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Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
TLDR
It is shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. Expand
Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization
In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where theExpand
Constrained Consensus and Optimization in Multi-Agent Networks
TLDR
A distributed "projected subgradient algorithm" which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set, and it is shown that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution. Expand
The Convex Geometry of Linear Inverse Problems
TLDR
This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. Expand
Semidefinite programming relaxations for semialgebraic problems
  • P. Parrilo
  • Mathematics, Computer Science
  • Math. Program.
  • 1 May 2003
TLDR
It is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility and provide a constructive approach for finding bounded degree solutions to the Positivstellensatz. Expand
SOSTOOLS: Sum of squares optimization toolbox for MATLAB
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of theExpand
Rank-Sparsity Incoherence for Matrix Decomposition
TLDR
This paper decomposes a matrix formed by adding an unknown sparse matrix to an unknown low-rank matrix into its sparse and low- rank components. Expand
Introducing SOSTOOLS: a general purpose sum of squares programming solver
TLDR
The paper provides an overview on sum of squares programming, describes the primary features of SOSToolS, and shows how SOSTOOLS is used to solve sum of square programs. Expand
Latent variable graphical model selection via convex optimization
Suppose we have samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent andExpand
Latent variable graphical model selection via convex optimization
TLDR
The modeling framework can be viewed as a combination of dimensionality reduction and graphical modeling (to capture remaining statistical structure not attributable to the latent variables) and it consistently estimates both the number of hidden components and the conditional graphical model structure among the observed variables. Expand
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