#### DMCA

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

Venue: | IEEE Control Systems Magazine |

Citations: | 8 - 0 self |

### Citations

1101 | Semidefinite programming
- Vandenberghe, Boyd
- 1994
(Show Context)
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... |

858 | A new polynomial-time algorithm for linear programming, Combinatorica 4
- Karmarkar
- 1984
(Show Context)
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... |

604 | Primal-Dual Interior-Point Methods.
- Wright
- 1997
(Show Context)
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... |

543 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(Show Context)
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... |

437 |
Nonlinear Programming, Sequential Unconstrained Minimization Techniques
- Fiacco, McCormick
- 1968
(Show Context)
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... |

311 |
Interior Point Polynomial Methods in Convex Programming.
- Nesterov, Nemirovskii
- 1994
(Show Context)
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 ... |

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

223 | Determinant maximization with linear matrix inequality constraints
- Vandenberghe, Boyd, et al.
- 1998
(Show Context)
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... |

205 | A C library for semidefinite programming
- CSDP
- 1999
(Show Context)
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... |

203 | Primal-dual interior-point methods for self-scaled cones
- E, Todd
- 1998
(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... |

181 |
Lower bounds for the partitioning of graphs.
- Donath, Hoffman
- 1973
(Show Context)
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... |

164 | Primal-dual path-following algorithms for semidefinite programming
- Monteiro
- 1997
(Show Context)
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 ... |

131 | On the Nesterov-Todd direction in semidefinite programming
- Todd, Toh, et al.
- 1998
(Show Context)
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... |

117 |
Iterative solution of problems of linear and quadratic programming," Soviet Mathematics Doklady 8(3
- Dikin
- 1967
(Show Context)
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... |

114 | SDPA (semidefinite programming algorithm) useraAZs manual, version 4.10.
- Fujisawa, Kojima, et al.
- 1998
(Show Context)
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... |

100 | S.: A primal-dual potential reduction method for problems involving matrix inequalities - Vandenberghe, Boyd - 1995 |

95 | Interior methods for constrained optimization
- Wright
- 1992
(Show Context)
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)... |

84 | Large-scale optimization of eigenvalues.
- Overton
- 1992
(Show Context)
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... |

78 | Implementation of interior - point methods for large scale linear programs,
- Andersen, Gondzio, et al.
- 1996
(Show Context)
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... |

75 | On minimizing the maximum eigenvalue of a symmetric matrix,
- Overton
- 1988
(Show Context)
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... |

72 | Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions, - Monteiro, Tsuchiya - 2000 |

70 |
S.: Interior point methods for the monotone linear complementarity problem in symmetric matrices
- Kojima, Shindoh, et al.
- 1997
(Show Context)
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... |

68 |
State-space solutions to standard H 2 and H1 control problems
- Doyle, Glover, et al.
- 1989
(Show Context)
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... |

64 |
Semi-definite matrix constraints in optimization,
- Fletcher
- 1985
(Show Context)
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... |

63 | Strong duality for semidefinite programming
- Ramana, Tun, et al.
- 1997
(Show Context)
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, ... |

59 | Geometric Algorithms - Grötschel, Lovász, et al. - 1988 |

56 | Symmetric primal-dual path following algorithms for semidefinite programming
- Sturm, Zhang
- 1999
(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... |

56 | M.: Local convergence of predictor-corrector infeasible interior-point algorithm for SDPs and SDLCPs - Kojima, Shida, et al. - 1998 |

53 |
A unified analysis for a class of pathfollowing primal-dual interior-point algorithms for semidefinite programming, Working paper
- Monteiro, Zhang
- 1996
(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 ... |

52 | A quadratically convergent predictor-corrector method for solving linear programs from infeasible starting points - Potra - 1994 |

52 |
Tutuncu, ”SDPT3– a MATLAB software package for semidefinite programming”,
- Toh, Todd, et al.
- 1996
(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... |

50 |
The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
- Cullum, Donath, et al.
- 1975
(Show Context)
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... |

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 |

44 | 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 ... |

38 |
sp: Software for Semidefinite Programming. User’s Guide, Beta Version
- 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... |

36 |
Robust synthesis via bilinear matrix inequalities, Int
- 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... |

36 |
Control system synthesis via bilinear matrix inequalities,
- Safonov, Goh, 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... |

35 | Homogeneous interior-point algorithms for semidefinite programming - Potra, Sheng - 1995 |

34 | A predictor-corrector interior-point algorithms for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction - Kojima, Shida, et al. - 1996 |

33 |
A primal-dual interior point method for linear programming
- Kojima, Mizuno, et al.
- 1989
(Show Context)
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... |

31 |
MAXDET: Software for determinant maximization problems” [Online]. Available://www.stanford.edu/∼boyd /MAXDET.html
- Wu, Vandenberghe, 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... |

31 |
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... |

28 | Self-scaled cones and interior-point methods in nonlinear programming, Working paper
- Nesterov, Todd
- 1994
(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... |

27 | A predictor-corrector method for semi-definite linear programming, Working paper - Lin, Saigal - 1995 |

22 | the projective method for solving linear matrix inequalities”, - Gahinet, Nemirovski - 1997 |

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

22 |
Optimization over Positive Semidefinite Matrices: Mathematical Background and User’s Manual
- NESTEROV, NEMIROVSKII
- 1990
(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 ... |

22 |
An interior-pointmethod for generalized linear-fractional programming,”
- Nemirovskii
- 1995
(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... |

19 |
LMITOOL: A front-end for LMI optimization, user's guide. Laboratoire de Math'ematiques Appliqu'ees
- Ghaoui, Nikoukhah, et al.
- 1995
(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 |
Biaffine Matrix Inequality properties and computational methods,
- Goh, Turan, et al.
- 1994
(Show Context)
Citation Context |

16 |
A general approach to polynomial-time algorithms design for convex programming
- Nesterov, Nemirovsky
- 1988
(Show Context)
Citation Context ...rmick [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 knowledge of a barrier function with a ce... |

16 | Sdpha: A MATLAB implementation of homogeneous interiorpoint algorithms for semidefinite programming
- Brixius, Potra
- 1998
(Show Context)
Citation Context ...ilable 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 described above. SDPT3 [74] is a Matlab implementation of the most important infeasible primal-dual path-f... |

16 |
Synthesis of Fixed-structure Controllers via Numerical Optimization,”
- Ghaoui, Balkrishnan
- 1994
(Show Context)
Citation Context ...luding 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 algorithms described in the above ref... |

15 |
A nonlinear programming problem in statistics (educational testing
- FLETCHER
- 1981
(Show Context)
Citation Context |

15 | On a matrix generalization of affine--scaling vector fields - Faybusovich - 1995 |

15 |
SDPpack user’s guide— version 0.9 Beta
- Alizadeh, Haeberly, et al.
- 1997
(Show Context)
Citation Context ...n 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 parts written in C to increase the efficiency. It also provides the useful feature of handling quadratic and line... |

13 |
Robust Control Synthesis via Bilinear Matrix Inequalities,
- Goh
- 1995
(Show Context)
Citation Context |

12 | A long--step path following algorithm for semidefinite programming problems. Working Paper - Anstreicher, Fampa - 1996 |

9 |
Optimization over the positive-definite cone: interior point methods and combinatorial applications
- Alizadeh
- 1992
(Show Context)
Citation Context ...convex constraints for which readily computable self-concordant barrier functions are known, and, therefore, interior-point methods are applicable. Independently of Nesterov and Nemirovskii, Alizadeh =-=[26]-=- and Kamath and Karmarkar [27, 28] generalized interior-point methods from linear programming to semidefinite programming. Vast progress has been made in the last two years, and today almost all inter... |

8 |
A continuous approach to compute upper bounds in quadratic maximization problems with integer constraints.
- Kamath, Karmarkar
- 1991
(Show Context)
Citation Context ...eadily computable self-concordant barrier functions are known, and, therefore, interior-point methods are applicable. Independently of Nesterov and Nemirovskii, Alizadeh [26] and Kamath and Karmarkar =-=[27, 28]-=- generalized interior-point methods from linear programming to semidefinite programming. Vast progress has been made in the last two years, and today almost all interior-point methods for linear progr... |

8 | On a general class of interior-point algorithms for semidefinite programming with polynomial complexity and superlinear convergence, - Sheng, Potra, et al. - 1996 |

8 | Polynomiality of primal-dual algorithms for semidefinite linear complementarity problems based on the Kojima-Shindoh- Hara family of directions, manuscript - Monteiro, Tsuchiya - 1996 |

8 |
Ghaoui, "Method of Centers for Minimizing Generalized Eigenvalues
- Boyd, El
- 1992
(Show Context)
Citation Context ...0 as a special case. Problem (16) 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 =-=[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... |

7 |
Increased roles of linear algebra in control education
- Skelton, Iwasaki
- 1995
(Show Context)
Citation Context ...st comprehensive list can be found in the book [3]. Since its publication, a number of papers have appeared chronicling further applications of SDP in control; we cite for instance the survey article =-=[5]-=- that appeared in this magazine, and the special issue of the International Journal of Robust and Nonlinear Control on Linear Matrix Inequalities in Control Theory and Applications, published recently... |

7 |
editors. Linear matrix inequalities in control theory and applications
- BALAKRISHNAN, FERON
- 1996
(Show Context)
Citation Context ...ne, and the special issue of the International Journal of Robust and Nonlinear Control on Linear Matrix Inequalities in Control Theory and Applications, published recently, in November-December, 1996 =-=[6]-=-. The growing popularity of LMI methods for control is also evidenced by the large number of publications in recent control conferences. Special classes of the SDP have a long history in optimization ... |

7 | Semi-definite programming: a path-following algorithm for a linear-quadratic functional - Faybusovich - 1995 |

7 | A long step primal--dual path following method for semidefinite programming - Jiang - 1996 |

7 | Optimizing eigenvalues of symmetric definite pencils
- HAEBERLY, OVERTON
- 1994
(Show Context)
Citation Context ...m (16) 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 [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 a... |

6 | An O(nL) iteration algorithm for computing bounds in quadratic optimization problems
- Kamath, Karmarkar
- 1993
(Show Context)
Citation Context ...eadily computable self-concordant barrier functions are known, and, therefore, interior-point methods are applicable. Independently of Nesterov and Nemirovskii, Alizadeh [26] and Kamath and Karmarkar =-=[27, 28]-=- generalized interior-point methods from linear programming to semidefinite programming. Vast progress has been made in the last two years, and today almost all interior-point methods for linear progr... |

6 | Long-step method of analytic centers for fractional problems
- Nemirovski
- 1994
(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... |

5 |
Algorithms and software tools for LMI problems in control. Session overview,” presented at the
- Vandenberghe, Balakrishnan
- 1996
(Show Context)
Citation Context .... In other words, SDPs are convex optimization problems with a linear objective function and linear matrix inequality (LMI) constraints. This paper is an updated version of the conference publication =-=[1]-=-, which was intended as an introduction to the special session Algorithms and Software Tools for LMI Problems in Control at the 1996 IEEE CACSD symposium in Dearborn. y Dr. Vandenberghe is with the De... |

4 | Connections between duality in control theory and convex optimization
- Balakrishnan, Vandenberghe
- 1995
(Show Context)
Citation Context ...onger 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, and the solutions are characterized by the optimality conditions Xs0; Zs0 TrA i X +... |

4 | A note on the Nesterov-Todd and the Kojima-Shindo-Hara search directions in semidefinite programming - Kojima, Shida, et al. - 1996 |

4 | Polynomial primal-dual cone affine scaling for semidefinite programming - Berkelaar, Sturm, et al. - 1999 |

3 |
La Methode des Centres dans un Espace Topologique.
- HUARD, LIEU
- 1966
(Show Context)
Citation Context ...ckness. 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] showed that those interior... |

1 |
Eigenvalue optimization", Acta Numerica
- Lewis, Overton
- 1996
(Show Context)
Citation Context ...e 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 formulations, we will refer to (1) as an SDP in ineq... |

1 | Geromel, "Interior point methods and LMI: Improvements and benchmark - Oliveira, C - 1996 |

1 | On primal-dual path-following algorithms fo semidefinite programming - Klerk, Roos, et al. - 1996 |

1 | An interior point method, based on rank-one updates, for linear programming - Sturm, Zhang - 1996 |

1 |
The Induced Norm Control Toolbox. User's manual
- Beran
- 1995
(Show Context)
Citation Context ...erface 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 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 alg... |