A Smooth Approximation To Probability Constrained Optimization Model In Compressed Sensing

Compressed sensing has high practical application value and is widely used in radar detection,image processing,wireless sensor networks and other fields.Signal reconstruction model with noise in compressed sensing can be expressed as an l₁-norm problem.In order to reconstruct high-precision images with a small amount of observation data,it is necessary to design the observation matrix by using restricted isometry property（RIP）and incoherence.However,the RIP of the matrix is difficult to judge.Due to the uncertainty of the observation matrix,the l₁-norm problem can be transformed into a stochastic optimization model with probability constrained optimization.When the parameter is small enough,the smooth approximation function is equivalent to the probability constraint function,and the corresponding smooth approximation problem is established.This paper mainly discusses the probability constrained optimization model in compressed sensing and its smooth approximation method.The main contents are as follows.Chapter 1 introduces the background of probabilistic constrained optimization problems and compressed sensing problems,and gives related definitions,theorems and propositions.In chapter 2,a probabilistic constrained optimization model is constructed based on D.C.function.Because the measurement matrix is random,the problem of signal reconstruction with noise can be expressed as an l₁-norm problem.A smooth approximation function φ（z,t） of characteristic function 1[0,+∞)（z） is established and a smooth D.C.approximation problem（Pδ）is constructed,and the convergence is analyzed.In chapter 3,the sequential convex approximation（SCA）method for solving D.C.approximation problem（Pδ）is discussed,and the convergence of the algorithm is analyzed.In chapter 4,the sample average approximation（SAA）method for solving D.C.approximation problem（Pδ）is presented,and the corresponding sample average approximation problem is built.Convergence of the algorithm is analyzed.