Probability Constrain Optimization Model And Its D.C. Approximation In Compressed Sensing

Compressed sensing signal reconstruction with noise can be represented as l1-norm problem,The typical algorithm is convex optimization algorithm.One of the most important theory is the selection of the observation matrix.In order to reconstruct a high-precision with a small amount of observations,it is necessary to satisfy the restricted isometric property(RIP)and non-coherence when setting the observation matrix.However,it is difficult to judge the RIP of a matrix.In view of the uncertainty of the observation matrix,the l1-norm problem is transformed into a stochastic optimization model with probability constraint in this paper.That is,the minimum l1-norm problem is solved when the constraint is satisfied with large probability.In chapter 1,we introduce research background and development status compressed sensing and probability constraint optimization problem.The preliminary knowledge is provided.In chapter 2,the probability constraint optimization problem of compressed sensing is established.We define a D.C.approximation function ?(z,?,t)of characteristic function.The properties of the functions are discussed,and the equivalent D.C.approximation problem(P)is presented.In some cases,we prove the equivalence between problem(P)and the probability constraint optimization problem,and the convergence analysis is also carried out.chapter 3 has discussed sequential convex approximation(SCA)method for solving problem(Pt).We also introduce the sequential convex approximation algorithm,and the properties of the algorithm is analyzed.In chapter 4,the sample average approximation(SAA)method for solving problem(Pt)is introduced.We also define the sample average approximation function pN(x,?,t)of D.C.approximation function ?(z,?,t)and construct relevant the sample average approximation problem(PN).The optimal value,optimal solution set of problem(PN)converge to the optimal value,optimal solution set of problem(Pt)when the sample size is sufficiently large.