semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite...
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semidefinite optimization, commonly referred to as semidefinite programming, has been a remarkably active area of research in optimization during the last decade. For combinatorial problems in particular, semidefinite programming has had a truly significant impact. This paper surveys some of the results obtained in the application of semidefinite programming to satisfiability and maximum-satisfiability problems. The approaches presented in some detail include the ground-breaking approximation algorithm of Goemans and Williamson for MAX-2-SAT, the Gap relaxation of de Klerk, van Maaren and Warners, and strengthenings of the Gap relaxation based on the Lasserre hierarchy of semidefinite liftings for polynomial optimization problems. We include theoretical and computational comparisons of the aforementioned semidefinite relaxations for the special case of 3-SAT, and conclude with a review of the most recent results in the application of semidefinite programming to SAT and MAX-SAT.
The last few years witnessed an increasing interest in the problem of control synthesis of nonlinear systems. A recently derived stability criterion for nonlinear systems-which has a remarkable convexity property- and...
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The last few years witnessed an increasing interest in the problem of control synthesis of nonlinear systems. A recently derived stability criterion for nonlinear systems-which has a remarkable convexity property- and the development of numerical methods for verification of positivity allows the computation-via semidefinite programming- of stabilizing controllers for the case of systems with polynomial or rational vector fields. Using the theory of semialgebraic sets these computational tools are extended in this paper for the case of polynomial or rational systems with uncertainty parameters.
The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show h...
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The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.
In this article, we propose a feasible direction algorithm for solving the semidefinite programming (SDP) relaxations of quadratic {-1, 1} programming problems. This algorithm's distinguishing features are that it...
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In this article, we propose a feasible direction algorithm for solving the semidefinite programming (SDP) relaxations of quadratic {-1, 1} programming problems. This algorithm's distinguishing features are that it uses a low rank factorization and searches with a constant step-size. Its convergence is also proven. Finally, we report some numerical examples to compare our method with the low rank factorization method of Burer and Monteiro on the SDP relaxation of the max-cut problem.
We discuss a partially augmented Lagrangian method for optimization programs with matrix inequality constraints. A global convergence result is obtained. Applications to hard problems in feedback control are presented...
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We discuss a partially augmented Lagrangian method for optimization programs with matrix inequality constraints. A global convergence result is obtained. Applications to hard problems in feedback control are presented to validate the method numerically.
The conventional way to treat integral quadratic constraint (IQC) problems is to transform them into semi-definite programs (SDPs). SDPs can then be solved using interior point methods which have been proven efficient...
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The conventional way to treat integral quadratic constraint (IQC) problems is to transform them into semi-definite programs (SDPs). SDPs can then be solved using interior point methods which have been proven efficient. This approach, however, is not always the most efficient since it introduces additional decision variables to the SDP, and the additional decision variables sometimes largely increase the complexity of the problem. In this paper, we demonstrate how to solve IQC problems by other alternatives. More specifically, we consider two cutting plane algorithms. We will show that in certain cases these cutting plane algorithms can solve IQC problems much faster than the conventional approach. Numerical examples, as well as some explanations from the point of view of computational complexity, are provided to support our point. (C) 2003 Elsevier Ltd. All rights reserved.
In this paper, we address the maximum-likelihood (ML) multiuser detection problem for asynchronous code-division multiple access (CDMA) systems with multiple receiver antennas in frequency-selective fading environment...
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In this paper, we address the maximum-likelihood (ML) multiuser detection problem for asynchronous code-division multiple access (CDMA) systems with multiple receiver antennas in frequency-selective fading environments. Multiuser ML detection (MLD) in this case provides attractive symbol error performance, but it requires the solution of a large-scale combinatorial optimization problem. To deal with the computational complexity of this problem, we propose an efficient approximation method based on a block alternating likelihood maximization (BALM) principle. The idea behind BALM is to decompose the large-scale MLD problem into smaller subproblems. Assuming binary or quaternary phase shift keying (BPSK or QPSK) (which are often employed in CDMA), the combinatorial subproblems are then accurately and efficiently approximated by the semidefinite relaxation (SDR) algorithm-an algorithm that has been recently shown to lead to quasi-ML performance in synchronous CDMA scenarios. Simulation results indicate that this BALM detector provides close-to-optimal bit error rate (BER) performance. The BALM principle is quite flexible, and we demonstrate this flexibility by extending BALM to multicarrier (MC) multiuser systems. By exploiting the special signal correlation structure of MC systems, we develop a variation of BALM in which dynamic programming (DP) is used to solve the subproblems. It is shown using simulations that the BER performance of this DP-based BALM detector is as promising as that of the SDR-based BALM detector.
PID control is well known and widely applied in industry and many design algorithms are readily available in the literature. However, systematic design of multi-loop or decentralized PID control for multivariable, pro...
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PID control is well known and widely applied in industry and many design algorithms are readily available in the literature. However, systematic design of multi-loop or decentralized PID control for multivariable, processes to meet certain objectives simultaneously is still a challenging task. Designing multi-loop PID controllers such that the process variables satisfy the generalized covariance constraints is studied in this paper. A convergent computational algorithm is proposed to calculate the multi-loop PID controller for a process with stable disturbances. This algorithm is then extended to a process with random-walk disturbances. The feasibility of the proposed algorithm is verified by applying it to several simulation examples. (C) 2004 ISA-The Instrumentation, Systems, and Automation Society.
Engineering design problems, especially in signal and image processing, give rise to linear least squares problems arising from discretization of some inverse problem. The associated data are typically subject to erro...
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Engineering design problems, especially in signal and image processing, give rise to linear least squares problems arising from discretization of some inverse problem. The associated data are typically subject to error in these applications while the computed solution may only be implemented up to limited accuracy digits, i.e., quantized. In the present paper, we advocate the use of the robust counterpart approach of Ben-Tal and Nemirovski to address these issues simultaneously. Approximate robust counterpart problems are derived, which leads to semidefinite programming problems yielding stable solutions to overdetermined systems of linear equations affected by both data uncertainty and implementation errors, as evidenced by numerical examples from stochastic signal modeling. (C) 2003 Elsevier Inc. All rights reserved.
Let Q = (x\s(i)(x) greater than or equal to 0, i = 1,...,k} where s(i)(x) = a(i)(T) x-x(T) Q(i)x-c(i), and Q(i) is an n x n positive semidefinite matrix. We prove that the volumetric and combined volumetric-logarithmi...
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Let Q = (x\s(i)(x) greater than or equal to 0, i = 1,...,k} where s(i)(x) = a(i)(T) x-x(T) Q(i)x-c(i), and Q(i) is an n x n positive semidefinite matrix. We prove that the volumetric and combined volumetric-logarithmic barriers for Q are O (rootkn) and O(rootkn) self-concordant, respectively. Our analysis uses the semidefinite programming (SDP) representation for the convex quadratic constraints defining Q, and our earlier results on the volumetric barrier for SDP. The self-concordance results actually hold for a class of SDP problems more general than those corresponding to the SDP representation of Q.
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