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A Survey of Computational Complexity Results in Systems and Control
, 2000
"... The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fi ..."
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Cited by 116 (21 self)
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The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NPcompleteness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
A Global Optimization Method, αBB, for General TwiceDifferentiable Constrained NLPs: I  Theoretical Advances
, 1997
"... In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the constru ..."
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Cited by 52 (3 self)
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In this paper, the deterministic global optimization algorithm, αBB, (αbased Branch and Bound) is presented. This algorithm offers mathematical guarantees for convergence to a point arbitrarily close to the global minimum for the large class of twicedifferentiable NLPs. The key idea is the construction of a converging sequence of upper and lower bounds on the global minimum through the convex relaxation of the original problem. This relaxation is obtained by (i) replacing all nonconvex terms of special structure (i.e., bilinear, trilinear, fractional, fractional trilinear, univariate concave) with customized tight convex lower bounding functions and (ii) by utilizing some α parameters as defined by Maranas and Floudas (1994b) to generate valid convex underestimators for nonconvex terms of generic structure. In most cases, the calculation of appropriate values for the α parameters is a challenging task. A number of approaches are proposed, which rigorously generate a set of α par...
Rigorous Convex Underestimators for General TwiceDifferentiable Problems
 Journal of Global Optimization
, 1996
"... . In order to generate valid convex lower bounding problems for nonconvex twicedifferentiable optimization problems, a method that is based on second order information of general twicedifferentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues ..."
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Cited by 35 (15 self)
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. In order to generate valid convex lower bounding problems for nonconvex twicedifferentiable optimization problems, a method that is based on second order information of general twicedifferentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues of such functions are obtained and used in constructing convex underestimators. By solving several nonlinear example problems, it is shown that the lower bounds are sufficiently tight to ensure satisfactory convergence of the ffBB, a branch and bound algorithm which relies on this underestimation procedure [3]. Key words: convex underestimators; twicedifferentiable; interval anlysis; eigenvalues 1. Introduction The mathematical description of many physical phenomena, such as phase equilibrium, or of chemical processes generally requires the introduction of nonconvex functions. As the number of local solutions to a nonconvex optimization problem cannot be predicted a priori, the identifi...
Rankone LMI Approach to Simultaneous Stabilization of Linear Systems
, 1998
"... Following a polynomial approach to control design, the costabilization by a fixed controller of a family of SISO linear systems is interpreted as an LMI feasibility problem with a rankone constraint. An LMI relaxation algorithm and a potential reduction heuristic are then proposed for addressing t ..."
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Cited by 21 (17 self)
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Following a polynomial approach to control design, the costabilization by a fixed controller of a family of SISO linear systems is interpreted as an LMI feasibility problem with a rankone constraint. An LMI relaxation algorithm and a potential reduction heuristic are then proposed for addressing this key optimization problem. This work was supported by the Barrande Project No. 97/00597/026, by the Grant Agency of the Czech Republic under contract No. 102/97/0861, by the Ministry of Education of the Czech Republic under contract No. VS97/034 and by the French Ministry of Education and Research under contract No. 10INSA96. y Corresponding author. Email henrion@laas.fr. FAX 33 5 61 33 69 69. 1 Introduction We consider the problem of simultaneously stabilizing, or costabilizing, a family of singleinput singleoutput (SISO) linear systems by one fixed controller of given order. This fundamental problem, recognized as one of the difficult open issues in linear system theory, ar...
Bounds on real eigenvalues and singular values of interval matrices
 SIAM J. Matrix Anal. Appl
"... Abstract. We study bounds on real eigenvalues of interval matrices, and our aim is to develop fast computable formulae that produce assharpaspossible bounds. We consider two cases: general and symmetric interval matrices. We focus on the latter case, since on the one hand such interval matrices h ..."
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Cited by 4 (2 self)
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Abstract. We study bounds on real eigenvalues of interval matrices, and our aim is to develop fast computable formulae that produce assharpaspossible bounds. We consider two cases: general and symmetric interval matrices. We focus on the latter case, since on the one hand such interval matrices have many applications in mechanics and engineering, and on the other hand many results from classical matrix analysis could be applied to them. We also provide bounds for the singular values of (generally nonsquare) interval matrices. Finally, we illustrate and compare the various approaches by a series of examples.
Globally Optimal Affine Epipolar Geometry from Apparent Contours Gang Li
"... We study the problem of estimating the epipolar geometry from apparent contours of smooth curved surfaces with affine camera models. Since apparent contours are viewpoint dependent, the only true image correspondences are projections of the frontier points, i.e., surface points whose tangent planes ..."
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We study the problem of estimating the epipolar geometry from apparent contours of smooth curved surfaces with affine camera models. Since apparent contours are viewpoint dependent, the only true image correspondences are projections of the frontier points, i.e., surface points whose tangent planes are also their epipolar planes. However, frontier points are unknown a priori and must be estimated simultaneously with epipolar geometry. Previous approaches to this problem adopt local greedy search methods which are sensitive to initialization, and may get trapped in local minima. We propose the first algorithm that guarantees global optimality for this problem. We first reformulate the problem using a separable form that allows us to search effectively in a 2D space, instead of on a 5D hypersphere in the classical formulation. Next, in a branchandbound algorithm we introduce a novel lower bounding function through interval matrix analysis. Experimental results on both synthetic and real scenes demonstrate that the proposed method is able to quickly obtain the optimal solution. 1.
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"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: