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16
Convergent relaxations of polynomial matrix inequalities and static output feedback
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Modelchecking Markov chains in the presence of uncertainties
 Tools and Algorithms for the Construction and Analysis of Systems (TACAS), volume 3920 of LNCS
, 2006
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BranchandCut Algorithms for the Bilinear Matrix Inequality Eigenvalue Problem
 Comput. Optim. Appl
, 1999
"... The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years. This inequality permits to reduce in a elegant way various problems of robust control into its form. However, on ..."
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Cited by 20 (1 self)
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The optimization problem with the Bilinear Matrix Inequality (BMI) is one of the problems which have greatly interested researchers of the control and system theory in the last few years. This inequality permits to reduce in a elegant way various problems of robust control into its form. However, on the contrary of the Linear Matrix Inequality (LMI) which can be solved by interiorpointmethods, the BMI is a computationally difficult object in theory and in practice. This article improves the branchandbound algorithm of Goh, Safonov and Papavassilopoulos (1995) by applying a better convex relaxation of the BMI Eigenvalue Problem (BMIEP), and proposes new BranchandBound and BranchandCut Algorithms. Numerical experiments were conducted in a systematic way over randomly generated problems, and they show the robustness and the efficiency of the proposed algorithms. Keywords: Bilinear Matrix Inequality, BranchandCut Algorithm, Convex Relaxation, Cut Polytope. y Author supported b...
A nonlinear SDP algorithm for static output feedback problems in COMPlib. LAASCNRS research report no. 04508
, 2004
"... Abstract: We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPleib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the p ..."
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Abstract: We present an algorithm for the solution of static output feedback problems formulated as semidefinite programs with bilinear matrix inequality constraints and collected in the library COMPleib. The algorithm, based on the generalized augmented Lagrangian technique, is implemented in the publicly available general purpose software PENBMI. Numerical results demonstrate the behavior of the code.
Towards Implementations of Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
, 1999
"... Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically ..."
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Cited by 11 (6 self)
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Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically by solving a finite number of linear programs, several important implementation issues remain unsolved. In this paper, we discuss those issues, present practically implementable algorithms and report numerical results.
Efficient Spectrum Sensing for Cognitive Radio Networks via Joint Optimization of Sensing Threshold and Duration
"... Abstract—Cognitive radio networks require fast and reliable spectrum sensing to achieve high network utilization by secondary users. Optimization approaches to spectrum sensing todate have largely focused on maximizing throughput for secondary users while considering only a single parameter variable ..."
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Abstract—Cognitive radio networks require fast and reliable spectrum sensing to achieve high network utilization by secondary users. Optimization approaches to spectrum sensing todate have largely focused on maximizing throughput for secondary users while considering only a single parameter variable pertinent to sensing notably the threshold or duration, but not both. In this work, we investigate the impact of true joint minimization under two performance criteria: a) minimization of the average time to detection of a spectrum hole and b) joint maximization of the aggregate opportunistic throughput. We show that the resulting nonconvex problem is actually biconvex under practical conditions for which effective algorithms can be developed that yields reliable numerical procedures to solve the resulting optimization problem. The results show that the proposed approach can considerably improve system performance (in terms of the mean time to detect a spectrum hole and also the aggregate opportunistic throughput of both primary and secondary users), relative to the scenarios with only a single sensing variable or a suboptimal adhoc optimization approach used for two variable case. Index Terms—Cognitive radio, spectrum sensing, biconvex, joint optimization. I.
Overcoming nonconvexity in polynomial robust control design
, 2004
"... When developing efficient and reliable computeraided control system design (CACSD) tools for loworder robust control systems analysis and synthesis, the main issue faced by theoreticians and practitioners is the nonconvexity of the stability domain in the space of polynomial coefficients, or equi ..."
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Cited by 3 (1 self)
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When developing efficient and reliable computeraided control system design (CACSD) tools for loworder robust control systems analysis and synthesis, the main issue faced by theoreticians and practitioners is the nonconvexity of the stability domain in the space of polynomial coefficients, or equivalently, in the space of design parameters. In this paper, we survey some of the recently developed techniques to overcome this nonconvexity, underlining their respective pros and cons. We also enumerate some related open research problems which, in our opinion, deserve particular attention. Keywords: Computeraided control system design (CACSD), Robust control, Loworder controller design, Polynomials, Convex optimization, Linear matrix inequalities (LMI) 1
Robust Structured Control Design via LMI Optimization ⋆
"... Abstract: This paper presents a new procedure for discretetime robust structured control design. Parameterdependent nonconvex conditions for stabilizable and induced L2norm performance controllers are solved by an iterative linear matrix inequalities (LMI) optimization. A wide class of controller ..."
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Abstract: This paper presents a new procedure for discretetime robust structured control design. Parameterdependent nonconvex conditions for stabilizable and induced L2norm performance controllers are solved by an iterative linear matrix inequalities (LMI) optimization. A wide class of controller structures including decentralized of any order, fixedorder dynamic output feedback, static output feedback can be designed robust to polytopic uncertainties. Stability is proven by a parameterdependent Lyapunov function. Numerical examples on robust stability margins shows that the proposed procedure can obtain less conservative results than traditional stability criteria.
Semidefinite relaxations for semiinfinite polynomial programming
 INFINITE MOMENT PROBLEM 19 1Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK. S7N 5E6, Canada Email address: mehdi.ghasemi@usask.ca, marshall@math.usask.ca 2Fachbereich Mathematik und Statistik, Universität Konstanz 78
, 2014
"... Abstract. This paper studies how to solve semiinfinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP ..."
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Abstract. This paper studies how to solve semiinfinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.