Study Of Several Nonsmooth,Nonconvex And Nonlinear Problems In Optimization Signal Processing

In the information age,effective signal analysis and processing methods are of great importance in the core theories and technologies of signal processing,communication and computer.Optimization is a subject that studies the optimal selection characteristics of decision problem,constructs the calculation methods to find the optimal solution,and studies their theoretical properties and calculation performance.The study of sparsity and optimization theory is an important part of the research field recently,which is widely used in signal processing.In this thesis,some problems of sparse optimization,clustering,classification and optimal design of Sigma Delta modulator are studied,and the related optimization theory and methods about nonsmooth,nonconvex and nonlinear optimization are also researched.The research work is summarized as follows:1.For nonsmooth optimization model in sparsity problem,an optimization problem with l1 norm form objective function subject to l? norm form inequality constraint is researched.The range of inequality constraint bound is firstly determined by solving a linear programming problem.Then the original problem is reformulated and analyzed theoretically.The piecewise linear relationship between the l1 norm objective function values and l? norm inequality constraint bounds is proved mathematically.Also,computer numerical experiments on a set of signals verify the validity of this result.2.For nonsmooth clustering optimization problem,this thesis proposes l2 norm,l1 norm and l? norm of clustering optimization algorithms based on dictionary learning.By solving an optimization problem to assign each feature to a cluster and solving another optimization problem to re-calculating the vectors representing the clusters,the algorithms all keeps iterating until it converges.This work can provide new thinking for the research of clustering optimization.For the intraclass separation and the interclass separation nonconvex classification optimization problem,to find the globally optimal linear projection hyperplane vector without satisfying the eigenvalue equation,this thesis reformulates the objective function as a weighted sum of the intraclass separation and the interclass separation subject to a quadratic equality constraint on the square of the l2 norm of the decision vector.As to the problem,it shows that the global minimum of the objective functional value is equal to the minimum singular value of the Hessian matrix of the objective function.Also,the globally optimal solution of the optimization problem is in the null space of the Hessian matrix minus this singular value multiplied by the identity matrix.The analytical form of global optimal solution is given,and the computer results show the effectiveness of the method.3.For some theories of nonlinear constrained optimization,the thesis studies and proves a corollary on the optimization problem with a lp(p>1)penalty function based on the sequential quadratic programming approach.It shows that the corollary defines the inequality condition for the search direction of the optimization problem.Furthermore,it proves that the derived search direction is a descent direction of the penalty function in the original problem by determining the range of penalty parameter.The corresponding algorithm is given and the effectiveness of this method is verified by numerical experiments,which can provide a new idea for the study of nonlinear constraint optimization methods.For the application of nonlinear constrained optimization,an optimal design of a multi-bit interpolative sigma delta modulator(SDM)based on the absolute stability criterion is proposed.First,the uniform mid-tread quantizer is chosen and proved the optimal one mathematically in the sense of minimizing the maximum output input ratio of the mid-tread quantizer.Second,a loop filter of the multibit interpolative SDM is designed with criteria and conditions in order to obtain a loop filter with a good noise shaping characteristic for achieving a high-signal-to-noise ratio.The filter is designed based on the minimization of the energy of the noise transfer function in signal band subject to the strictly stable or the marginally stable condition of the loop filter,the absolute stability criterion,and the specifications on both the noise transfer function in the signal band and the magnitude response of the loop filter outside the signal band.It can be described as a nonlinear constrained optimization problem.Actually it is also a nonconvex problem.The approximate global optimal solution of the problem is obtained by genetic algorithm.The result shows that the proposed multi-bit interpolative SDM achieves a broader stability margin and a higher signal to noise ratio compared to the state of art designs,which has certain theoretical value and practical significance.In summary,the thesis combines nonsmooth,nonconvex and nonlinear optimization theory to study some problems in signal processing.This is an innovative research that integrates the intersection innovation of information science and mathematic science.The content of this thesis is specific and clear.It's of certain significance for discovering theory of rules and characteristics as well as solving some practical application problems.