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Projections Onto Convex Sets (POCS) Based Optimization by Lifting
, 1306
"... Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some nonconvex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function ..."
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Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some nonconvex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in RN the corresponding set is a convex set in RN+1. The iterative optimization approach starts with an arbitrary initial estimate in RN+1 and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in totalvariation, filtered variation, l1, and entropic cost functions. It is also experimentally observed that cost functions based on lp, p < 1 can be handled by using the supporting hyperplane concept. 1
Denoising Using Projection Onto Convex Sets (POCS) Based Framework
"... Abstract—Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a con ..."
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Abstract—Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in RN the corresponding set is also a convex set in RN+1. The iterative optimization approach starts with an arbitrary initial estimate in RN+1 and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in totalvariation (TV), filtered variation (FV), `1, and entropic cost functions. A new denoising algorithm using the TV framework is developed. The new algorithm does not require any of the regularization parameter adjustment. Simulation examples are presented. Index Terms—Projection onto Convex Sets, Bregman Projections, Iterative Optimization, Lifting
Projection Onto Convex Sets (POCS) Based Signal Reconstruction Framework with an
, 2014
"... A new signal processing framework based on the projections onto convex sets (POCS) is developed for solving convex optimization problems. The dimension of the minimization problem is lifted by one and the convex sets corresponding to the epigraph of the cost function are defined. If the cost functi ..."
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A new signal processing framework based on the projections onto convex sets (POCS) is developed for solving convex optimization problems. The dimension of the minimization problem is lifted by one and the convex sets corresponding to the epigraph of the cost function are defined. If the cost function is a convex function in RN the corresponding epigraph set is also a convex set in RN+1. The iterative optimization approach starts with an arbitrary initial estimate in RN+1 and orthogonal projections are performed onto epigraph set in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in totalvariation (TV), filtered variation (FV), `1, `1, and entropic cost functions. New denoising and compressive sensing algorithms using the TV cost function are developed. The new algorithms do not require any of the regularization parameter adjustment. Simulation examples are presented. 1