## On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty (2002)

Venue: | SIAM Journal on Optimization |

Citations: | 38 - 11 self |

### BibTeX

@ARTICLE{Ben-tal02ontractable,

author = {Aharon Ben-tal and Arkadi Nemirovski},

title = {On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty},

journal = {SIAM Journal on Optimization},

year = {2002},

volume = {12},

pages = {811--833}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We present efficiently verifiable sufficient conditions for the validity of specific NPhard semi-infinite systems of Linear Matrix Inequalities (LMI’s) arising from LMI’s with uncertain data and demonstrate that these conditions are “tight ” up to an absolute constant factor. In particular, we prove that given an n × n interval matrix Uρ = {A | |Aij − A ∗ ij | ≤ ρCij}, one can build a computable lower bound, accurate within the factor π, on the supremum of those ρ for which 2 all instances of Uρ share a common quadratic Lyapunov function. We then obtain a similar result for the problem of Quadratic Lyapunov Stability Synthesis. Finally, we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube. Key words. Robust semidefinite optimization, data uncertainty, Lyapunov stability synthesis, relaxations of combinatorial problems AMS subject classifications. 90C05, 90C25, 90C30

### Citations

971 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...the factor π π 2 is known; it is the “ 2 Theorem” of Nesterov [12] established originally via a construction based on the famous MAXCUT-related “random hyperplane” technique of Goemans and Williamson =-=[7]-=-. Surprisingly, the alternative proof we have developed, although exploits randomization, seemingly uses nothing like the random hyperplane technique. 5. Maximizing homogeneous polynomial of degree 3 ... |

291 | Robust convex optimization
- Ben-Tal, Nemirovski
- 1998
(Show Context)
Citation Context ...ρDij ∀i, j} (A ∗ is the “nominal” matrix, D = [Dij ≥ 0]i,j is a “perturbation scale”, and ρ > 0 is a “perturbation level”). The Lyapunov Analysis example, same as other examples which can be found in =-=[1, 2, 4, 6]-=-, demonstrate the importance of robust solutions to semidefinite problems affected by data uncertainty, in particular, an interval one. Theoretically speaking, the major difficulty with this concept i... |

123 |
Quality of semidefinite relaxation for nonconvex quadratic optimization
- Nesterov
- 1997
(Show Context)
Citation Context ... the Matrix Cube problem and the problem of maximizing a positive definite quadratic form over the unit cube; in particular, we demonstrate that (N) allows to re-derive the “ π 2 Theorem” of Nesterov =-=[12]-=- stating that the standard semidefinite bound on the maximum of a positive definite quadratic form over the unit cube is tight within the factor π 2 . In concluding Section 5, we apply our techniques ... |

89 |
H∞ Design with pole placement constraints: an LMI approach
- Chilali, Gahinet
- 1996
(Show Context)
Citation Context ...s θ ; (z − ¯z) cos θ (z + ¯z) sin θsLINEAR MATRIX INEQUALITIES WITH INTERVAL UNCERTAINTY 15 � � 2h1 − (z + ¯z) 0 4. The stripe {z | h1 < ℜ(z) < h2}: fH(z) = . 0 (z + ¯z) − 2h2 It is known (see, e.g., =-=[5]-=-) that the spectrum Σ(A) of a real n × n matrix A belongs to H if and only if there exists X ∈ S m , X ≻ 0, such that the k × k block matrix M[X, A] with the m × m blocks Mij[X, A] = PijX + QijAX + Qj... |

84 | Robust solutions to uncertain semidefinite programs
- Ghaoui, Oustry, et al.
- 1998
(Show Context)
Citation Context ...ρDij ∀i, j}. (A ∗ is the “nominal” matrix, D =[Dij ≥ 0]i,j is a “perturbation scale,” and ρ>0is a “perturbation level.”) The Lyapunov analysis example, as well as other examples which can be found in =-=[1, 2, 4, 6]-=-, demonstrates the importance of robust solutions to semidefinite problems affected by data uncertainty, in particular by interval uncertainty. Theoretically speaking, the major difficulty with this c... |

62 | Robust solutions to uncertain semidefinite programs
- Gahoui, Oustry, et al.
- 1998
(Show Context)
Citation Context ...ρDij ∀i, j} (A ∗ is the “nominal” matrix, D = [Dij ≥ 0]i,j is a “perturbation scale”, and ρ > 0 is a “perturbation level”). The Lyapunov Analysis example, same as other examples which can be found in =-=[1, 2, 4, 6]-=-, demonstrate the importance of robust solutions to semidefinite problems affected by data uncertainty, in particular, an interval one. Theoretically speaking, the major difficulty with this concept i... |

56 | Robust Truss Topology Design via Semidefinite Programming”, Research Report # 4/95
- Ben-Tal, Nemirovski
- 1995
(Show Context)
Citation Context ...ρDij ∀i, j}. (A ∗ is the “nominal” matrix, D =[Dij ≥ 0]i,j is a “perturbation scale,” and ρ>0is a “perturbation level.”) The Lyapunov analysis example, as well as other examples which can be found in =-=[1, 2, 4, 6]-=-, demonstrates the importance of robust solutions to semidefinite problems affected by data uncertainty, in particular by interval uncertainty. Theoretically speaking, the major difficulty with this c... |

38 | A.: Robust semidefinite programming
- Ben-Tal, El-Ghaoui, et al.
- 2000
(Show Context)
Citation Context ...ρDij ∀i, j} (A ∗ is the “nominal” matrix, D = [Dij ≥ 0]i,j is a “perturbation scale”, and ρ > 0 is a “perturbation level”). The Lyapunov Analysis example, same as other examples which can be found in =-=[1, 2, 4, 6]-=-, demonstrate the importance of robust solutions to semidefinite problems affected by data uncertainty, in particular, an interval one. Theoretically speaking, the major difficulty with this concept i... |

21 |
A.: Stable Truss Topology Design via Semidefinite Programming
- Ben-Tal, Nemirovski
- 1997
(Show Context)
Citation Context |

11 |
Geometric methods in combinatorial optimization
- Grotschel, Lovász, et al.
- 1984
(Show Context)
Citation Context ... a semi-infinite system of LMI’s and as such it can be computationally intractable. However, the set X of solutions to (2) clearly is closed and convex; it follows that, essentially (for details, see =-=[8]-=-), “computational tractability” of (2) (i.e., ability to find efficiently a point in X or to maximize efficiently a linear function over X ) is equivalent to a possibility to solve efficiently the ass... |

11 |
Several NP-hard problems arising in robust stability analysis
- Nemirovski
- 1993
(Show Context)
Citation Context ...IES WITH INTERVAL UNCERTAINTY 3 Unfortunately, the Matrix Cube problem in general is NP-hard. This is so even in the case when all “edges” B 1 , ..., B L of the “matrix box” C[ρ] are of rank ≤ 2 (see =-=[9]-=- or Section 4 below). Consequently, one is enforced to look for verifiable sufficient conditions for the inclusion C[ρ] ⊂ S m + . The simplest condition of this type is evident: (S) Assume that there ... |

10 |
Interior Point Algorithms
- Nesterov, Nemirovskii
- 1993
(Show Context)
Citation Context ...homogeneous polynomial of degree 3 over the unit cube. In what follows, we frequently use the semidefinite duality; for reader’s convenience, we list here the relevant results (for proofs, see, e.g., =-=[11]-=-). Consider a semidefinite problem ⎧ ⎨ min x ⎩ cT ⎫ n� ⎬ x : xjAj − A0 � 0 ; (Pr) ⎭ j=1 here x ∈ R n , A0, ..., An ∈ S m . It is assumed that no nontrivial linear combination of the matrices A1, ..., ... |

10 |
A.: The Ellipsoid Method and Combinatorial Optimization
- Grötschel, Lovasz, et al.
- 1988
(Show Context)
Citation Context ... a semi-infinite system of LMIs, and as such it can be computationally intractable. However, the set X of solutions to (2) is clearly closed and convex; it follows that, essentially (for details, see =-=[8]-=-), the “computational tractability” of (2) (i.e., the ability to find efficiently a point in X or to maximize efficiently a linear function over X ) is equivalent to the possibility of solving efficie... |

6 | Long-step method of analytic centers for fractional problems
- Nemirovski
- 1994
(Show Context)
Citation Context ... ≤ Xij , −Bij[X] ≤ Xij , ∀i, j; Biℓ [Z] ≤ Y iℓ , −Biℓ [Z] ≤ Y iℓ , ∀i, ℓ; � � ρ Xij + � Y iℓ � � B[X, Z]. i,j i,ℓ ⎫ ⎪⎬ ⎪⎭ (ALS) Note that both (ALA) and (ALS) are Generalized Eigenvalue problems (see =-=[3, 10]-=-) and as such are “computationally tractable”. Taking into account that the ranks of the matrices Aij[X], A iℓ [X], Bij[X], B iℓ [Z] never exceed 2 and applying Corollary 2.4, we come to the result as... |

1 |
El Ghaoui,and A. Nemirovski, Robust semidefinite programming
- Ben-Tal, L
- 2000
(Show Context)
Citation Context ...ρDij ∀i, j}. (A ∗ is the “nominal” matrix, D =[Dij ≥ 0]i,j is a “perturbation scale,” and ρ>0is a “perturbation level.”) The Lyapunov analysis example, as well as other examples which can be found in =-=[1, 2, 4, 6]-=-, demonstrates the importance of robust solutions to semidefinite problems affected by data uncertainty, in particular by interval uncertainty. Theoretically speaking, the major difficulty with this c... |