On A Class Of Nonsmooth Optimization Problem And Its Application On Envelope Constrained Problem

Nonsmooth optimization theory is a very important branch of optimization theory.It is very common in practice and has a wide applications.In this thesis, we consider a special class of nonsmooth optimization problems, where the nonsmooth terms existed in the constraint function are not the variables, but the absolute values of the variables. We can ?nd many examples of this kind of problem in practice, for example, the envelope constrained problem. For this kind of problem,we develop a smoothing technique to transfer it into an equivalent smooth optimization problem, and then the gradient based algorithms can be applied to solve this problem.Next, we apply this technique to envelope constrained problem, where the processed signal can be discrete and continuous. We apply the smoothing technique to transfer the problems into equivalent optimization problems and solve them by gradient based method.There are 5 chapters in this thesis. It begins with the introductions of nonsmooth analysis and smoothing methods. We have discussed the difference between the nonsmooth function and smooth function, nonsmooth optimization problem and smooth optimization problem. We also discuss the classi?cation of nonsmooth optimization problems and their corresponding smoothing techniques.In Chapter 2, we consider a special class of nonsmooth optimization problem which appear in envelope constrained problem. We develop a smoothing technique for this problem and solve it by corresponding method.In Chapter 3, the nonsmooth optimization problem in Chapter 2 is generalized,where the linear nonsmooth term is replaced by a nonlinear nonsmooth term. We apply a smoothing technique to transfer this problem into an equivalent smooth optimization problem. We also introduce some monotonicity condition such that the complementary condition in the problem can be removed, and then the problem can be simpli?ed.We consider the envelope constrained problem in Chapter 4. First, we introduce the principle of envelope constrained problem, that is, we design the ?lter such that the input signal can be converted to an output signal contained in a given envelope. Next,considering the case that the signal contain the noise, the envelope constrained problem is transferred into a nonsmooth optimization problem. We divide this problem into two cases, where the input signal is discrete and continuous, respectively. Then, we develop the corresponding smoothing technique to deal with these two cases.The envelope constrained problem in Chapter 4 can be generalized in Chapter 5,where the ?lter is replaced by ?lter bank. We analyze the necessity of this generalization,and then discuss two kinds of nonsmooth optimization problems when the signal is discrete and continuous, respectively. We develop the corresponding smoothing technique to transfer the problems into two smooth optimization problems, and then solve them respectively.