## Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization (2000)

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@TECHREPORT{Parrilo00structuredsemidefinite,

author = {Pablo A. Parrilo},

title = {Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization},

institution = {},

year = {2000}

}

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### Citations

11209 |
Computers and Intractability: A Guide to the Theory of NPcompleteness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...t will solve every possible instance in the problem class, what can be said about its computational complexity? The answer to this question turns out to be delicate, and the theory of NP-completeness =-=[36] is the -=-best attempt so far to answer these issues. The foundations of the NP-completeness theory lie in the definition of “solving” a yes/no decision problem as a Turing machine “recognizing” a certa... |

4824 |
Matrix Analysis
- Horn, Johnson
- 1986
(Show Context)
Citation Context ...ent-wise product of two matrices A =[aij] andB=[bij] of the same dimensions is defined as A ◦ B ≡ [aijbij]. An important property of this product is the following: Theorem 2.1 (Schur product theor=-=em, [44]) If A and B are-=- positive semidefinite matrices, then A ◦ B is also positive semidefinite. Moreover, if both A and B are positive definite, so is A ◦ B. AsetS⊆R n isasaidtobeacone if λ ≥ 0,x∈S ⇒λx ∈ S... |

1389 |
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
- Guckenheimer, Holmes
- 1983
(Show Context)
Citation Context ...oped in previous chapters can be applied to rigorously prove bounds on the distance to the bifurcation surface. The conditions for a vector field f(x,µ) to have a saddle-node bifurcation at (x0,µ0)a=-=re[38]: f=0 w ∗-=- Dxf=0 w ∗ Dµf�=0 w ∗ D 2 xf(v,v) �=0s94 where v,w are the right and left eigenvectors, respectively, of the jacobian J := Dxf, corresponding to the simple eigenvalue zero. The two conditions... |

1313 |
Combinatorial Optimization: Algorithms and Complexity
- Papadimitriou, Steiglitz
- 1982
(Show Context)
Citation Context ... for copositive matrices is hard, in general. It has been shown that checking if a given matrix is not copositive is an NP-complete problem [65]. Equivalently, checking copositivity is in co-NPC (see =-=[36, 70]-=- for background material on computational complexity). This implies that, unless co-NP=NP (a consequence of P=NP), in general it is not possible to construct polynomial time certificates of copositivi... |

1030 |
Nonlinear Systems
- Khalil
- 2002
(Show Context)
Citation Context ...condition can be checked efficiently, using for instance interior point methods. For nonlinear systems, in the general case there are no systematic methodologies for the search for Lyapunov functions =-=[51]-=-. Nevertheless, in the presence of additional structure, such as the case of mechanical systems, sometimes it is possible to find natural energy-based Lyapunov functions. Alternative approaches use an... |

966 | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
- Goemans, Williamson
- 1995
(Show Context)
Citation Context ...ts consistent with this probability density (given by the matrix Y ), and rounding them to the values ±1. In this specific case, good bounds can be obtained on the expected value of the resulting cut=-= [37]-=-. In principle, in certain instances we can do so in our case too. However, there are some important differences. In the quadratic case, any positive semidefinite matrix is a valid candidate for a set... |

587 |
Approximation Algorithms for NP-hard Problems
- Hochbaum
- 1996
(Show Context)
Citation Context ... time verifiable certificates of infeasibility (i.e., when the answer of the decision problem is “no”). Furthermore, the important practical issue of approximability is just beginning to be addres=-=sed [42]-=-. In this respect, we should emphasize that apparently similar NP-complete problems (for example, MAX CUT and MAX CLIQUE), can have completely different approximability properties. We mentioned earlie... |

571 |
Nonlinear control systems
- Isidori
- 1995
(Show Context)
Citation Context ...quations that achieves the computed value of γ 2 , corresponding to P ≈ 0.7025, Q ≈ 0.8766. 7.5 Zero dynamics stability When studying the global feedback linearization procedure for nonlinear sys=-=tems [46]-=-, a problem that appears is that of the zero dynamics stability. This question, that extends the linear concepts of minimum phase, deals with the stability of the system, when the outputs is constrain... |

553 | The Linear Complementarity Problem
- Cottle, Pang, et al.
- 1992
(Show Context)
Citation Context ...d mathematics, especially in optimization. It is a critical ingredient in the characterization of local solutions of constrained optimization problems [65], such as the linear complementarity problem =-=[25]-=-. Also, it has been recently shown that its use can notably improve certain convex relaxation bounds in quadratic programming problems withs60 linear constraints [75]. As we have seen in the past chap... |

503 |
Nonlinear and Adaptive Control Design
- Krstic, Kanellakopoulos, et al.
- 1995
(Show Context)
Citation Context ...rocedures to be no easier than the corresponding analysis questions. However, the presence of additional properties, such as a triangular structure of the vector field in simple cases of backstepping =-=[56]-=-, usually helps in the complexity reduction. The extent to which the presented results can be applied in synthesis procedures still remains to be fully determined. 7.7 Conclusions The sum of squares d... |

271 | Cones of matrices and set-functions and 0–1 optimization
- Lovasz, Schrijver
- 1991
(Show Context)
Citation Context ...se operations are carried over by the optimization procedure. It would be interesting to expand the connections with related ideas that have been explored in the context of “lift-and-project” meth=-=ods [59, 58, 82]-=- for deriving valid inequalities in zero-one combinatorial optimization problems. In those papers, the authors develop tractable approximations to the convex hull of zero-one points in a given convex ... |

254 |
Interior-point polynomial methods in convex programming
- Nesterov, Nemirovsky
- 1994
(Show Context)
Citation Context ...correspond to the particular case of the convex set being the intersection of an affine family of matrices and the positive semidefinite cone. As shown in the seminal work of Nesterov and Nemirovskii =-=[67]-=-, where a general theory of interior-point polynomial time methods for convex programming is developed, semidefinite programs can be efficiently solved both theoretically and practically. The critical... |

251 |
Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. New York-Heidelberg-Berlin
- Cox, Little, et al.
- 1992
(Show Context)
Citation Context ...und on the degree, and a parameterization of the unknown polynomials gi, then a solution can be obtained by solving a system of linear equations. The other procedure is based on Gröbner basis methods=-= [26, 64]. B-=-y Hilbert’s Basis theorem, every polynomial ideal is finitely generated. Gröbner bases provide a computationally convenient representation for a set of generating polynomials of an ideal. For examp... |

180 |
The Classical Moment Problem
- Akhiezer
- 1965
(Show Context)
Citation Context ...riate normal distribution with that preassigned covariance. However, for higher order moments, not every set of numbers obtained from the relaxation necessarily correspond to the moments of a measure =-=[1, 8]-=-. The root of this problem, it turns out, is again the distinction between the conditions of nonnegativity of a polynomial and being a sum of squares.s79 A notable exception is the one dimensional cas... |

166 | Computation of piecewise quadratic Lyapunov functions for hybrid systems
- Johansson, Rantzer
- 1998
(Show Context)
Citation Context ...s provably hard. For this reason, having semidefinite programming conditions that guarantee copositivity would allow for enhanced bounds for this type of problems. The other application, presented in =-=[48, 49]-=-, deals with the analysis of piecewise linear systems using piecewise quadratic Lyapunov functions. One of the basic issues in that problem is checking nonnegativity of the Lyapunov function, in a reg... |

154 | Introductory Real Analysis - Kolmogorov - 1975 |

124 | The complex structured singular value
- Packard, Doyle
- 1993
(Show Context)
Citation Context ...ee the well-posedness of the feedback interconnection of a constant matrix M and a diagonal uncertainty block ∆ = diag{δ1,δ2,...,δn}, δi ∈ C, that satisfies �n i=1 |δi| 2 ≤ 1. As in the s=-=tandard case [69],-=- necessary and sufficient conditions are computationally hard, and therefore approximation methods should be used instead. Sufficient conditions (given by µ upper bounds) are usually computed using L... |

121 | System Analysis via Integral Quadratic Constraints
- Megretsky, Rantzer
- 1997
(Show Context)
Citation Context ...of classical linear control results, such as the bounded real and positive real lemma. It is also a fundamental tool in the practical application of the IQC (integral quadratic constraints) framework =-=[61] -=-to the analysis of uncertain systems. The theorem replaces an infinite family of LMIs, parameterized by ω, by a finite dimensional problem. This is extremely useful from a practical viewpoint, since ... |

97 |
Squared functional systems and optimization problems,” in High Performance Optimization
- Nesterov
- 2000
(Show Context)
Citation Context ...st the problem as an LMI, or is it possible to solve the problem directly in the original space. After all, the set of sum of squares polynomials is a “nice” closed convex cone. In this direction,=-= in [66]-=- it has been shown that the natural self-concordant barrier for the cone of positive definite univariate polynomials is essentially optimal. In the general Positivstellensatz approach, another importa... |

96 |
Algorithmic Algebra
- Mishra
- 1993
(Show Context)
Citation Context ...resented in Chapter 7, are Lyapunov function computation, output feedback stabilization, multidimensional system stability, etc.s39 As mentioned in Chapter 1, the Tarski-Seidenberg decision procedure =-=[12, 64, 13]-=- provides in this case an explicit algorithm for deciding if (4.1) holds, so we know that the problem is decidable. There are also a few alternative approaches, also based in decision algebra; see [13... |

85 |
Linear operators leaving invariant a cone in a Banach space
- Krein, Rutman
- 1962
(Show Context)
Citation Context ...ings, the existence of a componentwise nonnegative eigenvector. The Perron-Frobenius theory has been extended considerably, with some generalizations to general Banach spaces (due to Krein and Rutman =-=[55]). -=-We are interested here in a particular finite dimensional version.s13 Theorem 2.2 ([9]) Assume that the linear operator L : R n → R n maps the proper cone K into itself. Then 1. ρ(L) is an eigenval... |

83 |
Some NP-complete problems in quadratic and nonlinear programming
- Murty, Kabadi
- 1987
(Show Context)
Citation Context ...numerous applications in diverse fields of applied mathematics, especially in optimization. It is a critical ingredient in the characterization of local solutions of constrained optimization problems =-=[65]-=-, such as the linear complementarity problem [25]. Also, it has been recently shown that its use can notably improve certain convex relaxation bounds in quadratic programming problems withs60 linear c... |

81 | A linear matrix inequality approach to H∞ control
- Gahinet, Apkarian
- 1994
(Show Context)
Citation Context ...ities, appearing in H∞ control. For these problems, under appropriate regularity hypotheses, the feasibility of the Riccati matrix inequality implies the solvability of the algebraic Riccati equatio=-=n [34]-=-. In this case, it is not necessary to solve LMIs, but instead just solve Riccati equations, at a lower computational cost. Similarly, the results in this chapter show that for a certain class of LMIs... |

77 |
A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L∞-norm
- Boyd, Balakrishnan
- 1990
(Show Context)
Citation Context ... then updated by a mechanism reminiscent of those used in H∞ norm computation. Previous related work includes of course the literature on the computation of H∞ system norms. In particular, referen=-=ces [16, 20, 15]-=- developed quadratically convergent algorithms, based explicitly on the Hamiltonian approach. Also, a somewhat related approach in [60] implements a cutting-plane based algorithm, where linear constra... |

76 |
Sharp effective Nullstellensatz
- Kollár
- 1988
(Show Context)
Citation Context ...aches to effectively find polynomials gi. The first one depends on having explicit bounds on the degree of the products figi. A number of such bounds are available in the literature, see for instance =-=[19, 53, 7]. Fo-=-r example, if the polynomials fi(x) have maximum degree d, andx∈C n ,then the bound degfigi ≤ max(3,d) n holds. The bound is tight, in the sense that there exist specific examples of systems for w... |

69 |
Piecewise linear control systems
- Johansson
- 2003
(Show Context)
Citation Context ...many important results in robustness analysis. A recent example of an application of copositive matrices in a control setting is in the stability analysis using piecewise quadratic Lyapunov functions =-=[48]-=-. From a computational complexity viewpoint, the recognition problem for copositive matrices is hard, in general. It has been shown that checking if a given matrix is not copositive is an NP-complete ... |

63 |
Bounds for the degrees in the Nullstellensatz
- Brownawell
- 1987
(Show Context)
Citation Context ...aches to effectively find polynomials gi. The first one depends on having explicit bounds on the degree of the products figi. A number of such bounds are available in the literature, see for instance =-=[19, 53, 7]. Fo-=-r example, if the polynomials fi(x) have maximum degree d, andx∈C n ,then the bound degfigi ≤ max(3,d) n holds. The bound is tight, in the sense that there exist specific examples of systems for w... |

59 |
A bisection method for computing the H∞ norm of a transfer matrix and related problems
- Boyd, Balakrishnan, et al.
- 1989
(Show Context)
Citation Context ... then updated by a mechanism reminiscent of those used in H∞ norm computation. Previous related work includes of course the literature on the computation of H∞ system norms. In particular, referen=-=ces [16, 20, 15]-=- developed quadratically convergent algorithms, based explicitly on the Hamiltonian approach. Also, a somewhat related approach in [60] implements a cutting-plane based algorithm, where linear constra... |

57 |
A linear matrix inequality approach to H1 control
- Gahinet, Apkarian
- 1994
(Show Context)
Citation Context ...ities, appearing in H1 control. For these problems, under appropriate regularity hypotheses, the feasibility of the Riccati matrix inequality implies the solvability of the algebraic Riccati equation =-=[34]-=-. In this case, it is not necessary to solve LMIs, but instead just solve Riccati equations, at a lower computational cost. Similarly, the results in this chapter show that for a certain class of LMIs... |

47 |
Output feedback stabilization and related problems - solutions via decision methods
- Anderson, Bose, et al.
- 1975
(Show Context)
Citation Context ...ski’s results on the existence of a decision procedure for elementary algebra over the reals, settles the decidability question for this quite large class of problems. This theory has been applied i=-=n [3]-=-, for example, to show the decidability of the static output feedback problem. Since many propositions in systems theory can be formulated on a first order logic (where quantifiers only affect variabl... |

47 |
Sums of squares of real polynomials
- Choi, Lam, et al.
- 1995
(Show Context)
Citation Context ... form as a sum of squares of rational functions. For notational simplicity, we will use the notation psd for “positive semidefinite” and sos for “sum of squares.” Following the notation in ref=-=erences [24, 80], l-=-et Pn,m be the set of psd forms of degree m in n variables, and Σn,m the set of forms p such that p = � k h2 k , where hk are forms of degree m/2. Hilbert himself noted that not every psd polynomia... |

45 |
G.P.: On the rank minimization problem over a positive semidefinite linear matrix inequality
- Mesbahi, Papavassilopoulos
- 1997
(Show Context)
Citation Context ...problem, with dimensions equal to those of M. Note that the matrix M T ◦M ∗ is simply the matrix whose elements are the square of the absolute value of the elements of M. Rank minimization problem=-= In [63, 62]-=-, Mesbahi and Papavassilopoulos show that for certain special cases, the rank minimization problem (which is computationally hard in general) can be reduced to a semidefinite program (an LMI). The str... |

42 |
Algebraic Geometry
- Real
- 1998
(Show Context)
Citation Context ...resented in Chapter 7, are Lyapunov function computation, output feedback stabilization, multidimensional system stability, etc.s39 As mentioned in Chapter 1, the Tarski-Seidenberg decision procedure =-=[12, 64, 13]-=- provides in this case an explicit algorithm for deciding if (4.1) holds, so we know that the problem is decidable. There are also a few alternative approaches, also based in decision algebra; see [13... |

42 | sdpsol: A Parser/Solver for Semidefinite Programs With Matrix Structure
- Boyd, Wu
- 1996
(Show Context)
Citation Context ... added to the problem at each iteration. Note that this can also be interpreted as having a dual feasible starting point, which is useful in case we are using a primal-dual LMI solver (such as SDPSOL =-=[18]-=-). For the frequency domain inequalities arising from IQC optimization, the dual problem has been extensively analyzed in [50]. It has been shown there that upper bounds, or even the optimal value, of... |

36 |
A new bound for Pólya’s Theorem with applications to polynomials positive on polyhedra
- Powers, Reznick
- 2001
(Show Context)
Citation Context ... lower bounds for r usually involve a “condition number” for the form P:s67 the minimum r grows as the form tends to degeneracy (nontrivial solutions). Some of these effective bounds are presented=-= in [28, 27, 73]-=-. However, these bounds can also be conservative: even if P has nontrivial zeros, it might be possible to prove copositivity with a small value of r, as the examples we present shows. Some interesting... |

34 | Trends in multidimensional systems theory - BOSE - 1985 |

34 | Nonlinear control system design by quantifier elimination
- Jirstrand
- 1997
(Show Context)
Citation Context ...very possible instance will have unacceptable behavior for a problem with a large number of variables. This is the main drawback of theoretically powerful methodologies such as quantifier elimination =-=[31, 47]-=-. If we want to avoid the inherent complexity problems related with the exact solution, the question arises: are there any conditions, that can be tested in polynomial time, to guarantee global positi... |

33 | An algorithm for sums of squares of real polynomials
- Powers, Wörmann
- 1998
(Show Context)
Citation Context ... the underlying machinery in Shor’s global bound for polynomial functions [91], as is explicitly mentioned in [83]. It has also been presented as the “Gram matrix” method in [24] and more recent=-=ly in [74]-=-, although no mention to interior point methods is made: the resulting LMIs are solved via decision methods. A related scheme also appears in [41] (note also the important correction in [33]).s41 The ... |

31 |
A fast algorithm to compute the H∞-norm of a transfer function matrix
- Bruinsma, Steinbuch
- 1990
(Show Context)
Citation Context ... then updated by a mechanism reminiscent of those used in H∞ norm computation. Previous related work includes of course the literature on the computation of H∞ system norms. In particular, referen=-=ces [16, 20, 15]-=- developed quadratically convergent algorithms, based explicitly on the Hamiltonian approach. Also, a somewhat related approach in [60] implements a cutting-plane based algorithm, where linear constra... |

30 |
A Bisection Method for Computing the H1 Norm of a Transfer Matrix and Related Problems
- Boyd, Balakrishnan, et al.
- 1989
(Show Context)
Citation Context ... then updated by a mechanism reminiscent of those used in H1 norm computation. Previous related work includes of course the literature on the computation of H1 system norms. In particular, references =-=[16, 20, 15]-=- developed quadratically convergent algorithms, based explicitly on the Hamiltonian approach. Also, a somewhat related approach in [60] implements a cutting-plane based algorithm, where linear constra... |

29 |
On nonnegative forms in real variables some or all of which are nonnegative
- Diananda
- 1962
(Show Context)
Citation Context ...a positive semidefinite and an elementwise nonnegative matrix, i.e., M = P + N, P ≥ 0, nij ≥ 0. (5.3) As mentioned earlier, this is a well-known sufficient condition for copositivity (see for exam=-=ple [29]-=-). The equivalence between these two tests has also been noticed in [23, Lemma 3.5]. Note that condition (5.3) can be obtained by considering the enhanced Shor relaxation, where new quadratic constrai... |

29 | Robust multi-objective feedback design by quantifier elimination
- Dorato, Yang, et al.
- 1997
(Show Context)
Citation Context ...very possible instance will have unacceptable behavior for a problem with a large number of variables. This is the main drawback of theoretically powerful methodologies such as quantifier elimination =-=[31, 47]-=-. If we want to avoid the inherent complexity problems related with the exact solution, the question arises: are there any conditions, that can be tested in polynomial time, to guarantee global positi... |

28 |
The multidimensional moment problem and semi-groups
- Berg
- 1980
(Show Context)
Citation Context ...riate normal distribution with that preassigned covariance. However, for higher order moments, not every set of numbers obtained from the relaxation necessarily correspond to the moments of a measure =-=[1, 8]-=-. The root of this problem, it turns out, is again the distinction between the conditions of nonnegativity of a polynomial and being a sum of squares.s79 A notable exception is the one dimensional cas... |

27 |
Computation of Closest Bifurcations in Power Systems
- Dobson
- 1994
(Show Context)
Citation Context ...o a singularity, not just feasible solutions. In other words, if we find a bifurcation “nearby,” then we need to be absolutely sure that there are no other points that are even closer. The results=-= in [30, 2]-=- do not fully address this issue: a Monte Carlo approach is employed, where the optimization is restarted from multiple initial conditions. The techniques developed in previous chapters can be applied... |

20 |
Linear transformations with invariant cones
- Birkhoff
- 1967
(Show Context)
Citation Context ...There are several proofs of this theorem in the literature. Some use Brouwer’s fixed point theorem (as in the infinite dimensional case), or properties of the Jordan canonical form (Birkhoff’s pro=-=of, [10]-=-). In order to present the main theorem, we will have to introduce certain technical concepts, to deal with the subtleties of strict vs. nonstrict order inequalities. In particular, the concept of irr... |

20 |
On pre-conditioning of matrices
- Osborne
- 1960
(Show Context)
Citation Context ... small, the optimal value of the LMI (2.13) is 1 + O(ε), but the fast upper bound is approximately √ n. Another available procedure for computing fast solutions of the µ LMI is the one due to Osbo=-=rne [68]-=-. A preliminary comparison made with random, normallys24 distributed matrices gives a slight advantage to the Osborne procedure. However, the algorithm proposed can give better upper bounds (the oppos... |

18 |
Computing a closest bifurcation instability in multidimensional parameter space
- Dobson
- 1992
(Show Context)
Citation Context ...ay from the hypersurface where bifurcations occur. Despite its practical importance, there does not seem to be many systematic approaches to the problem of computing bifurcation margins. In reference =-=[30]-=-, Dobson proposed two methods for computing locally closest bifurcations to a given set of nominal parameters. These methods (iterative and direct) aim to numerically solve the equations characterizin... |

17 | Copositive relaxations for general quadratic programming
- Quist, Klerk, et al.
(Show Context)
Citation Context ...he linear complementarity problem [25]. Also, it has been recently shown that its use can notably improve certain convex relaxation bounds in quadratic programming problems withs60 linear constraints =-=[75]-=-. As we have seen in the past chapters, these convex relaxations are the underlying basis of many important results in robustness analysis. A recent example of an application of copositive matrices in... |

16 | A strengthened sdp relaxation via a second lifting for the max-cut problem
- ANJOS, WOLKOWICZ
- 1999
(Show Context)
Citation Context ...be 12. The solution of the standard semidefinite relaxation for this problem is equal to 12.5. When applying the new relaxation to this problem, we are able to obtain the exact value 12. In the paper =-=[4]-=-, a different strengthened SDP relaxation for MAX CUT is presented. Even though the results in that paper provide improved bounds over the standard relaxation, in neither the case of the 5-cycle nor t... |

16 |
A Fast Algorithm to Compute the H1 Norm of a Transfer Function
- 23Bruinsma, Steinbuch
- 1990
(Show Context)
Citation Context ... then updated by a mechanism reminiscent of those used in H1 norm computation. Previous related work includes of course the literature on the computation of H1 system norms. In particular, references =-=[16, 20, 15]-=- developed quadratically convergent algorithms, based explicitly on the Hamiltonian approach. Also, a somewhat related approach in [60] implements a cutting-plane based algorithm, where linear constra... |