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Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
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.
Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization
- P. Parrilo
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 the…
Constrained Consensus and Optimization in Multi-Agent Networks
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.
Semidefinite programming relaxations for semialgebraic problems
- P. Parrilo
- Mathematics, Computer ScienceMath. Program.
- 1 May 2003
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.
The Convex Geometry of Linear Inverse Problems
- V. Chandrasekaran, B. Recht, P. Parrilo, A. Willsky
- Computer ScienceFound. Comput. Math.
- 3 December 2010
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.
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 the…
Rank-Sparsity Incoherence for Matrix Decomposition
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.
Latent variable graphical model selection via convex optimization
- V. Chandrasekaran, P. Parrilo, A. Willsky
- Computer Science48th Annual Allerton Conference on Communication…
- 6 August 2010
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.
Introducing SOSTOOLS: a general purpose sum of squares programming solver
- S. Prajna, A. Papachristodoulou, P. Parrilo
- Computer ScienceProceedings of the 41st IEEE Conference on…
- 10 December 2002
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.
Semidefinite Optimization and Convex Algebraic Geometry
This book provides a self-contained, accessible introduction to the mathematical advances and challenges resulting from the use of semidefinite programming in polynomial optimization. This quickly…