## Algorithms and Software for LMI Problems in Control (1997)

Venue: | IEEE Control Systems Magazine |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Vandenberghe97algorithmsand,

author = {Lieven Vandenberghe and Venkataramanan Balakrishnan},

title = {Algorithms and Software for LMI Problems in Control},

booktitle = {IEEE Control Systems Magazine},

year = {1997},

pages = {89--95}

}

### OpenURL

### Abstract

this article is to provide an overview of the state of the art of

### Citations

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Citation Context ...m. For more extensive surveys on the theory and applications of SDP, we refer to Alizadeh [15], Boyd et al. [3], Lewis and Overton [16], Nesterov and Nemirovskii [17, x6.4], and Vandenberghe and Boyd =-=[18]-=-. We have already defined an SDP formally in (1). To distinguish it from other formulations, we will refer to (1) as an SDP in inequality form. The optimization problem maximize TrCX subject to Xs0 Tr... |

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Citation Context ... the sixties; see e.g., Fiacco and McCormick [21], Lieu and Huard [22], and Dikin [23]). Interest in them was revived in 1984, when Karmarkar introduced a polynomial-time interior-point method for LP =-=[24]-=-. In 1988 Nesterov and Nemirovskii [25] showed that those interior-point methods for linear programming can, in principle, be generalized to all convex optimization problems. The key element is the kn... |

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Citation Context ...emidefinite programming In this section we provide a brief introduction to the semidefinite programming problem. For more extensive surveys on the theory and applications of SDP, we refer to Alizadeh =-=[15]-=-, Boyd et al. [3], Lewis and Overton [16], Nesterov and Nemirovskii [17, x6.4], and Vandenberghe and Boyd [18]. We have already defined an SDP formally in (1). To distinguish it from other formulation... |

483 | Primal-Dual Interior-Point Methods
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Citation Context ...mming. This recent research has largely concentrated on primal-dual methods in the hope of emulating the excellent performance of primal-dual interior-point methods for large-scale linear programming =-=[29, 30]-=-. The remainder of this section will concentrate on this recent work. We should mention however that other methods have been used successfully, e.g., the ellipsoid algorithm, the method of alternating... |

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Citation Context ...led complementary slackness. Interior-point methods Brief history The ideas underlying interior-point methods for convex optimization can be traced back to the sixties; see e.g., Fiacco and McCormick =-=[21]-=-, Lieu and Huard [22], and Dikin [23]). Interest in them was revived in 1984, when Karmarkar introduced a polynomial-time interior-point method for LP [24]. In 1988 Nesterov and Nemirovskii [25] showe... |

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Citation Context ... equations become " A T 0 \GammaX \Gamma2 A # " ffix ffiy # = " 0 z \Gamma X \Gamma1 e # : The first SDP methods were based on these primal or dual scalings (see for example, Nesterov a=-=nd Nemirovskii [17]-=-, Alizadeh [26], and Vandenberghe and Boyd [4]). In linear programming, however, the primal and dual scalings are rarely used in practice. Instead, one usually prefers a primal-dual symmetric scaling ... |

212 | An interior-point method for semidefinite programming
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Citation Context ...very rapid progress has been made in the last two years. Among the proposed symmetric primal-dual algorithms, three variations seem to be the most promising. Helmberg, Rendl, Vanderbei, and Wolkowicz =-=[33]-=-, Kojima, Shidoh and Hara [34], and Monteiro [35] solve (10) and (11) and linearize the resulting ffiX. Alizadeh, Haeberly and Overton [36] first write XZ = I as XZ + ZX = 2I and then linearize this a... |

186 | Primal-dual interior-point methods for self-scaled cones
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Citation Context ...t write XZ = I as XZ + ZX = 2I and then linearize this as X ffiZ + ffiXZ + Z ffiX + ffiZX = 2I \Gamma XZ \Gamma ZX: The resulting ffiX and ffiZ are automatically symmetric. Finally, Nesterov and Todd =-=[37, 38]-=-, and recently Sturm and Zhang [39], have defined a third direction, obtained as follows. First a matrix R is computed such that R T XR =s1=2 and R T Z \Gamma1 R =s1=2 , wheresis a diagonal matrix wit... |

180 | Determinant maximization with linear matrix inequality constraints
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Citation Context ...html). 7 Extensions The determinant maximization problem In their survey of LMI problems in control, Boyd et al. [3] also considered an extension of the SDP (1), which was discussed in more detail in =-=[76]-=-. This extension can be written in the following form: minimize b T y \Gamma log det / \GammaD \Gamma m X i=1 y i B i ! subject to C + m X i=1 y i A is0 D + m X i=1 y i B i ! 0: (14) We will call this... |

151 | Csdp, a c library for semidefinite programming
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Citation Context ...in turn based on LMITOOL. Several implementations of the most recent primal-dual methods are also available now. SDPA [71] is a C++ code, based on the algorithm of Kojima, Shindoh and Hara [34]. CSDP =-=[72]-=- is a C implementation of the algorithm of Helmberg, Rendl, Vanderbei, and Wolkowicz [33]. SDPHA [73] is a Matlab implementation of a homogeneous formulation of the different primal-dual methods descr... |

148 | Primal-dual path following algorithms for semide¯nite programming," Working Paper
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Citation Context ... years. Among the proposed symmetric primal-dual algorithms, three variations seem to be the most promising. Helmberg, Rendl, Vanderbei, and Wolkowicz [33], Kojima, Shidoh and Hara [34], and Monteiro =-=[35]-=- solve (10) and (11) and linearize the resulting ffiX. Alizadeh, Haeberly and Overton [36] first write XZ = I as XZ + ZX = 2I and then linearize this as X ffiZ + ffiXZ + Z ffiX + ffiZX = 2I \Gamma XZ ... |

146 |
Lower Bounds for the Partitioning of Graphs
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Citation Context ...zation as well. For example, certain eigenvalue minimization problems that can be cast as SDPs have been used for obtaining bounds and heuristic solutions for combinatorial optimization problems (see =-=[7, 8]-=- and [9, Chapter 9]). The efficiency of recent interior-point methods for SDP, which is directly responsible for the popularity of SDP in control, has therefore also attracted a great deal of interest... |

114 | On the Nesterov–Todd direction in semidefinite programming
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Citation Context ... ; m (12) \GammaRR T ffiXRR T + m X i=1 ffiy i A i = Z \Gamma X \Gamma1 : (13) to obtain the search directions ffiX, ffiZ, ffiy. Numerical details on this method can be found in Todd, Toh and Tutuncu =-=[40]-=-. Finally, Kojima, Shindoh and Hara [34], Monteiro [35], and Monteiro and Zhang [41] have presented unifying frameworks for primal-dual methods. Some other important recent articles and reports are li... |

97 | SDPA (SemiDefinite Programming Algorithm) User's Manual | Version 6.00
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Citation Context ...he Induced-Norm Control Toolbox [70] is a toolbox for robust and optimal control, in turn based on LMITOOL. Several implementations of the most recent primal-dual methods are also available now. SDPA =-=[71]-=- is a C++ code, based on the algorithm of Kojima, Shindoh and Hara [34]. CSDP [72] is a C implementation of the algorithm of Helmberg, Rendl, Vanderbei, and Wolkowicz [33]. SDPHA [73] is a Matlab impl... |

91 |
Iterative solution of problems of linear and quadratic programming
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Citation Context ...-point methods Brief history The ideas underlying interior-point methods for convex optimization can be traced back to the sixties; see e.g., Fiacco and McCormick [21], Lieu and Huard [22], and Dikin =-=[23]-=-). Interest in them was revived in 1984, when Karmarkar introduced a polynomial-time interior-point method for LP [24]. In 1988 Nesterov and Nemirovskii [25] showed that those interior-point methods f... |

89 | A primal-dual potential reduction method for problems involving matrix inequalities - Vandenberghe, Boyd - 1995 |

86 | Interior methods for constrained optimization
- Wright
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Citation Context ...veral researchers have demonstrated that methods that use this primal-dual symmetric scaling can achieve a higher accuracy than methods based on the the primal or dual scaling (see for example Wright =-=[32]-=-), and therefore the symmetric scaling is the basis of all practical LP interior-point methods. The extension of this symmetric primal-dual scaling to SDP is not straightforward: The linearization (9)... |

81 | Large-scale optimization of eigenvalues
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Citation Context ...larity of SDP in control, has therefore also attracted a great deal of interest in optimization circles, overshadowing earlier solution methods based on techniques from nondifferentiable optimization =-=[8, 10, 11, 12, 13]-=-. At every major optimization conference, there are workshops and special sessions devoted exclusively to SDP, and a special issue of Mathematical Programming has recently been devoted to SDP [14]. Th... |

73 |
On minimizing the maximum eigenvalue of a symmetric matrix
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Citation Context ...larity of SDP in control, has therefore also attracted a great deal of interest in optimization circles, overshadowing earlier solution methods based on techniques from nondifferentiable optimization =-=[8, 10, 11, 12, 13]-=-. At every major optimization conference, there are workshops and special sessions devoted exclusively to SDP, and a special issue of Mathematical Programming has recently been devoted to SDP [14]. Th... |

70 | Implementation of Interior-Point methods for large scale linear programs
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Citation Context ...mming. This recent research has largely concentrated on primal-dual methods in the hope of emulating the excellent performance of primal-dual interior-point methods for large-scale linear programming =-=[29, 30]-=-. The remainder of this section will concentrate on this recent work. We should mention however that other methods have been used successfully, e.g., the ellipsoid algorithm, the method of alternating... |

68 |
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Citation Context ...esign, gain-scheduled controller design, and many others. For a few very special cases there are "analytical solutions" to SDPs (via Riccati equations for the ones encountered with H 2 and H=-=1 control [2]-=-, for example), but in general they can be solved numerically very efficiently. In many cases---for example, with multi-model control [3]---the LMIs encountered in SDPs in systems and control theory h... |

65 |
Interior-point methods for the monotone linear complementarity problem in symmetric matrices
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Citation Context ...ade in the last two years. Among the proposed symmetric primal-dual algorithms, three variations seem to be the most promising. Helmberg, Rendl, Vanderbei, and Wolkowicz [33], Kojima, Shidoh and Hara =-=[34]-=-, and Monteiro [35] solve (10) and (11) and linearize the resulting ffiX. Alizadeh, Haeberly and Overton [36] first write XZ = I as XZ + ZX = 2I and then linearize this as X ffiZ + ffiXZ + Z ffiX + ff... |

55 | Symmetric Primal-Dual Path-Following Algorithms for Semidefinite Programming
- Sturm, Zhang
- 1995
(Show Context)
Citation Context ... linearize this as X ffiZ + ffiXZ + Z ffiX + ffiZX = 2I \Gamma XZ \Gamma ZX: The resulting ffiX and ffiZ are automatically symmetric. Finally, Nesterov and Todd [37, 38], and recently Sturm and Zhang =-=[39]-=-, have defined a third direction, obtained as follows. First a matrix R is computed such that R T XR =s1=2 and R T Z \Gamma1 R =s1=2 , wheresis a diagonal matrix with as diagonal elements the eigenval... |

54 |
Semi-definite matrix constraints in optimization
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Citation Context ...larity of SDP in control, has therefore also attracted a great deal of interest in optimization circles, overshadowing earlier solution methods based on techniques from nondifferentiable optimization =-=[8, 10, 11, 12, 13]-=-. At every major optimization conference, there are workshops and special sessions devoted exclusively to SDP, and a special issue of Mathematical Programming has recently been devoted to SDP [14]. Th... |

51 | Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs - Kojima, Shida, et al. - 1998 |

50 | Strong duality for semi-definite programming
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Citation Context ...en u ? = ` ? . The result follows from standard convex optimization duality. (A stronger duality theory that does not require strict feasibility was recently developed by Ramana, Tuncel and Wolkowicz =-=[19]-=-.) Some connections between SDP duality and duality in control are explored in [20]. If we assume that both (1) and (2) are strictly feasible, then the optimal values in both 3 problems are attained, ... |

50 | Polynomial Convergence of Primal-Dual Algorithms for the Second-Order - Monteiro, Tsuchiya |

49 |
A unified analysis for a class of path-following primaldual interior point algorithms for semidefinite programming
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- 1998
(Show Context)
Citation Context ...tain the search directions ffiX, ffiZ, ffiy. Numerical details on this method can be found in Todd, Toh and Tutuncu [40]. Finally, Kojima, Shindoh and Hara [34], Monteiro [35], and Monteiro and Zhang =-=[41]-=- have presented unifying frameworks for primal-dual methods. Some other important recent articles and reports are listed in the references of this paper 2 . Software packages Several researchers have ... |

48 | An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices - JARRE - 1993 |

47 | On extending primal-dual interior-point algorithms from linear programming to semidefinite programming - Zhang - 1998 |

46 | Geometric Algorithms and - Grötschel, Lovász, et al. - 1993 |

45 |
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
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Citation Context ...zation as well. For example, certain eigenvalue minimization problems that can be cast as SDPs have been used for obtaining bounds and heuristic solutions for combinatorial optimization problems (see =-=[7, 8]-=- and [9, Chapter 9]). The efficiency of recent interior-point methods for SDP, which is directly responsible for the popularity of SDP in control, has therefore also attracted a great deal of interest... |

42 | sdpsol: A Parser/Solver for Semidefinite Programs With Matrix Structure
- Boyd, Wu
- 1996
(Show Context)
Citation Context ...ode SP [67] is based on a primal-dual potential reduction method with the Nesterov and Todd scaling. The code is written in C with calls to BLAS and LAPACK and includes an interface to Matlab. SDPSOL =-=[68]-=- and LMITOOL [69] offer user-friendly interfaces to SP that simplify the specification of SDPs where the variables have matrix structure. The Induced-Norm Control Toolbox [70] is a toolbox for robust ... |

41 | A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with O( p nL)-iteration complexity - Potra |

37 |
Tütüncü, SDPT3 - a matlab software package for semidefinite programming
- Toh, Todd, et al.
- 1999
(Show Context)
Citation Context ...tion of the algorithm of Helmberg, Rendl, Vanderbei, and Wolkowicz [33]. SDPHA [73] is a Matlab implementation of a homogeneous formulation of the different primal-dual methods described above. SDPT3 =-=[74]-=- is a Matlab implementation of the most important infeasible primal-dual path-following methods. SDPPACK [75] is an implementation of the algorithm of [36]. It is written in Matlab, with critical part... |

36 |
SP: Software for Semidefinite Programming. User’s Guide
- Vandenberghe, Boyd
- 1994
(Show Context)
Citation Context ...sing the projective algorithm [17]. Matlab's LMI Control Toolbox [66] is based on the same algorithm, and offers a graphical user interface and extensive support for control applications. The code SP =-=[67]-=- is based on a primal-dual potential reduction method with the Nesterov and Todd scaling. The code is written in C with calls to BLAS and LAPACK and includes an interface to Matlab. SDPSOL [68] and LM... |

34 | Homogeneous interior-point algorithms for semidefinite programming - Potra, Sheng - 1998 |

31 | Self-scaled cones and interior-point methods in nonlinear programming
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(Show Context)
Citation Context ...t write XZ = I as XZ + ZX = 2I and then linearize this as X ffiZ + ffiXZ + Z ffiX + ffiZX = 2I \Gamma XZ \Gamma ZX: The resulting ffiX and ffiZ are automatically symmetric. Finally, Nesterov and Todd =-=[37, 38]-=-, and recently Sturm and Zhang [39], have defined a third direction, obtained as follows. First a matrix R is computed such that R T XR =s1=2 and R T Z \Gamma1 R =s1=2 , wheresis a diagonal matrix wit... |

30 |
Control system synthesis via bilinear matrix inequalities
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Citation Context ... BMIs include a wide variety of control problems, including synthesis with structured uncertainty, fixed-order controller design, decentralized controller synthesis etc. (see Safonov, Goh, and others =-=[82, 83, 84, 85, 86, 87]-=-, El Ghaoui and Balakrishnan [88], etc). The fundamental difference with LMIs is that BMI problems are non-convex, and no non-exponentialtime algorithms for their solution are known to exist. The algo... |

29 |
A primal-dual interior point method for linear programming
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Citation Context ...d [4]). In linear programming, however, the primal and dual scalings are rarely used in practice. Instead, one usually prefers a primal-dual symmetric scaling introduced by Kojima, Mizuno and Yoshise =-=[31]. For linear pr-=-ogramming the resulting equations for the search directions have the form " A T 0 \GammaX \Gamma1 Z A # " ffix ffiy # = " 0 z \Gamma X \Gamma1 e: # : (8) These equations are obtained by... |

28 | A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh–Haeberly–Overton search direction - Kojima, Shida, et al. - 1999 |

28 |
MAXDET: software for determinant maximization problems. Software package. Available from http://www.stanford.edu/∼boyd/MAXDET.html
- Vandenberghe, Boyd, et al.
(Show Context)
Citation Context ... the resulting algorithms have wide application. A list of applications and an interior-point method for the maxdet-problem are described in [76]. Software for solving maxdet-problems is available in =-=[77]-=-, and has been incorporated in SDPSOL [68]. The generalized eigenvalue minimization problem A third standard problem from [3] is the generalized eigenvalue minimization problem. Suppose we have a pair... |

28 |
Robust synthesis via bilinear matrix inequalities
- Goh, Safonov, et al.
- 1996
(Show Context)
Citation Context ... BMIs include a wide variety of control problems, including synthesis with structured uncertainty, fixed-order controller design, decentralized controller synthesis etc. (see Safonov, Goh, and others =-=[82, 83, 84, 85, 86, 87]-=-, El Ghaoui and Balakrishnan [88], etc). The fundamental difference with LMIs is that BMI problems are non-convex, and no non-exponentialtime algorithms for their solution are known to exist. The algo... |

26 | A Predictor-Corrector Method for Semi-Definite Programming, Working Paper - Lin, Saigal - 1995 |

24 |
G.P.: A global optimization approach for the BMI problem
- Goh, Safonov, et al.
- 1994
(Show Context)
Citation Context ... BMIs include a wide variety of control problems, including synthesis with structured uncertainty, fixed-order controller design, decentralized controller synthesis etc. (see Safonov, Goh, and others =-=[82, 83, 84, 85, 86, 87]-=-, El Ghaoui and Balakrishnan [88], etc). The fundamental difference with LMIs is that BMI problems are non-convex, and no non-exponentialtime algorithms for their solution are known to exist. The algo... |

20 | Duality and self-duality for conic convex programming - Luo, Sturm, et al. - 1996 |

19 |
Nemirovsky A., “Optimization over positive semidefinite matrices
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(Show Context)
Citation Context ...er 2 . Software packages Several researchers have made available software for semidefinite programming. The first implementation of an interior-point method for SDP was by Nesterov and Nemirovskii in =-=[65]-=-, using the projective algorithm [17]. Matlab's LMI Control Toolbox [66] is based on the same algorithm, and offers a graphical user interface and extensive support for control applications. The code ... |

18 |
lmitool: a front-end for LMI optimization, user's guide
- Ghaoui, Nikoukha, et al.
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(Show Context)
Citation Context ...sed on a primal-dual potential reduction method with the Nesterov and Todd scaling. The code is written in C with calls to BLAS and LAPACK and includes an interface to Matlab. SDPSOL [68] and LMITOOL =-=[69]-=- offer user-friendly interfaces to SP that simplify the specification of SDPs where the variables have matrix structure. The Induced-Norm Control Toolbox [70] is a toolbox for robust and optimal contr... |

17 | The projective method for solving linear matrix inequalities - Nemirovski, Gahinet - 1994 |

17 |
An interior point method for generalized linearfractional programming
- Nesterov, Nemirovsky
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(Show Context)
Citation Context ...m, 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 [78], Haeberly and Overton [79], and Nesterov and Nemirovskii =-=[17, 80, 81]-=- for details on specialized algorithms, and [3] for applications of this problem in control. An implementation of the Nesterov and Nemirovskii algorithm is also provided in the LMI Control toolbox [66... |