Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to , one recovers the constrained solutio...
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Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to , one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and the solution is recovered for a finite value of the penalty constant. In practice, the kinks in the penalty and the unknown magnitude of the penalty constant prevent wide application of the exact penalty method in nonlinear programming. In this article, we examine a strategy of path following consistent with the exact penalty method. Instead of performing optimization at a single penalty constant, we trace the solution as a continuous function of the penalty constant. Thus, path following starts at the unconstrained solution and follows the solution path as the penalty constant increases. In the process, the solution path hits, slides along, and exits from the various constraints. For quadraticprogramming, the solution path is piecewise linear and takes large jumps from constraint to constraint. For a general convex program, the solution path is piecewise smooth, and path following operates by numerically solving an ordinary differential equation segment by segment. Our diverse applications to (a) projection onto a convex set, (b) nonnegative least squares, (c) quadratically constrained quadratic programming, (d) geometric programming, and (e) semidefinite programming illustrate the mechanics and potential of path following. The final detour to image denoising demonstrates the relevance of path following to regularized estimation in inverse problems. In regularized estimation, one follows the solution path as the penalty constant decreases from a large value.
Semidefinite programming is well-known for providing relaxations of quadratic programs. In practice, only few real-world applications of this approach have been reported. This can be explained by the fact that the sta...
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Semidefinite programming is well-known for providing relaxations of quadratic programs. In practice, only few real-world applications of this approach have been reported. This can be explained by the fact that the standard semidefinite relaxation must generally be tightened with cuts, which increases the substantial computational cost of the semidefinite program. Then, the challenge is to come up with the most effective cuts. In this paper, we present a systematic approach based on a polynomial separation problem to compute such cuts. Then, we apply this technique to a well-known problem of energy management, i.e., the scheduling of the nuclear outages which is a combinatorial problem with quadratic objective and non-convex quadratic constraints. This leads to the identification of some relevant cutting planes for this problem, allowing an average enhancement of 25% of the semidefinite relaxation compared to the linear relaxation. (C) 2014 Elsevier Ltd. All rights reserved.
In this paper, we introduce a procedure for the null broadening algorithm design with respect to the perturbation of the interference location. This method is based on maximizing the array output signal-to-interferenc...
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In this paper, we introduce a procedure for the null broadening algorithm design with respect to the perturbation of the interference location. This method is based on maximizing the array output signal-to-interference-plus-noise-ratio(SINR)subject to quadratic constraints. The design problem can be cast as a fractional quadratically constrained quadratic programming(QCQP) problem that can be solved efficiently using the semidefinite programming(SDP) techniques, the semidefinite relaxation can be used to obtain a lower bound on the optimal objective function. This proposed approach imposes broadened nulls towards the interference region while possesses well-maintained pattern. Theoretical analysis and numerical results demonstrate that the performance of proposed adaptive beamformer is almost always close to optimal.
This paper outlines an optimization-based method to estimate the reach set of a system while acknowledging unknown-but-bounded input and parametric uncertainty in the underlying dynamical model. The approach is ground...
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ISBN:
(纸本)9781479978861
This paper outlines an optimization-based method to estimate the reach set of a system while acknowledging unknown-but-bounded input and parametric uncertainty in the underlying dynamical model. The approach is grounded in a second-order Taylor-series expansion of the system's state variables along the solution trajectories as a function of the uncertain elements. Subsequently, over the time horizon of interest, quadraticallyconstrainedquadratic Programs (QCQPs) are formulated to estimate maximum and minimum bounds on the state variables to recover the reach set. To contend with the nonconvexity of the QCQPs, Lagrangian relaxations are leveraged to formulate Semidefinite Programs (SDPs) that provide guaranteed bounds to the solutions of the QCQPs. Applications of the method to quantify the impact of uncertain power injections in power-system dynamic models are demonstrated with numerical examples.
Identification of network topology is to estimate the topology of a controllable and observable network with given number of nodes such that the identified network will satisfy the response between specified input and...
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ISBN:
(纸本)9781479986842
Identification of network topology is to estimate the topology of a controllable and observable network with given number of nodes such that the identified network will satisfy the response between specified input and observed output. This paper examines the network topology identification (NTI) problems to find the original graph Laplacian from input-output data. A 'similar' set of state-space matrices satisfying the input-output response is firstly constructed through system identification procedure. Based on the similarity relationship, we reformulate the NTI problems as general quadratically constrained quadratic programming (QCQP) problems. The QCQP problem is then transformed into semidefinite programming (SDP) problem with a rank one constraint. An iterative rank minimization method is proposed to gradually approach the optimal solution. Examples are presented to verify the convergence of the proposed method.
Based on an augmented Lagrangian line search function, a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems. Compared to quadratic p...
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Based on an augmented Lagrangian line search function, a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems. Compared to quadraticprogramming solved in the traditional SQP methods, a convex quadratically constrained quadratic programming is solved here to obtain a search direction, and the Maratos effect does not occur without any other corrections. The "active set" strategy used in this subproblem can avoid recalculating the unnecessary gradients and (approximate) Hessian matrices of the constraints. Under certain assumptions, the proposed method is proved to be globally, superlinearly, and quadratically convergent. As an extension, general problems with inequality and equality constraints as well as nonmonotone line search are also considered. (c) 2007 Published by Elsevier B.V.
quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in...
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quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In this paper, we focus on the case of quadraticallyconstrainedquadratic 0-1 programs. The variant of QCR previously proposed for this case involves the addition of a quadratic number of auxiliary continuous variables. We show that, in fact, at most one additional variable is needed. Some computational results are also presented.
The performance of adaptive beamforming degrades in presence of signal model mismatches. In particular, when the desired signal is present in training snapshots, the adaptive beamforming is quite sensitive to desired ...
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ISBN:
(纸本)9781479943548
The performance of adaptive beamforming degrades in presence of signal model mismatches. In particular, when the desired signal is present in training snapshots, the adaptive beamforming is quite sensitive to desired signal steering vector mismatch. Therefore, a robust adaptive beamforming for actual system is proposed based on the reconstruction of interference covariance matrix and mismatched steering vector compensation. In the proposed method, the interference covariance matrix is firstly reconstructed by using Root-MUSIC method to estimate Direction-of-Arrival (DOA) of signals, where the desired signal component is removed from the training snapshots. Subsequently, the mismatched desired signal steering vector is compensated by solving a quadratically constrained quadratic programming problem. Simulation results show that the performance of proposed algorithm outperforms the existing robust adaptive beamforming, and the output signal-to-interference-plus-noise ratio (SINR) is close to optimal values. Therefore, the proposed algorithm could be significantly effective for actual system.
Finite impulse response (FIR) filters have been a primary topic of digital signal processing since their inception. Although FIR filter design is an old problem, with the developments of fast convex solvers, convex mo...
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ISBN:
(纸本)9781479948741
Finite impulse response (FIR) filters have been a primary topic of digital signal processing since their inception. Although FIR filter design is an old problem, with the developments of fast convex solvers, convex modelling approach for FIR filter design has become an active research topic. In this work, we propose a new method based on convex programming for designing FIR filters with the desired frequency characteristics. FIR filter design problem, which is modelled as a non-convex quadraticallyconstrainedquadratic program (QCQP), is transformed to a semidefinite program (SDP). By relaxing the constraints, a convex programming problem, which we call RSDP(Relaxed Semidefinite Program), is obtained. Due to the relaxation, solution to the RSDPs fails to be rank-1. Typically used rank-1 approximations to the obtained RSDP solution does not satisfy the constraints. To overcome this issue, an iterative algorithm is proposed, which provides a sequence of solutions that converge to a rank-1 matrix. Conducted experiments and comparisons demonstrate that proposed method successfully designs FIR filters with highly flexible frequency characteristics.
We consider the class of quadratically-constrainedquadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J...
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We consider the class of quadratically-constrainedquadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949-978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098-1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-More type condition.
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