S. The resulting search directions are known as the HRVW/KSH/M, dual HRVW/KSH/M, and NT directions. The first was introduced by Helmberg, Rendl, Vanderbei, and Wolkowicz =-=[28]-=-, and independently Kojima, Shindoh, and Hara [33], using di#erent motivations, and then rediscovered from the perspective above by Monteiro [39]. The second was also introduced by Kojima, Shindoh, an... |

192 | Primal-Dual Interior-Point Methods for Self-Scaled Cones
- Nesterov, Todd
- 1998
(Show Context)
Citation Context ...more general conic optimization problems. Nesterov and Nemirovski [44, 45] consider general convex cones, with the sole proviso that a self-concordant barrier is known for the cone. Nesterov and Todd =-=[46, 47]-=- consider the subclass of self-scaled cones, which admit symmetric primal-dual algorithms (these cones turn out to coincide with symmetric (homogeneous self-dual) cones). Another viewpoint is that of ... |

183 |
Self-scaled barriers and interior-point methods for convex programming
- Nesterov, Todd
- 1997
(Show Context)
Citation Context ...ima, Shindoh, and Hara [33] and rediscovered by Monteiro; since it arises by switching the roles of X and S, it is called the dual of the first direction. The last was introduced by Nesterov and Todd =-=[46, 47]-=-, from yet another motivation, and shown to be derivable in this form by Todd, Toh, and Tutuncu [60]. These and several other search directions are discussed in Kojima et al. [32] and Todd [59]. In th... |

162 |
Interior-point methods for the monotone semidefinite linear complementarity problems
- Kojima, Shindoh, et al.
- 1997
(Show Context)
Citation Context ...rch directions are known as the HRVW/KSH/M, dual HRVW/KSH/M, and NT directions. The first was introduced by Helmberg, Rendl, Vanderbei, and Wolkowicz [28], and independently Kojima, Shindoh, and Hara =-=[33]-=-, using di#erent motivations, and then rediscovered from the perspective above by Monteiro [39]. The second was also introduced by Kojima, Shindoh, and Hara [33] and rediscovered by Monteiro; since it... |

151 | Primal-dual path following algorithms for semidefinite programming
- Monteiro
- 1997
(Show Context)
Citation Context ...(This does not seem to cause di#culties in practice.) A more general approach is to apply a similarity to XS before symmetrizing it. This was discussed for a specific pair of similarities by Monteiro =-=[39]-=-, and then in general by Zhang [70]. So let P be nonsingular, and let us replace the last part of # P by 1 2 (PXSP -1 + P -T SXP T ) - #I. (14) (Zhang showed that this is zero exactly when XS = #I as ... |

147 | A spectral bundle method for semidefinite programming
- Helmberg, Rendl
(Show Context)
Citation Context ...# , and the set of them all is called the #-subdi#erential #g # (y). In our case this turns out to be #g # (y) = {AW : (A # y - C) . W # # max (A # y - C) - #, trace W = 1, W # 0}. Helmberg and Rendl =-=[27]-=- develop a very e#cient algorithm, the spectral bundle method, by modifying the classical bundle method to exploit this structure. From the result above, it is easy to see that g(y # ) # (A # y - C) .... |

146 |
Lower bounds for the partitioning of graphs
- Donath, Hoffman
(Show Context)
Citation Context ...k of Lure, Postnikov, and Yakubovich in the Soviet Union in the '40s, '50s, and '60s established the importance of LMIs in control theory (see Boyd at al. [9]). In the early '70s, Do2 nath and Ho#man =-=[13]-=- and then Cullum, Donath, and Wolfe [12] showed that some hard graph-partitioning problems could be attacked by considering a related eigenvalue optimization problem -- as we shall see, these are clos... |

121 | On the Nesterov-Todd direction in semidefinite programming
- Todd, Toh, et al.
- 1998
(Show Context)
Citation Context ...n explicit form for the inverse of E ; instead, to find E -1 U we need to solve a Lyapunov system. Also, the su#cient conditions of Theorem 5.3 do not hold for this choice, and Todd, Toh, and Tutuncu =-=[60]-=- give an example where the Newton direction is not well-defined at a pair of strictly feasible solutions. (This does not seem to cause di#culties in practice.) A more general approach is to apply a si... |

110 | Solving large-scale sparse semidefinite programs for combinatorial optimization
- Benson, Ye, et al.
(Show Context)
Citation Context ... be shown to give a constant decrease in the potential function. It follows that we can attain the iteration complexity bound given in Theorem 6.1. Details can be found in, for example, Benson et al. =-=[8]-=-, which describes why this method is attractive for SDP problems arising in combinatorial optimization problems and gives some excellent computational results. Now let us consider a symmetric primal-d... |

95 | Eigenvalue optimization
- Lewis, Overton
- 1996
(Show Context)
Citation Context ...tions, especially in control theory; the book of Boyd et al. which describes the latter in much further detail and gives the history of SDP in control theory; the excellent paper of Lewis and Overton =-=[34]-=- in this journal on the very closely related topic of eigenvalue optimization; and the aforementioned handbook edited by Wolkowicz et al. [67]. We also mention that SDP is both an extension of LP and ... |

93 |
An O( √ nL)–iteration homogeneous and self–dual linear programming algorithm
- Ye, Todd, et al.
- 1994
(Show Context)
Citation Context ...l system that always has strictly feasible solutions at hand and whose solution gives the required information about the original problems: see [31, 38, 53], based on the work of Ye, Todd, and Mizuno =-=[69]-=- for linear programming. 6.1 Path-following methods These methods are motivated by Theorem 5.4, and attempt to track points on the central path as the parameter # is decreased to zero. We mention firs... |

68 | Exploiting sparsity in semidefinite programming via matrix completion I: General framework
- Fukuda, Kojima, et al.
(Show Context)
Citation Context ...while X may well be dense. Of course, S -1 is likely to be dense, but we may be able to perform operations cheaply with this matrix using a sparse Cholesky factorization of S. Recently, Fukuda et al. =-=[20]-=- have investigated ways in which the primal-dual methods discussed next can exploit this form of sparsity. Now we turn to primal-dual path-following methods. Here we maintain (X, y, S), and our steps ... |

65 | On extending some primal-dual interior–point algorithms from linear programming to semidefinite programming
- Zhang
- 1998
(Show Context)
Citation Context ...ties in practice.) A more general approach is to apply a similarity to XS before symmetrizing it. This was discussed for a specific pair of similarities by Monteiro [39], and then in general by Zhang =-=[70]-=-. So let P be nonsingular, and let us replace the last part of # P by 1 2 (PXSP -1 + P -T SXP T ) - #I. (14) (Zhang showed that this is zero exactly when XS = #I as long as X and S are symmetric.) An ... |

62 | Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices
- Overton, Womersley
- 1993
(Show Context)
Citation Context ...0, 1)-programming problems than can LP. Fletcher [17, 18] revived interest in SDP among nonlinear programmers in the '80s, and this led to a series of papers by Overton and Overton and Womersley; see =-=[50]-=- and the references therein. The key contributions of Nesterov and Nemirovski [44, 45] and Alizadeh [1] showed that the new generation of interior-point methods pioneered by Karmarkar [30] for LP coul... |

58 | An exact duality theory for semidefinite programming and its complexity implications
- Ramana
- 1997
(Show Context)
Citation Context ...p, we can ask whether there is a perhaps more complicated dual problem for which strong duality always holds, without any additional regularity assumptions. The answer is in the a#rmative: see Ramana =-=[54]-=- and Ramana, Tuncel, and Wolkowicz [55]. Finally, if we assume that strong duality holds, then we have as necessary and su#cient optimality conditions the following: A # y + S = C, S # 0, AX = b, X # ... |

56 | Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions
- Monteiro, Tsuchiya
(Show Context)
Citation Context ...S) # N F (# ). Choose # = # for some # # (0, 1), compute the search direction chosen from the AHO, HRVW/KSH/M, dual HRVW/KSH/M, and NT search directions, and take a full Newton step. Repeat. Monteiro =-=[40]-=- showed that such an algorithm, with # = .1 and # = 1 - .1/ # n, generates a sequence of iterates all in the narrow neighbourhood, and produces a strictly feasible point with duality gap at most # tim... |

55 |
Laplacian eigenvalues and the maximum cut problem
- Delorme, Poljak
- 1993
(Show Context)
Citation Context ...of the next section guaranteeing strong duality, so either provides a relaxation of the original max-cut problem. These bounds on the value of a maximum weight cut were obtained by Delorme and Poljak =-=[14]-=-. Since we have a relaxation, the optimal value of the SDP problem provides an upper bound on the value of the max cut. But in this case, we can also use the solution of the primal problem to generate... |

55 |
Semi.definite matrix constraints in optimization
- FLETCHER
- 1985
(Show Context)
Citation Context ...o approximate the solution of both LP and SDP relaxations. Lovasz and Schrijver [36] later showed how SDP problems can provide tighter relaxations of (0, 1)-programming problems than can LP. Fletcher =-=[17, 18]-=- revived interest in SDP among nonlinear programmers in the '80s, and this led to a series of papers by Overton and Overton and Womersley; see [50] and the references therein. The key contributions of... |

53 | Superlinear convergence of a symmetric primal-dual path-following algorithm for semidefinite programming
- Luo, Sturm, et al.
- 1998
(Show Context)
Citation Context ... associated to X(#) and (y(#), S(#)) is n#, and this approaches zero as # tends to zero. (In fact, the central path approaches optimal solutions to the primal and dual problems as # decreases to zero =-=[37, 23]-=-, but we shall not prove this here.) To show that (P ) has a bounded nonempty set of optimal solutions, we proceed as in the proof of Theorem 5.2, again choosing (y, S) # F 0 (D). Clearly, the set of ... |

53 | Strong duality for semidefinite programming
- Ramana, Tuncel, et al.
- 1997
(Show Context)
Citation Context ...s more complicated dual problem for which strong duality always holds, without any additional regularity assumptions. The answer is in the a#rmative: see Ramana [54] and Ramana, Tuncel, and Wolkowicz =-=[55]-=-. Finally, if we assume that strong duality holds, then we have as necessary and su#cient optimality conditions the following: A # y + S = C, S # 0, AX = b, X # 0, XS = 0. (Here the natural last condi... |

47 | Linear systems in Jordan algebras and primal-dual interior-point algorithms
- Faybusovich
- 1997
(Show Context)
Citation Context ...dmit symmetric primal-dual algorithms (these cones turn out to coincide with symmetric (homogeneous self-dual) cones). Another viewpoint is that of Euclidean Jordan Algebras, developed by Faybusovich =-=[15, 16]-=- and now investigated by a number of authors: see Alizadeh and Schmieta [5]. Since the area is receiving so much attention, it is hard to keep abreast of recent developments, but this is immeasurably ... |

47 |
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
- CULLUM, DONATH, et al.
- 1975
(Show Context)
Citation Context ...he Soviet Union in the ’40s, ’50s, and ’60s established the importance of LMIs in control theory (see Boyd at al. [9]). In the early ’70s, Do2snath and Hoffman [13] and then Cullum, Donath, and Wolfe =-=[12]-=- showed that some hard graph-partitioning problems could be attacked by considering a related eigenvalue optimization problem – as we shall see, these are closely connected with SDP. Then in 1979, Lov... |

46 |
Centered Newton method for mathematical programming, in System Modeling and Optimization
- Tanabe
- 1988
(Show Context)
Citation Context ...void the need to adjust a parameter. Such potential functions were first introduced by Karmarkar in his seminal work on interior-point methods for linear programming [30]. Consider the Tanabe-Todd-Ye =-=[58, 61]-=- primal-dual potential function # # (X, y, S) := (n + #) ln X . S - ln det X - ln det S - n ln n, defined for strictly feasible points (X, y, S). If # := #(X 1 2 SX 1 2 ), then it is easy to see that ... |

40 |
A Centered Projective Algorithm for Linear Programming
- Todd, Ye
- 1989
(Show Context)
Citation Context ...void the need to adjust a parameter. Such potential functions were first introduced by Karmarkar in his seminal work on interior-point methods for linear programming [30]. Consider the Tanabe-Todd-Ye =-=[58, 61]-=- primal-dual potential function # # (X, y, S) := (n + #) ln X . S - ln det X - ln det S - n ln n, defined for strictly feasible points (X, y, S). If # := #(X 1 2 SX 1 2 ), then it is easy to see that ... |

39 | A second-order bundle method to minimize the maximum eigenvalue function
- Oustry
(Show Context)
Citation Context ...d, but also attains asymptotic quadratic convergence under suitable regularity conditions. These bundle methods are further discussed, and improved computational results given, in Helmberg and Oustry =-=[26]-=-. Fukuda and Kojima [19] have recently proposed an interior-point method for the same class of problems, working just in the space of y to avoid di#culties for large-scale problems. This paper also ha... |

38 |
Euclidean Jordan algebras and interior-point algorithms
- Faybusovich
- 1997
(Show Context)
Citation Context ...dmit symmetric primal-dual algorithms (these cones turn out to coincide with symmetric (homogeneous self-dual) cones). Another viewpoint is that of Euclidean Jordan Algebras, developed by Faybusovich =-=[15, 16]-=- and now investigated by a number of authors: see Alizadeh and Schmieta [5]. Since the area is receiving so much attention, it is hard to keep abreast of recent developments, but this is immeasurably ... |

38 |
A recipe for semidefinite relaxation for 0-1 quadratic programming
- Polijak, Rendl, et al.
- 1995
(Show Context)
Citation Context ...xed unit vector (1), then we recover (IQP). The third way to derive the SDP relaxation is by taking the dual twice. (This approach was apparently first considered by Shor [57]; see also Poljak et al. =-=[51]-=-.) Given any optimization problem max{f(x) : g(x) = b, x # #}, where we have distinguished a certain set of m equality constraints and left the rest as an abstract set restriction, the Lagrangian dual... |

37 | Initialization in semidefinite programming via a self-dual skew-symmetric embedding
- Klerk, Roos, et al.
- 1997
(Show Context)
Citation Context ...roblems (P) and (D) can be embedded in a larger self-dual system that always has strictly feasible solutions at hand and whose solution gives the required information about the original problems: see =-=[31, 38, 53]-=-, based on the work of Ye, Todd, and Mizuno [69] for linear programming. 6.1 Path-following methods These methods are motivated by Theorem 5.4, and attempt to track points on the central path as the p... |

36 | On homogeneous interior-point algorithms for semidefinite programming
- Potra, Sheng
- 1998
(Show Context)
Citation Context ...roblems (P) and (D) can be embedded in a larger self-dual system that always has strictly feasible solutions at hand and whose solution gives the required information about the original problems: see =-=[31, 38, 53]-=-, based on the work of Ye, Todd, and Mizuno [69] for linear programming. 6.1 Path-following methods These methods are motivated by Theorem 5.4, and attempt to track points on the central path as the p... |

32 | Interior point trajectories in semidefinite programming
- Goldfarb, Scheinberg
- 1998
(Show Context)
Citation Context ... associated to X(#) and (y(#), S(#)) is n#, and this approaches zero as # tends to zero. (In fact, the central path approaches optimal solutions to the primal and dual problems as # decreases to zero =-=[37, 23]-=-, but we shall not prove this here.) To show that (P ) has a bounded nonempty set of optimal solutions, we proceed as in the proof of Theorem 5.2, again choosing (y, S) # F 0 (D). Clearly, the set of ... |

31 |
On the Search Directions in Interior-Point Methods for Semidefinite Programming
- Todd
- 1997
(Show Context)
Citation Context ...dd [46, 47], from yet another motivation, and shown to be derivable in this form by Todd, Toh, and Tutuncu [60]. These and several other search directions are discussed in Kojima et al. [32] and Todd =-=[59]-=-. In the first case, the Newton direction can be obtained from the solution of a linear system as in (10) with E = I, F = X # S -1 ; in the second case with E = S #X -1 , F = I; and in the third case ... |

28 |
Dual quadratic estimates in polynomial and Boolean programming
- Shor
- 1990
(Show Context)
Citation Context ...ow of P to a multiple of a fixed unit vector (1), then we recover (IQP). The third way to derive the SDP relaxation is by taking the dual twice. (This approach was apparently first considered by Shor =-=[57]-=-; see also Poljak et al. [51].) Given any optimization problem max{f(x) : g(x) = b, x # #}, where we have distinguished a certain set of m equality constraints and left the rest as an abstract set res... |

22 | Conic convex programming and self-dual embedding
- Luo, Sturm, et al.
(Show Context)
Citation Context ...X # 0. If X 16 satisfies both these conditions, the maximum reduces to just C . X, and we retrieve (P). Next we present a number of examples, from Vandenberghe and Boyd [63] and Luo, Sturm, and Zhang =-=[38]-=-, showing how strong duality can fail. Further examples can be found in the latter reference. Consider first max -y 1 , # -1 0 0 0 # y 1 + # 0 0 0 -1 # y 2 # # 0 1 1 0 # . Equivalently, we require tha... |

21 | A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming
- Monteiro, Zhang
- 1981
(Show Context)
Citation Context ... the longest step that keeps the iterate in N-# (# ). Here it is not certain that the AHO search direction will be well-defined, so our theoretical results are for the other cases. Monteiro and Zhang =-=[43]-=- showed that such an algorithm, with any # and # in (0, 1) and independent of n, and using the NT search direction, generates a strictly feasible point with duality gap at most # times that of the ori... |

20 |
On systems of linear inequalities in Hermitian matrix variables
- Bellman, Fan
- 1963
(Show Context)
Citation Context ...l solutions, and there is no duality gap. We will give an alternative proof of this corollary in the next section. We note the historical fact that Corollary 4.2 was proved in 1963 by Bellman and Fan =-=[6]-=- for the following pair of SDP problems: min # j C j . X j # j (A ij X j +X j A T ij ) = B i , for all i X j # 0, for all j, max # i B i . Y i # i (Y i A ij +A T ij Y i ) # C j , for all j. 20 Here al... |

19 | Solving semidefinite programs via nonlinear programming part ii: Interior point methods for a subclass of sdps
- Burer, Monteiro, et al.
- 1999
(Show Context)
Citation Context ... constraints specifying the diagonal entries of X: (P ) : minC . X, diag(X) = d, AX = b, X # 0, 35 with dual problem (D) : max d T z + b T y, Diag(z) +A # y + S = C, S # 0. Burer, Monteiro, and Zhang =-=[10]-=- suggest solving (D) by an equivalent nonlinear programming problem obtained by eliminating variables. In fact, they only consider strictly feasible solutions of (D). Their procedure is based on a the... |

18 | Interior-point algorithms for semidefinite programming based on a nonlinear programming formulation
- Burer, Monteiro, et al.
- 1999
(Show Context)
Citation Context ...n suggest algorithms to solve this problem: a log-barrier method and a potential-reduction method. A subsequent paper relaxes the requirement that the diagonal of X be fixed. Instead, they require in =-=[11]-=- that the diagonal be bounded below, so the first constraint becomes diag(X) # d. This constraint can be without loss of generality, since it holds for any positive semidefinite matrix if we choose th... |

17 |
Path following methods
- Monteiro, Todd
- 2000
(Show Context)
Citation Context ...urhood.) Predictor-corrector methods, which alternate taking # = 1 (with a line search) and # = 0, and use two sizes of narrow neighbourhood, also have the same complexity. Also see Monteiro and Todd =-=[41]-=-. A typical long-step primal-dual path-following algorithm assumes given an initial strictly feasible point (X, y, S) # N-# (# ). Choose # = # for some # # (0, 1), compute the search direction chosen ... |

15 |
Symmetric cones, potential reduction methods
- Alizadeh, Schmieta
- 2000
(Show Context)
Citation Context ...metric (homogeneous self-dual) cones). Another viewpoint is that of Euclidean Jordan Algebras, developed by Faybusovich [15, 16] and now investigated by a number of authors: see Alizadeh and Schmieta =-=[5]-=-. Since the area is receiving so much attention, it is hard to keep abreast of recent developments, but this is immeasurably assisted by three web sites, those of Helmberg [25] and Alizadeh [2] on sem... |

14 |
Goemans, Semidefinite programming in combinatorial optimization
- X
- 1997
(Show Context)
Citation Context ...gives the same value as the previous ones: we refer to Grotschel, Lovasz, and Schrijver [24] for a proof of this (and several 13 other definitions of # = #(G)). See also the survey article of Goemans =-=[21]-=-. We just note here the relationship between positive semidefinite matrices and sets of vectors. If V gives an optimal solution to the problem defining # as a maximum eigenvalue, then #I - V - ee T is... |

14 | The U-Lagrangian of the maximum eigenvalue function
- Oustry
- 1999
(Show Context)
Citation Context ...ch as the spectral bundle method, but it is not clear how second-order information can be incorporated in nonsmooth optimization. However, for the maximum eigenvalue problem, this is possible: Oustry =-=[48, 49]-=- devises the so-called U-Lagrangian of the maximum eigenvalue function, uses this to get a quadratic approximation to the latter along a manifold where the maximum eigenvalue has a fixed multiplicity,... |