## Semidefinite Programming (1996)

### Cached

### Download Links

Venue: | SIAM REVIEW |

Citations: | 766 - 46 self |

### BibTeX

@MISC{Vandenberghe96semidefiniteprogramming,

author = {Lieven Vandenberghe and Stephen Boyd},

title = {Semidefinite Programming},

year = {1996}

}

### OpenURL

### Abstract

### Citations

3257 | Convex Analysis - Rockafellar - 1970 |

1139 | Geometric Algorithms and Combinatorial Optimization - Grötschel, Lovász, et al. - 1981 |

934 | Improved approximation algorithms for maximum cuts and satisfiability problems using semidefinite programming. Journal of the Association for Computing Machinery, 42:1115–1145
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...AX-CUT problem, which is a specific case of (20) where b = 0 and the diagonal of A is zero, Goemans and Williamson have proved that the lower bound from (21) is at most 14% suboptimal (see [GW94] and =-=[GW95]-=-). This is much better than any previously known bound. Similar strong results on semidefinite programming relaxations of NP-hard problems have been obtained by Karger, Motwani, and Sudan [KMS94]. The... |

710 |
Nonlinear Programming : Theory and Algorithms
- Bazaraa, Shetty
- 1979
(Show Context)
Citation Context ...hen an initial primal strictly feasible or dual strictly feasible point is not known. 6.1 Big-M method The `big-M ' method is standard in nonlinear programming; see, e.g., Bazaraa, Sherali and Shetty =-=[BSS93]-=-, or Anstreicher [Ans91]. We distinguish three cases. Case 1. A strictly feasible x is known, but no strictly feasible Z. Case 2. A strictly feasible Z is known, but no strictly feasible x. Case 3. Ne... |

646 | A New Polynomial-Time Algorithm for Linear Programming
- Karmarkar
- 1984
(Show Context)
Citation Context ..., Ringertz [Rin91], Fan and Nekooie [FN92], Fan [Fan93], Hiriart-Urruty and Ye [HUY95], Shapiro and Fan [SF94], and Pataki [Pat94]. Interior-point methods for LPs were introduced by Karmarkar in 1984 =-=[Kar84]-=-, although many of the underlying principles are older (see, e.g., Fiacco and McCormick [FM68], Lieu and Huard [LH66], and Dikin [Dik67]). Karmarkar's algorithm, and the interior-point methods develop... |

494 | Iterative Solution Methods - Axelsson - 1994 |

470 | Interior point methods in semidefinite programming with applications in combinatorial optimization
- Alizadeh
- 1995
(Show Context)
Citation Context ...g. 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 Den Hertog [... |

430 |
Convex Analysis and Minimization Algorithms
- Hiriart-Urruty, Lemaréchal
- 1993
(Show Context)
Citation Context ...ial time. In practice, however, the ellipsoid method is slow. Some general methods for nondi erentiable convex optimization are described by Shor [100], Kiwiel [57], and Hiriart-Urruty and Lemarechal =-=[47]-=-. These methods are more e cient in practice than the ellipsoid method, and can be used to solve semide nite programs. In this paper we consider recently developed interior-point methods for semidefin... |

332 | LSQR: An algorithm for sparse linear equations and sparse least squares
- Paige, Saunders
- 1982
(Show Context)
Citation Context ...niques. Iterative techniques A second group of methods solves the equations (66), (62) or (86) iteratively. For (66) or (65) the conjugate gradients method or the LSQR algorithm of Paige and Saunders =-=[PS82]-=- appear to be very well suited. In exact arithmetic, these algorithms solve (65) in m+1 iterations, where each iteration requires an evaluation of the two (adjoint) linear mappings (v 1 ; : : : ; vm )... |

318 |
Solution of Sparse Indefinite Systems of Linear Equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ...methods for exploiting structure in semidefinite programs arising in engineering. One can also consider solving the symmetric systems (62) or (86) iteratively, using Paige and Saunders' SYMMLQ method =-=[PS75]-=-, or Freund and Nachtigal's symmetric QMR method [FN94]. Working on (62) or (86) has the advantage of allowing more freedom in the selection of preconditioners [GMPS92]. In practice, i.e., with roundo... |

308 |
Non Linear Programming: Sequential Unconstrained Minimisation Technique
- Fiacco, McCormick
- 1966
(Show Context)
Citation Context ... and Fan [SF94], and Pataki [Pat94]. Interior-point methods for LPs were introduced by Karmarkar in 1984 [Kar84], although many of the underlying principles are older (see, e.g., Fiacco and McCormick =-=[FM68]-=-, Lieu and Huard [LH66], and Dikin [Dik67]). Karmarkar's algorithm, and the interior-point methods developed afterwards, combine a very low, polynomial, worst-case complexity with excellent behavior i... |

273 |
Minimization Methods for Nondifferentiable Functions
- Shor
- 1985
(Show Context)
Citation Context ...(see [YN77, Sho77]) to solve problem (1) in polynomial time. In practice, however, the ellipsoid method is slow. 3 Some general methods for nondifferentiable convex optimization are described by Shor =-=[Sho85]-=-, Kiwiel [Kiw85], and Hiriart-Urruty and Lemar'echal [HUL93]. These methods are more efficient in practice than the ellipsoid method, and can be used to solve semidefinite programs. In this paper we c... |

260 | Cones of matrices and set-functions and 0–1 optimization
- Lovász, Schrijver
- 1991
(Show Context)
Citation Context ...onath and Hoffman [DH73]). Many people seem to have developed similar ideas independently. We should however stress the importance of the work 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 power... |

247 |
Interior-point polynomial methods in convex programming, Volume 13
- Nesterov, Nemirovsky
- 1994
(Show Context)
Citation Context ...rF (x) \Gamma1 can be substituted for det F (x) \Gamma1 in (45)), but this one enjoys many special properties. In particular, when F (x) ? 0, it is analytic, strictly convex, and self-concordant (see =-=[NN94]-=-). Figure 5 shows the contour lines of the barrier function for the semidefinite program of r x ? Figure 5: Contour lines of the barrier function (incremented in unit steps). x ? is the minimizer of t... |

202 | An interior-point method for semidefinite programming
- Helmberg, Rendl, et al.
- 1996
(Show Context)
Citation Context ...ent 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 ... |

178 | Approximate graph coloring by semidefinite programming
- Karger, Motwani, et al.
- 1998
(Show Context)
Citation Context ...] and [GW95]). This is much better than any previously known bound. Similar strong results on semidefinite programming relaxations of NP-hard problems have been obtained by Karger, Motwani, and Sudan =-=[KMS94]-=-. The usefulness of semidefinite programming in combinatorial optimization was recognized more than twenty years ago (see, e.g., Donath and Hoffman [DH73]). Many people seem to have developed similar ... |

166 |
Optimal Design of Experiments
- Pukelsheim
- 1993
(Show Context)
Citation Context ...the reliability of the test. By solving the semidefinite program (14) one can compute a lower bound for ae. Semidefinite programming also has applications in optimal experiment design (see Pukelsheim =-=[Puk93]-=-). 11 Geometrical problems involving quadratic forms Many geometrical problems involving quadratic functions can be expressed as semidefinite programs. We will give one simple example. Suppose we are ... |

143 |
Lower bounds for the partitioning of graphs
- Donath, Hoffman
- 1973
(Show Context)
Citation Context ... been obtained by Karger, Motwani, and Sudan [KMS94]. The usefulness of semidefinite programming in combinatorial optimization was recognized more than twenty years ago (see, e.g., Donath and Hoffman =-=[DH73]-=-). Many people seem to have developed similar ideas independently. We should however stress the importance of the work by Grotschel, Lov'asz, and Schrijver [GLS88, Chapter 9], [LS91] who have demonstr... |

142 | A Unified Approach to Interior Point Algorithms for Linear Complementarity
- KOJIMA, MEGIDDO, et al.
- 1991
(Show Context)
Citation Context ...onzaga [Gon92]). Interior-point methods were subsequently extended to handle convex quadratic programming, and to certain linear complementarity problems (see, e.g., Kojima, Megiddo, Noma and Yoshise =-=[KMNY91]-=-). An important breakthrough was achieved by Nesterov and Nemirovsky in 1988 [NN88, NN90b, NN90a, NN91a, NN91a]. They showed that interior-point methods for linear programming can, in principle, be ge... |

121 |
Path-following methods for linear programming
- Gonzaga
- 1992
(Show Context)
Citation Context ...al, worst-case complexity with excellent behavior in practice. Karmarkar's paper has had an enormous impact, and several 4 variants of his method have been developed (see, e.g., the survey by Gonzaga =-=[Gon92]-=-). Interior-point methods were subsequently extended to handle convex quadratic programming, and to certain linear complementarity problems (see, e.g., Kojima, Megiddo, Noma and Yoshise [KMNY91]). An ... |

119 |
Interior point methods for linear programming: Ready for production use
- MARSTEN, SHANNO
- 1990
(Show Context)
Citation Context ...ency. It is now generally accepted that interior-point methods for LPs are competitive with the simplex method and even faster for problems with more than 10; 000 variables or constraints (see, e.g., =-=[LMS94]-=-). Similarly, our experience with system and control applications suggests that interior-point methods for semidefinite programs are competitive with other methods for small problems, and substantiall... |

101 | Complementarity and nondegeneracy in semidefinite programming
- Alizadeh, Haeberly, et al.
- 1997
(Show Context)
Citation Context ...its dual (30). Taking Z = diag(z), and F (x) = diag(Ax+b), we see that ZF (x) = 0 if and only if z i (Ax+ b) i = 0 for i = 1; : : : ; n, i.e., the zero patterns in z and Ax+ b are complementary. (See =-=[AHO95]-=- for a detailed analysis of complementarity in semidefinite programming.) Theorem 1 gives us optimality conditions for the semidefinite program (1) if we assume strict primal and dual feasibility: x i... |

90 |
Iterative solution of problems of linear and quadratic programming", Soviet Mathematics Doklady 8
- Dikin
- 1967
(Show Context)
Citation Context ...rior-point methods for LPs were introduced by Karmarkar in 1984 [Kar84], although many of the underlying principles are older (see, e.g., Fiacco and McCormick [FM68], Lieu and Huard [LH66], and Dikin =-=[Dik67]-=-). Karmarkar's algorithm, and the interior-point methods developed afterwards, combine a very low, polynomial, worst-case complexity with excellent behavior in practice. Karmarkar's paper has had an e... |

84 |
Methods of descent for nondifferentiable optimization
- Kiwiel
- 1985
(Show Context)
Citation Context ...7]) to solve problem (1) in polynomial time. In practice, however, the ellipsoid method is slow. 3 Some general methods for nondifferentiable convex optimization are described by Shor [Sho85], Kiwiel =-=[Kiw85]-=-, and Hiriart-Urruty and Lemar'echal [HUL93]. These methods are more efficient in practice than the ellipsoid method, and can be used to solve semidefinite programs. In this paper we consider recently... |

84 | A primal-dual potential reduction method for problems involving matrix inequalities
- Vandenberghe, Boyd
- 1995
(Show Context)
Citation Context ...lized 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... |

80 | Interior methods for constrained optimization
- Wright
- 1992
(Show Context)
Citation Context ... only in the scaling matrices S used in (62). In linear programming, the equivalent of method 3 is usually preferred, since it is more efficient and has better numerical properties (see, e.g., Wright =-=[Wri92]-=-). We should however mention two other possibilities that generalize (73). Alizadeh, Haeberly, and Overton [AHO94] have pointed out the potential numerical difficulties in (69) and (71) and proposed t... |

78 | Large-scale optimization of eigenvalues
- Overton
- 1992
(Show Context)
Citation Context ... tI \Gamma A(x)s0; with variables x 2 R k and t 2 R. Problems of this type arise in control theory, structural optimization, graph theory and combinatorial optimization, and other fields. See Overton =-=[Ove92]-=-, Mohar and Poljak [MP93], and Grotschel, Lov'asz and Schrijver [GLS88, Chapter 9] for surveys. Several interesting related problems can be solved using semidefinite programming. As an example, to min... |

71 | On minimizing the maximum eigenvalue of a symmetric matrix - Overton - 1988 |

69 |
878-approximation algorithm for max-cut and max-2sat
- GOEMANS, WILLIAMSON
- 1994
(Show Context)
Citation Context ...er on the MAX-CUT problem, which is a specific case of (20) where b = 0 and the diagonal of A is zero, Goemans and Williamson have proved that the lower bound from (21) is at most 14% suboptimal (see =-=[GW94]-=- and [GW95]). This is much better than any previously known bound. Similar strong results on semidefinite programming relaxations of NP-hard problems have been obtained by Karger, Motwani, and Sudan [... |

68 |
O(n 3 L) potential reduction algorithm for linear programming
- Ye
- 1991
(Show Context)
Citation Context .... It turns out that a complete primal-dual algorithm can be based on the primal system only, by choosing ffiZ p as dual search direction. In linear 35 programming this primal-dual method is due to Ye =-=[Ye91]-=-; the extension to semidefinite programs is due to Nesterov and Nemirovsky [NN94] and Alizadeh [Ali91]. Again it is possible to reduce ' by at least a fixed amount. Theorem 5 Let x (k) and Z (k) denot... |

63 |
Combinatorial optimization with interior point methods and semi-de nite matrices
- Alizadeh
- 1991
(Show Context)
Citation Context ...ing ffiZ p as dual search direction. In linear 35 programming this primal-dual method is due to Ye [Ye91]; the extension to semidefinite programs is due to Nesterov and Nemirovsky [NN94] and Alizadeh =-=[Ali91]-=-. Again it is possible to reduce ' by at least a fixed amount. Theorem 5 Let x (k) and Z (k) denote the values of x and Z after the kth iteration of the potential reduction algorithm with search direc... |

63 |
On a theorem of weyl concerning eigenvalues of linear transformations i
- Fan
- 1949
(Show Context)
Citation Context ...ell known that the sum of the r largest eigenvalues of a matrix A = A T 2 R p\Thetap can be expressed as maximum TrW T AW subject to W 2 R p\Thetar W T W = I: (41) This result is attributed to Ky Fan =-=[Fan49]. Ove-=-rton and Womersley [OW92] have observed that (41) can be expressed as the semidefinite program maximize TrAZ 11 subject to TrZ 11 = r Z 11 + Z 22 = I " Z 11 Z 12 Z T 12 Z 22 #s0: (42) The equival... |

62 | Optimality conditions and duality theory for minimizing sums of the Lqest eigenvalues of symmetric matrices. Mathemicd Programming - Overton, Wamersley - 1993 |

61 |
Interior-point methods for the monotone linear complementarity problem in symmetric matrices
- KOJIMA, SHINDOH, et al.
- 1997
(Show Context)
Citation Context ...or-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 Den Hertog [dH93]. 1.3 Outline In x2 we descri... |

60 | Ghaoui, \Method of centers for minimizing generalized eigenvalues
- Boyd, El
- 1993
(Show Context)
Citation Context ...0 as a special case. Problem (82) is not a semidefinite program, however, because of the bilinear term tB(x). It is a quasi-convex problem, and can still be efficiently solved. See Boyd and El Ghaoui =-=[BE93]-=-, Haeberly and Overton [HO94], and Nesterov and Nemirovsky [NN91b, Nem94] for details. 7.2 Determinant maximization In x4.3 we discussed the problem of minimizing the barrier function \Gamma log det F... |

57 |
An O(/-ffL) iteration potential reduction algorithm for linear complementarity problems
- KOJIMA, MIZUNO, et al.
- 1991
(Show Context)
Citation Context ...riate approximate plane search. In the case of an LP, with F = diag(Ax + b) and Z = diag(z), this symmetric scaling coincides with the primal-dual symmetric scaling used in Kojima, Mizuno and Yoshise =-=[KMY91], for example, -=-where search directions are computed from " FZ \Gamma1 A A T 0 # " ffiz sym ffix sym # = " \GammaaeF e + Z \Gamma1 e 0 # : (73) The three algorithms we discussed so far differ only in t... |

56 |
Interior Point Approach to Linear, Quadratic and Convex Programming, Algorithms and Complexity
- Hertog
- 1994
(Show Context)
Citation Context ...], 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 Den Hertog =-=[dH93]-=-. 1.3 Outline In x2 we describe several applications of semidefinite programming. Section x3 covers duality theory for semidefinite programs, emphasizing the similarities and differences with linear p... |

56 |
Preconditioners for Indefinite Systems Arising in Optimization
- Gill, Murray, et al.
- 1992
(Show Context)
Citation Context ...gy is to solve the sparse system (67) directly. Several researchers have argued that this method has better numerical properties (See Fourer and Mehrotra [FM91], Gill, Murray, Ponceleon, and Saunders =-=[GMPS92]-=-, and Vanderbei and Carpenter [VC93]). Moreover, directly solving (67) avoids the loss of sparsity caused by squaring A. Neither of these techniques works for semidefinite programs unfortunately, beca... |

55 | An exact duality theory for semidefinite programming and its complexity implications - Ramana - 1997 |

51 |
Semidefinite matrix constraints in optimization," SIAMJournal on Control and Optimization 23
- Fletcher
- 1985
(Show Context)
Citation Context ...n early paper on the theoretical properties of semidefinite programs is Bellman and Fan [BF63]. Other references discussing optimality conditions are Craven and Mond [CM81], Shapiro [Sha85], Fletcher =-=[Fle85]-=-, Allwright [All88], Wolkowicz [Wol81], and Kojima, Kojima and Hara [KKH94]. Many researchers have worked on the problem of minimizing the maximum eigenvalue of a symmetric matrix, which can be cast a... |

48 |
Programming in Infinite-Dimensional Spaces: Theory andApplications
- ANDERSON, NASH, et al.
- 1987
(Show Context)
Citation Context ... inequality when x and y are vectors, and matrix inequality when x and y are (symmetric) matrices. In this paper, the context will always make it clear which is meant. 2 See however Anderson and Nash =-=[AN87]-=- for simplex-like methods in semi-infinite linear progamming, and Pataki [Pat95] and Lasserre [Las95] for extensions of the simplex method to semidefinite programming. 2 where we assume that d T x ? 0... |

45 |
An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices
- Jarre
- 1993
(Show Context)
Citation Context ...d 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], an... |

45 |
Quadratic optimization problems
- Shor
- 1987
(Show Context)
Citation Context ... follows from convexity, along with the observation that we can construct, in polynomial time, a cutting plane for the constraint set through any given infeasible point (see, e.g., [BEFB94, x2.3], or =-=[Sho87]-=-). One can therefore apply the ellipsoid method of Yudin and Nemirovsky, and Shor (see [YN77, Sho77]) to solve problem (1) in polynomial time. In practice, however, the ellipsoid method is slow. 3 Som... |

45 | New algorithms in convex programming based on a notion of ’centre’ (for systems of analytic inequalities) and on rational expectations - Sonnevend - 1988 |

44 |
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
- Cullum, Donath, et al.
(Show Context)
Citation Context ...ny researchers have worked on the problem of minimizing the maximum eigenvalue of a symmetric matrix, which can be cast as a semidefinite program (see x2). See, for instance, Cullum, Donath and Wolfe =-=[CDW75]-=-, Goh and Teo [GT88], Panier [Pan89], Allwright [All89], Overton [Ove88, Ove92], Overton and Womersley [OW93, OW92], Ringertz [Rin91], Fan and Nekooie [FN92], Fan [Fan93], Hiriart-Urruty and Ye [HUY95... |

43 | Geometric Algorithms and - Grotschel, Lovasz, et al. - 1988 |

42 | SDPSOL: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure. User’s Guide, Version Beta - Wu, Boyd - 1996 |

41 |
C.: Convex analysis and minimization algorithms II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften. 306
- Hiriart-Urruty, Lemarchal
- 1993
(Show Context)
Citation Context ... In practice, however, the ellipsoid method is slow. 3 Some general methods for nondifferentiable convex optimization are described by Shor [Sho85], Kiwiel [Kiw85], and Hiriart-Urruty and Lemar'echal =-=[HUL93]-=-. These methods are more efficient in practice than the ellipsoid method, and can be used to solve semidefinite programs. In this paper we consider recently developed interior-point methods for semide... |

41 | Eigenvalues in combinatorial optimization
- MOHAR, POIAAK
- 1993
(Show Context)
Citation Context ...riables x 2 R k and t 2 R. Problems of this type arise in control theory, structural optimization, graph theory and combinatorial optimization, and other fields. See Overton [Ove92], Mohar and Poljak =-=[MP93]-=-, and Grotschel, Lov'asz and Schrijver [GLS88, Chapter 9] for surveys. Several interesting related problems can be solved using semidefinite programming. As an example, to minimize the sum of the r la... |

41 |
Symmetric indefinite systems for interior point methods
- Vanderbei, Carpenter
- 1993
(Show Context)
Citation Context ...directly. Several researchers have argued that this method has better numerical properties (See Fourer and Mehrotra [FM91], Gill, Murray, Ponceleon, and Saunders [GMPS92], and Vanderbei and Carpenter =-=[VC93]-=-). Moreover, directly solving (67) avoids the loss of sparsity caused by squaring A. Neither of these techniques works for semidefinite programs unfortunately, because they lead to systems with large ... |