SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.

Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones

We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interiorpoint methods for smooth or cone convex programs.

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Graph Implementations for Nonsmooth Convex Programs

We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the /spl lscr//sub 1/-norm. A number of recent theoretical results on sparsifying properties of /spl lscr//sub 1/ penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame/spl acute/r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization.

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A sparse signal reconstruction perspective for source localization with sensor arrays

http://galton.uchicago.edu/~lekheng/courses/310w11/boyd-socp.pdf

Applications of second-order cone programming

Recently, a lot of attention has been paid to regularization based methods for sparse signal reconstruction (e.g., basis pursuit denoising and compressed sensing) and feature selection (e.g., the Lasso algorithm) in signal processing, statistics, and related fields. These problems can be cast as -regularized least-squares programs (LSPs), which can be reformulated as convex quadratic programs, and then solved by several standard methods such as interior-point methods, at least for small and medium size problems. In this paper, we describe a specialized interior-point method for solving large-scale -regularized LSPs that uses the preconditioned conjugate gradients algorithm to compute the search direction. The interior-point method can solve large sparse problems, with a million variables and observations, in a few tens of minutes on a PC. It can efficiently solve large dense problems, that arise in sparse signal recovery with orthogonal transforms, by exploiting fast algorithms for these transforms. The method is illustrated on a magnetic resonance imaging data set.

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An Interior-Point Method for Large-Scale $\ell_1$ -Regularized Least Squares

We show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great efficiency by recently developed interior-point methods. The synthesis problems involve arrays with arbitrary geometry and element directivity, constraints on far- and near-field patterns over narrow or broad frequency bandwidth, and some important robustness constraints. We show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays.

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Antenna array pattern synthesis via convex optimization

Cette these montre que de tres nombreux problemes de synthese de diagrammes de reseaux d'antennes peuvent etre resolus par des techniques numeriques d'optimisation convexe. En effet, les reseaux d'antennes consideres peuvent avoir une geometrie quelconque et les diagrammes elementaires des antennes peuvent etre tres generaux. Ces reseaux peuvent egalement etre a fonctionnement large bande. Les questions importantes de robustesse sont egalement abordees. Enfin on peut noter que les methodes numeriques utilisees sont applicables a de nombreux autres techniques de l'ingenieur, notamment a la synthese de filtres. La convexite est une notion mathematique fondamentale qui n'a guere ete utilisee dans le domaine des reseaux d'antennes. Elle a pourtant deux proprietes essentielles qui sont exposees dans ce rapport: tout d'abord elle garantit toujours une optimalite globale, c'est a dire que tout minimum local d'une fonction convexe est un minimum global ; ensuite la convexite permet d'obtenir des informations detaillees sur l'optimisation grace a la theorie de la dualite. Bien que ces problemes n'aient pas de solution analytique, ils peuvent etre traites numeriquement, et de maniere plus efficace encore depuis l'apparition recente de nouveaux algorithmes: il s'agit de la famille des algorithmes de l'ellipsoide et des tres efficaces methodes de points interieurs. De nombreuses simulations effectuees avec ces deux groupes d'algorithmes sont presentees, en particulier le probleme classique de la minimisation des lobes secondaires. D'autres simulations traitent de la valeur des poids et de minimisation de puissance

Synthese de diagrammes de reseaux d'antennes par optimisation convexe

Antenna pattern synthesis deals with choosing control parameters (the complex weights) of an array of antennas, in order to achieve a set of given specifications. It appears that these problems can often be formulated as convex optimization problems, which can be numerically solved with algorithms such as the ellipsoid algorithm of interior point methods. After a brief presentation of antenna pattern synthesis and of convex optimization, we illustrate then with simulations results. We first minimize the sidelobe levels of a cosecant diagram. Then we show an example of interference cancellation and compare it to adaptive techniques. We will also introduce the important problem of robust antenna arrays.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Antenna pattern synthesis through convex optimization

We show that two antenna array pattern synthesis problems can be expressed as convex optimization problems. The first one deals with a symmetric planar array with real weights, which can be expressed as a linear program. The second one concerns a broadband acoustic array, which becomes a convex quadratically constrained quadratic program. Because these two problems are convex, they can be (numerically) solved with great efficiency by recently developed interior-point methods. Thanks to the efficiency of the interior point methods, we also built a computer-aided design tool for the first problem.

Hervé Lebret

Papers

Applications of second-order cone programming

Antenna array pattern synthesis via convex optimization

Synthese de diagrammes de reseaux d'antennes par optimisation convexe

Antenna pattern synthesis through convex optimization

Optimal beamforming via interior point methods