## Semidefinite Programming Relaxations and Algebraic Optimization in Control (2003)

### Cached

### Download Links

Citations: | 12 - 4 self |

### BibTeX

@MISC{Parrilo03semidefiniteprogramming,

author = {Pablo A. Parrilo and Sanjay Lall},

title = {Semidefinite Programming Relaxations and Algebraic Optimization in Control},

year = {2003}

}

### OpenURL

### Abstract

We present an overview of the essential elements of semide nite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certi cates. Our focus is on the exciting developments occurred in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sum-of-squares. These developments are illustrated with examples of applications to control systems.

### Citations

10922 |
Computers and Intractability: A Guide to the Theory of NP-Completeness,” Freeman and Co
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...olynomial-time algorithms to make their meaning more precise, these questions play an essential role in some of the central issues in computational complexity; whether P=NP and NP=co-NP, respectively =-=[20]-=-. Note also that convexity properties by themselves do not automatically imply polynomial-time solvability. The specific representation of the feasible set plays a crucial role here, and the existing ... |

3666 |
Convex Optimization
- Boyd, Vandenberghe
- 2004
(Show Context)
Citation Context ...urvey works, covering different complementary aspects are the early work by Alizadeh [1], Goemans [22], as well as the more recent ones due to Todd [58], and Laurent and Rendl [33]. The upcoming book =-=[11]-=- presents a beautiful and 2s2003.09.15.01 clear exposition of the theory, numerical approaches and broad applications of convex programming. Other works dealing more specifically with semidefinite pro... |

937 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...ocedure; see [10] and the references therein. Several important results are available regarding the performance of these relaxations; the celebrated Goemans-Williamson approximation scheme for MAXCUT =-=[23]-=- relies exactly on this relaxation, followed by a randomized rounding step. For many other related problems, it has been possible to prove a priori approximation guarantees. Combinatorial Optimization... |

770 | Semidefinite programming
- Vandenbergh, Boyd
- 1996
(Show Context)
Citation Context ...atrix eigenvalues. We are fortunate that many excellent works dealing with the development and applications of SDP are available. In particular, we mention the wellknown work of Vandenberghe and Boyd =-=[60]-=- as a wonderful survey of the basic theory and initial applications, and the handbook [64] for a comprehensive treatment of the many aspects of the subject. Other survey works, covering different comp... |

473 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(Show Context)
Citation Context ...itial applications, and the handbook [64] for a comprehensive treatment of the many aspects of the subject. Other survey works, covering different complementary aspects are the early work by Alizadeh =-=[1]-=-, Goemans [22], as well as the more recent ones due to Todd [58], and Laurent and Rendl [33]. The upcoming book [11] presents a beautiful and 2s2003.09.15.01 clear exposition of the theory, numerical ... |

319 | Global optimization with polynomials and the problem of moments
- Lasserre
- 2001
(Show Context)
Citation Context ...n general, testing feasibility of such sets is intractable. Recent developments have led to an approach using semidefinite programming to test feasibility of semialgebraic sets [43, 44, 45] (see also =-=[32]-=- for a dual approach, and [14]). By solving a semidefinite program, one may obtain a certificate of infeasibility for an infeasible semialgebraic set. The size of this certificate (i.e., the size of t... |

273 |
Minimization Methods for Nondifferentiable Functions
- Shor
- 1985
(Show Context)
Citation Context ...ot automatically imply polynomial-time solvability. The specific representation of the feasible set plays a crucial role here, and the existing results require the availability of either subgradients =-=[55]-=-, or a self-concordant barrier function [39]. There are many examples of optimization problems over (possibly implicitly given) convex sets, where all these operations, or even checking membership, ar... |

248 |
Interior-point polynomial methods in convex programming, ser
- Nesterov, Nemirovsky
- 1994
(Show Context)
Citation Context ...so provides analytical benefits, since in this case further algebraic or geometric properties may provide additional insight (for instance, the polyhedrality of LP). The language of conic programming =-=[39]-=- provides a convenient framework to analyze these issues. By posing an abstract optimization problem of the form minimize 〈C, X〉 subject to AX = b, 5 X ∈ K, (4)s2003.09.15.01 where X is an element of ... |

225 | Semidefinite programming relaxations for semialgebraic problems
- Parrilo
(Show Context)
Citation Context ...ed not be convex, and in general, testing feasibility of such sets is intractable. Recent developments have led to an approach using semidefinite programming to test feasibility of semialgebraic sets =-=[43, 44, 45]-=- (see also [32] for a dual approach, and [14]). By solving a semidefinite program, one may obtain a certificate of infeasibility for an infeasible semialgebraic set. The size of this certificate (i.e.... |

215 | SDPT3—a MATLAB software package for semidefinitequadratic-linear programming
- TOH, TODD, et al.
(Show Context)
Citation Context ...ice, combining good performance and accuracy, primal and dual solutions, as well as reasonable speed-ups depending on the problem sparsity. We mention a few of the best-known ones: SeDuMi [56], SDPT3 =-=[59]-=-, SDPA [18], CSDP [9], DSDP [7], among others. Other good pointers to the available SDP solvers are the SDP webpages of C. Helmberg and H. Wolkowicz. 18s2003.09.15.01 Several approaches other than int... |

201 | Algebraic Geometry - Bochnak, Coste, et al. - 1998 |

183 | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization
- Parrilo
- 2000
(Show Context)
Citation Context ...ed not be convex, and in general, testing feasibility of such sets is intractable. Recent developments have led to an approach using semidefinite programming to test feasibility of semialgebraic sets =-=[43, 44, 45]-=- (see also [32] for a dual approach, and [14]). By solving a semidefinite program, one may obtain a certificate of infeasibility for an infeasible semialgebraic set. The size of this certificate (i.e.... |

146 | Robust Solutions to Least-Squares Problems with Uncertain Data
- GHAOUI, LEBRET
- 1997
(Show Context)
Citation Context ...y = T u + Sw for all u, y satisfying the dynamics (11), and so we can write this problem as min u max �T u + Sw − ydes�2 (12) w∈W This problem is therefore a robust least-squares problem; recent work =-=[17]-=- in this area has produced solutions using semidefinite programming for this and more general versions, and certain classes of this problem may be solved using second-order cone programming [34]. Let ... |

139 | a C library for semidefinite programming
- Borchers, CSDP
- 1999
(Show Context)
Citation Context ...rformance and accuracy, primal and dual solutions, as well as reasonable speed-ups depending on the problem sparsity. We mention a few of the best-known ones: SeDuMi [56], SDPT3 [59], SDPA [18], CSDP =-=[9]-=-, DSDP [7], among others. Other good pointers to the available SDP solvers are the SDP webpages of C. Helmberg and H. Wolkowicz. 18s2003.09.15.01 Several approaches other than interior-point methods h... |

120 | The complex structured singular value
- Packard, Doyle
- 1993
(Show Context)
Citation Context ...n 5. Yet another related reformulation of the KYP lemma is given by the equivalence of the structured singular value µ and its upper bound in the case where there is a full block and one scalar block =-=[40]-=-. In the case of systems with large state dimension n, the KYP approach is not very efficient, since the matrix variable representing the storage function that appears in the LMI has (n 2 + n)/2 compo... |

114 |
Applications of second-order cone programming. Linear Algebra and its
- Lobo, Vandenberghe, et al.
- 1998
(Show Context)
Citation Context ...scale instances (of the order of tens of thousands of variables), that are currently out of reach for SDP. More details on the theory and applications of second-order cone programming can be found in =-=[39, 34, 2]-=-. We are confident that the application areas where SOCP techniques are used will increase dramatically in the future, either as the natural modeling framework, or as a computational efficient device ... |

104 | Semidefinite optimization
- Todd
- 2001
(Show Context)
Citation Context ...reatment of the many aspects of the subject. Other survey works, covering different complementary aspects are the early work by Alizadeh [1], Goemans [22], as well as the more recent ones due to Todd =-=[58]-=-, and Laurent and Rendl [33]. The upcoming book [11] presents a beautiful and 2s2003.09.15.01 clear exposition of the theory, numerical approaches and broad applications of convex programming. Other w... |

103 | A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
- Burer, Monteiro
(Show Context)
Citation Context ...18s2003.09.15.01 Several approaches other than interior-point methods have also been investigated, a few of them being bundle methods [26, 37], or nonlinear approaches based on special factorizations =-=[13]-=-. This research has increased steadily in the last few years. The codes based on these new developments are the only ones achieving satisfactory performance for some of the very large and structured p... |

102 |
Least squares stationary optimal control and the algebraic Riccati equation
- Willems
- 1971
(Show Context)
Citation Context ...cepted that matrix inequalities and duality are inextricably linked with the study of linear control systems; this is witnessed by the fundamental role played by the Lyapunov and Riccati inequalities =-=[63, 4]-=-. Very significant work by many researchers has continuously expanded these boundaries; robustness analysis techniques based on quadratic stability, multipliers, structured singular values, or integra... |

98 | Gloptipoly: global optimization over polynomials with matlab and sedumi
- Henrion, J
(Show Context)
Citation Context ...rt SDPSOL [12], as well as the more recent ones YALMIP [35], SeDuMi Interface [31], and LMILab translator [54], dealing with general SDPs, as well as the more domain-specific IQCbeta [36], Gloptipoly =-=[28]-=-, and SOSTOOLS [47]. Any of these parsers can make the task of posing and solving a specific problem a much simpler and enjoyable procedure than manual, error-prone methods. We strongly encourage the ... |

96 | Semidefinite programming in combinatorial optimization
- Goemans
- 1997
(Show Context)
Citation Context ...tions, and the handbook [64] for a comprehensive treatment of the many aspects of the subject. Other survey works, covering different complementary aspects are the early work by Alizadeh [1], Goemans =-=[22]-=-, as well as the more recent ones due to Todd [58], and Laurent and Rendl [33]. The upcoming book [11] presents a beautiful and 2s2003.09.15.01 clear exposition of the theory, numerical approaches and... |

96 | Computers and Intractability: A Guide to the Theory of NPCompleteness (W.H - MR, DS - 1979 |

94 | SDPA semidefinite programming algorithm ± Version
- Fujisawa, Kojima, et al.
(Show Context)
Citation Context ...ing good performance and accuracy, primal and dual solutions, as well as reasonable speed-ups depending on the problem sparsity. We mention a few of the best-known ones: SeDuMi [56], SDPT3 [59], SDPA =-=[18]-=-, CSDP [9], DSDP [7], among others. Other good pointers to the available SDP solvers are the SDP webpages of C. Helmberg and H. Wolkowicz. 18s2003.09.15.01 Several approaches other than interior-point... |

94 | Concrete aspects of Hilbert's 17th Problem. In: Real Algebraic Geometry and Ordered Structures
- Reznick
(Show Context)
Citation Context ...(x) �2 i=1 and in this case we may choose a factorization with ⎡ ⎤ 0 2 V = ⎣1 −3⎦ , 2 1 giving f(x) = (x 2 2 + 2x1x2) 2 + (2x 2 1 − 3x2 2 2 + x1x2) which is a sum-of-squares (SOS) decomposition of f (=-=[46, 51]-=-). From this decomposition, it is immediately clear that there cannot exist a feasible x, and the decomposition (characterized by Q) is clearly a certificate of infeasibility. Hence we may decide whet... |

72 |
H∞ design with pole placement constraints: an LMI approach
- Chilali, Gahinet
- 1996
(Show Context)
Citation Context ...y P = � z | P + Rz + R T ¯z ≻ 0 � , P ∈ S k , R ∈ R k×k , where the inequality should be interpreted as forcing the Hermitian matrix on the lefthand side to be positive semidefinite. It is well known =-=[10, 15, 53]-=- that the two desired design specifications �Tzw�2 ≤ γ and σ(A + BuK) ⊂ P can be expressed as the SDP conditions trace(Z) ≤ γ 2 � AclX1 + X1A , T � � � T cl X1C X1 Bz � 0, � 0, CX1 −I Z and P ⊗ X2 + R... |

67 | SOSTOOLS: Sum of squares optimization toolbox for MATLAB
- Prajna, Papachristodoulou, et al.
- 2004
(Show Context)
Citation Context ...re required. This result is called the Positivstellensatz. Software for testing feasibility of semialgebraic sets using the above methods is available in the form of a Matlab toolbox, called SOSTOOLS =-=[47]-=-. 16s2003.09.15.01 Applications. Testing feasibility of semialgebraic sets has important applications in control and combinatorial optimization. For example, the integer program of (8) can be formulat... |

63 |
Linear matrix inequality representation of sets
- Helton, Vinnikov
- 2002
(Show Context)
Citation Context ...e by an SDP. Without using additional variables this is not always possible, as there are nontrivial obstructions that must be avoided. We refer the reader to the recent result by Helton and Vinnikov =-=[27]-=- for a complete answer in the two-dimensional case. The results in Section 7 can be used to constructively build arbitrarily close approximations. 2.3 Other Classes of Optimization Problems Semidefini... |

47 | Semidefinite Programming and Integer Programming, Centrum voor Wiskunde en Informatica
- Laurent, Rendl
- 2002
(Show Context)
Citation Context ... of the subject. Other survey works, covering different complementary aspects are the early work by Alizadeh [1], Goemans [22], as well as the more recent ones due to Todd [58], and Laurent and Rendl =-=[33]-=-. The upcoming book [11] presents a beautiful and 2s2003.09.15.01 clear exposition of the theory, numerical approaches and broad applications of convex programming. Other works dealing more specifical... |

45 | On the construction of Lyapunov functions using the sum of squares decomposition
- Papachristodoulou, Prajna
- 2002
(Show Context)
Citation Context ...trol context as a convenient sufficient condition for polynomial nonnegativity. Recent work has applied the sum-of-squares approach to the problem of finding a Lyapunov function for nonlinear systems =-=[43, 41, 57]-=-. This approach allows one to search over affinely parametrized polynomial or rational Lyapunov functions for systems with dynamics of the form ˙xi(t) = fi(x(t)) for all i = 1, . . . , n where the fun... |

45 | Minimizing polynomial functions
- Parrilo, Sturmfels
(Show Context)
Citation Context ...ed not be convex, and in general, testing feasibility of such sets is intractable. Recent developments have led to an approach using semidefinite programming to test feasibility of semialgebraic sets =-=[43, 44, 45]-=- (see also [32] for a dual approach, and [14]). By solving a semidefinite program, one may obtain a certificate of infeasibility for an infeasible semialgebraic set. The size of this certificate (i.e.... |

42 | SDPSOL: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure. User’s Guide, Version Beta
- Wu, Boyd
- 1996
(Show Context)
Citation Context ...areas have written specific parsers that partially or fully automate the conversion tasks, when used within a problem-solving environment such as MATLAB. Among them we mention the early effort SDPSOL =-=[12]-=-, as well as the more recent ones YALMIP [35], SeDuMi Interface [31], and LMILab translator [54], dealing with general SDPs, as well as the more domain-specific IQCbeta [36], Gloptipoly [28], and SOST... |

42 |
On the Kalman-Yakubovich-Popov lemma
- Rantzer
- 1996
(Show Context)
Citation Context ...7s2003.09.15.01 An important subclass for which several customized algorithms are already available is that of optimization problems with an structure induced by the Kalman-YakubovichPopov lemma (see =-=[63, 49]-=- and the references therein). This fundamental result establishes the equivalence between a frequency domain inequality and the feasibility of a particular SDP. It is an important generalization of cl... |

41 |
Algebraic Geometry
- Real
- 1998
(Show Context)
Citation Context ...allowing for unrestricted degrees of the polynomials si and tij, and arbitrary products of fi, one may always construct a refutation for any given infeasible semialgebraic set P using SOS polynomials =-=[8]-=-. No assumptions whatsoever on the polynomials fi are required. This result is called the Positivstellensatz. Software for testing feasibility of semialgebraic sets using the above methods is availabl... |

35 | Tütüncü, SDPT3 – a MATLAB software package for semidefinite programming, version 1.3, Optim - Todd, Toh, et al. - 1999 |

34 | Hyperbolic polynomials and interior point methods for convex programming
- Guler
- 1997
(Show Context)
Citation Context ...ling flexibility. Yet another interesting class, that only in the last few years has begun to be explored, is that of optimization problems involving hyperbolic polynomials and their associated cones =-=[24, 5]-=-, which are of great theoretical interest. Research in this area is still at its early stage, with some open fundamental mathematical questions, and practical applications still under investigation. 3... |

32 | An algorithm for sums of squares of real polynomials
- Powers, Wörmann
- 1998
(Show Context)
Citation Context ...(x) �2 i=1 and in this case we may choose a factorization with ⎡ ⎤ 0 2 V = ⎣1 −3⎦ , 2 1 giving f(x) = (x 2 2 + 2x1x2) 2 + (2x 2 1 − 3x2 2 2 + x1x2) which is a sum-of-squares (SOS) decomposition of f (=-=[46, 51]-=-). From this decomposition, it is immediately clear that there cannot exist a feasible x, and the decomposition (characterized by Q) is clearly a certificate of infeasibility. Hence we may decide whet... |

32 | Review of LFTs, LMIs, and - JC, Packard, et al. - 1991 |

29 | An independent benchmarking of SDP and SOCP solvers
- Mittelmann
- 2003
(Show Context)
Citation Context ...mprehensive benchmarking effort of the performance of several solvers in a representative collection of problems (including some arising from control) has been recently reported by Hans Mittelmann in =-=[38]-=-; an up-to-date version of the results is available at the web page http://plato.la.asu.edu/bench.html. The consensus and experience among researchers, backed up by the hard data mentioned, seems to b... |

24 | Minimization Methods for Nondi#erentiable Functions - Shor - 1985 |

23 | Convex optimization problems involving finite autocorrelation sequences
- Alkire, Vandenberghe
(Show Context)
Citation Context ... speed-ups in the computation of the worst-case frequency response [52]. Additionally, methods for taking advantage of autocorrelation structure have also been developed by Alkire and Vandenberghe in =-=[3]-=-. 9 Available Implementations Despite the impressive advances on the theoretical and modeling aspects of SDP, much of its impact in applications has undoubtedly been a direct consequence of the effort... |

21 | Hyperbolic polynomials and convex analysis
- Bauschke, Güler, et al.
(Show Context)
Citation Context ...ling flexibility. Yet another interesting class, that only in the last few years has begun to be explored, is that of optimization problems involving hyperbolic polynomials and their associated cones =-=[24, 5]-=-, which are of great theoretical interest. Research in this area is still at its early stage, with some open fundamental mathematical questions, and practical applications still under investigation. 3... |

21 | The complex structured singular value. Automatica - Packard, JC - 1993 |

20 | Semidefinite Programming Duality and Linear System theory: Connections and Implications for Computation
- VANDENBERGHE, BALAKRISHNAN
- 1999
(Show Context)
Citation Context ...cepted that matrix inequalities and duality are inextricably linked with the study of linear control systems; this is witnessed by the fundamental role played by the Lyapunov and Riccati inequalities =-=[63, 4]-=-. Very significant work by many researchers has continuously expanded these boundaries; robustness analysis techniques based on quadratic stability, multipliers, structured singular values, or integra... |

20 | Saigal R., Vandenberghe L., eds: Handbook of Semidefinite Programming and Applications - Wolkowicz - 2000 |

17 | Interior point methods in semide programming with applications to combinatorial optimization - Alizadeh - 1995 |

16 |
SBmethod ± A C ‡‡ implementation of the spectral bundle method. Available from http:// www-user.tu-chemnitz.de/ helmberg/SBmethod
- Helmberg
(Show Context)
Citation Context ...SDP solvers are the SDP webpages of C. Helmberg and H. Wolkowicz. 18s2003.09.15.01 Several approaches other than interior-point methods have also been investigated, a few of them being bundle methods =-=[26, 37]-=-, or nonlinear approaches based on special factorizations [13]. This research has increased steadily in the last few years. The codes based on these new developments are the only ones achieving satisf... |

13 | H1 design with pole placement constraints: an LMI approach - Chilali, Gahinet - 1996 |

12 |
Review of LFTs, LMIs, and µ
- Doyle, Packard, et al.
- 1991
(Show Context)
Citation Context ...the subsequent [21]. Excellent available references that collect results at the different stages of the development of SDPs from a control perspective are the review paper of Doyle, Packard, and Zhou =-=[16]-=- and the lecture notes by Scherer and Weiland [53]. 1.1 Notation We make use of the following notation. The nonnegative orthant in Rn is denoted by Rn + , defined by R n + = � x ∈ R n | xi ≥ 0 for all... |

9 | Lebret H. Applications of second-order cone programming. Linear Algebra and its Applications - Lobo, Vandenberghe, et al. - 1998 |

8 | Efficient solution of linear matrix inequalities for integral quadratic constraints
- Hansson, Vandenberghe
- 2000
(Show Context)
Citation Context ...all rely on outer approximation methods based on a discretization of the frequency axis, with the former using linear programming and the latter using SDP cuts. The scheme by Hansson and Vandenberghe =-=[25]-=- is based on the interior point machinery, but cleverly exploits the Lyapunov structure at each iteration at the linear algebra level. Rotea and D’Amato exploit several properties of the µ SDP upper b... |