Nonlinear Programming Theory and Algorithms
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Abstract: (2007). Nonlinear Programming Theory and Algorithms. Technometrics: Vol. 49, No. 1, pp. 105-105.
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01 Jan 2018
2,291 citations
Cites background from "Nonlinear Programming Theory and Al..."
...General discussions of different types of nonlinear programming methods are provided in [23, 39]....
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TL;DR: A new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer is proposed.
Abstract: We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method. The proposed algorithm is an instance of the so-called alternating direction method of multipliers, for which convergence has been proved. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is faster than the current state of the art methods.
1,211 citations
Cites methods from "Nonlinear Programming Theory and Al..."
...It has been shown that, wheng is a linear function, i.e., g(u) = Gu, the Bregman iterative algorithm is equivalent to the augmented Lagrangian method [46], which is briefly reviewed in the following subsection....
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Book•
27 Jul 2017TL;DR: Predictive Control for Linear and Hybrid Systems is an ideal reference for graduate, postgraduate and advanced control practitioners interested in theory and/or implementation aspects of predictive control.
Abstract: Model Predictive Control (MPC), the dominant advanced control approach in industry over the past twenty-five years, is presented comprehensively in this unique book. With a simple, unified approach, and with attention to real-time implementation, it covers predictive control theory including the stability, feasibility, and robustness of MPC controllers. The theory of explicit MPC, where the nonlinear optimal feedback controller can be calculated efficiently, is presented in the context of linear systems with linear constraints, switched linear systems, and, more generally, linear hybrid systems. Drawing upon years of practical experience and using numerous examples and illustrative applications, the authors discuss the techniques required to design predictive control laws, including algorithms for polyhedral manipulations, mathematical and multiparametric programming and how to validate the theoretical properties and to implement predictive control policies. The most important algorithms feature in an accompanying free online MATLAB toolbox, which allows easy access to sample solutions. Predictive Control for Linear and Hybrid Systems is an ideal reference for graduate, postgraduate and advanced control practitioners interested in theory and/or implementation aspects of predictive control.
1,142 citations
Cites background or methods from "Nonlinear Programming Theory and Al..."
...2 Geometric interpretation of KKT conditions [27]....
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...The material of this part follows closely the presentation from the following books and lecture notes: “Convex optimization” by Boyd and Vandenberghe [65], “Nonlinear Programming Theory and Algorithms” by Bazaraa, Sherali and Shetty [27], “LMIs in Control” by Scherer and Weiland [258] and “Lectures on Polytopes” by Ziegler [296]....
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...A detailed discussion on constraint qualifications can be found in Chapter 5 of [27]....
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...The following example [27] shows a convex problem where the KKT conditions are not fulfilled at the optimum....
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TL;DR: This paper proposes a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications and shows that the proposed algorithm is a strong contender for the state-of-the-art.
Abstract: We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly non-smooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, non-smoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either total-variation or wavelet-based (or, more generally, frame-based) regularization. The proposed algorithm is an instance of the so-called "alternating direction method of multipliers", for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the state-of-the-art.
957 citations
Cites background from "Nonlinear Programming Theory and Al..."
.... Moreover, the fact thatf1 = 0 turns Step 3 of the algorithm into a simple quadratic minimization problem, which has a unique solution ifG has full column rank: arg min u ∥∥Gu− ζk ∥∥2 2 = ( GHG )−1 GHζk, (16) = [ J∑ j=1 (H(j))HH(j) ]−1 J∑ j=1 ( H(j) )H ζ (j) k , whereζk = vk + dk (and, naturally,ζ…...
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TL;DR: This paper provides a survey of all the architectures that have been presented in the literature so far, using a unified description that includes optimization problem statements, diagrams, and detailed algorithms.
Abstract: Multidisciplinary design optimization is a field of research that studies the application of numerical optimization techniques to the design of engineering systems involving multiple disciplines or components. Since the inception of multidisciplinary design optimization, various methods (architectures) have been developed and applied to solve multidisciplinary design-optimization problems. This paper provides a survey of all the architectures that have been presented in the literature so far. All architectures are explained in detail using a unified description that includes optimization problem statements, diagrams, and detailed algorithms. The diagrams show both data and process flow through the multidisciplinary system and computational elements, which facilitate the understanding of the various architectures, and how they relate to each other. A classification of the multidisciplinary design-optimization architectures based on their problem formulations and decomposition strategies is also provided, a...
868 citations