Introductory Lectures on Convex Optimization

Nesterov, Yurii

doi:10.1007/978-1-4419-8853-9

Cited by 3,773 publications

(4,866 citation statements)

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“…The theory and methods of unconstrained optimization are discussed in extensive detail in Dennis and Schnabel (1983), Gill, Murray and Wright (1981), Nash and Sofer (1996), Griva, Nash, and Sofer (2008), Nesterov (2004), and Nocedal and Wright (2006). For a guide to software for numerical optimization, see Moré and Wright (1993) and the online NEOS Optimization Software Guide.…”

Section: Discussionmentioning

confidence: 99%

Unary NP-Complete (NP-Hard)

2013

Encyclopedia of Operations Research and Management Science

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A transportation problem in which the total amount to be shipped (supply) is not equal to the total demand. The unbalanced problem can be stated as a standard transportation problem by the addition of a fictitious destination when the supply is greater than the demand, or by adding a fictitious origin if the demand is greater than the supply. In the first case, the demand at the fictitious destination is the difference between the total supply and total demand, while in the second case, the supply at the fictitious origin is the difference between the total demand and total supply. See ▶ Transportation Problem Unbounded Optimal SolutionA solution to a constrained optimization problem in which the objective function value can be shown to increase (or decrease) without bound on the feasible region. A real-world problem whose mathematical model exhibits an unbounded optimal solution must have an incorrect formulation. Unconstrained OptimizationAriela Sofer George Mason University, Fairfax, VA, USA IntroductionUnconstrained optimization is concerned with finding the minimizing or maximizing points of a nonlinear function, where the variables are free to take on any value. Unconstrained optimization problems occur in a wide range of applications in science and engineering. A rich source of unconstrained optimization problems are data-fitting problems, in which some model function with unknown parameters is fitted to data, using some criterion of best fit. This criterion may be the minimum sum of squared errors, or the maximum of a likelihood or entropy function. Unconstrained problems also arise from constrained optimization problems, since these are often solved by solving a sequence of unconstrained problems.In mathematical terms, an unconstrained minimization problem can be written in the form minimize f ðxÞ; where x ¼ ðx 1 ; . . . ; x n Þ T is a vector of unrestricted variables in the n-dimensional space < n . Ideally, one would like to find a global minimizer of the function, i.e., a point x Ã that yields the lowest value of f. Such a solution satisfies If the inequality above holds strictly, i.e., f ðx Ã Þ < f ðxÞ for all x, then x Ã is a strict global minimizer.In many cases, finding a global minimizer is extremely difficult. For this reason, most algorithms attempt only to find a local minimizer of the function, i.e., a point x Ã that satisfies f ðx Ã Þ f ðxÞ for all x in some neighborhood of x Ã . If the objective f is a convex function (see the next section), a local minimizer will also be a global minimizer; however, for nonconvex functions this property does not generally hold.There is no inherent difference between minimization problems and maximization problems, since maximizing f can be accomplished by minimizing À f and then multiplying the optimal objective values by À 1. For this reason it is sufficient to focus only on unconstrained minimization problems. BackgroundMuch of the research in unconstrained optimization has focused on functions with continuous derivatives. Throughout this discussion it ...

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Section: Discussionmentioning

confidence: 99%

Unary NP-Complete (NP-Hard)

2013

Encyclopedia of Operations Research and Management Science

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“…In this paper, we use an algorithm called NESTA to solve the convex optimization problem of (7). NESTA is developed by Stephen Becker, etc., for solving large-scale problems based on Nesterov's work [48][49][50]. It is a fast and robust first-order method that solves minimum 1 problems and a large number of extensions including TV minimization with the accelerated convergence rate of O(1/k 2 ).…”

Section: Compressive Sensing Approach Applied To Sairsmentioning

confidence: 99%

A Compressive Sensing Approach for Synthetic Aperture Imaging Radiometers

Li¹,

Zhou²,

Ren³

et al. 2013

PIER

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Abstract-The aperture synthesis technology represents a promising new approach to microwave radiometers for high-resolution observations of the Earth from geostationary orbit. However, the future application of the new technology may be limited by its large number of antennas, receivers, and correlators. In order to reduce significantly the complexity of the on-board hardware requirements, a novel method based on the recently developed theory of compressive sensing (CS) is proposed in this paper. Due to the fact that the brightness temperature distributions of the Earth have a sparse representation in some proper transform domain -for example, in terms of spatial finite-differences or their wavelet coefficients, we use the CS approach to significantly undersample the visibility function. Thus the number of antennas, receivers, and correlators can be further reduced than those based on the traditional methods that obey the Shannon/Nyquist sampling theorem. The reconstruction is performed by minimizing the 1 norm of a transformed image. The effectiveness of the proposed approach is validated by numerical simulations using the image corresponding to AMSU-A over the Pacific.

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“…However, it is not differentiable in general, which precludes gradient ascent methods for its maximization. We can, however, use the projected supergradient method [13].…”

Section: (6)mentioning

confidence: 99%

“…Specifically, the method is easy to implement, but in general has worse convergence properties than first-order gradient-based methods. We refer the reader to [13,16] for details.…”

Section: (6)mentioning

confidence: 99%

Optimization for Multi-Region Segmentation of Cardiac MRI

Ulén

Strandmark

Kahl

2012

Statistical Atlases and Computational Models of the Heart. Imaging and Modelling Challenges

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Abstract. We introduce a new multi-region model for simultaneous segmentation of the left and right ventricles, myocardium and the left ventricular papillary muscles in MRI. The model enforces geometric constraints such as inclusion and exclusion between the regions, which makes it possible to correctly segment different regions even though the intensity distributions are identical. We efficiently optimize the model using Lagrangian duality which is faster and more memory efficient than current state of the art. As the optimization is based on global techniques, the resulting segmentations are independent of initialization. We evaluate our approach on two benchmarks with competitive results.

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Introductory Lectures on Convex Optimization

Cited by 3,773 publications

References 0 publications

Unary NP-Complete (NP-Hard)

Unary NP-Complete (NP-Hard)

A Compressive Sensing Approach for Synthetic Aperture Imaging Radiometers

Optimization for Multi-Region Segmentation of Cardiac MRI

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