Models And Algorithms For Several Classes Of Nonsmooth Optimization Problems And Their Applications In Point Clouds Matching

In many fields,such as science and engineering,nonsmooth optimization problems exist widely.For nonsmooth optimization problems with large scale or complex structure,classical discrete optimization algorithms are often unable to solve such problems in real time.As a kind of artificial neural networks,neurodynamic algorithms can be realized based on the circuit in hardware,and can solve optimization problems in real-time.Compared with the traditional numerical algorithms,neurodynamic algorithms are more efficient in solving the large scale optimization problems with complex structure.As a powerful tool to measure probability distribution,optimal transport theory has become an important research field in recent years because of its significant practical value.In this paper,the neurodynamic approach and optimal transport theory are used to construct algorithms for several classes of nonsmooth optimization problems existing widely in practice.And then,we discuss the behavior of solutions to the proposed neurodynamic approaches and applications of optimal transport in point clouds matching problems.The main contents of this thesis are:1.For a class of nonsmooth distributed convex optimization problems with general constraints,a multiagent neurodynamic algorithm with continuous-time format is proposed.This algorithm can minimize the objective function of each agent individually and collectively,and the state solutions of all agents reach an output consensus under mild assumptions.And then,the boundedness and global existence of state solution of the proposed algorithm are proved.In addition,the uniqueness and “slow solution” property of state solution are obtained when there is no simple set constraint in the considered problem.Finally,the state solution converges to the feasible region of equivalent optimization problem asymptotically and the output solution of each agent is convergent to the optimal solution set of the primal distributed optimization problem.2.For a class of general constrained nonsmooth sparse convex optimization problems with l1 penalty,a projection-based neurodynamic algorithm with differential equation structure is established.Due to the existence of nonsmooth term l1,most existing neurodynamic algorithms for such problems are often with differential inclusion structures,which leads to the difficulty of subgradient selection in practice.Therefore,a sufficientand necessary condition for determining the elements in subdifferential of l1 norm is discussed firstly.Then,a sufficient and necessary global optimality condition of this sparse optimization problem is obtained,and hence a projection-based neurodynamic algorithm is proposed.Secondly,the positive invariance of the state solution to the set defined by the equality constraints is studied.And then,the boundedness,global existence and stability in the sense of Lyapunov of state solution are proved.Thirdly,we prove that the state solution can converge to an optimal solution of this sparse optimization problem from any initial point.3.For a class of nonsmooth pseudoconvex optimization problems with general convex constraints,a neurodynamic algorithm is given.Firstly,based on smoothing technique,we construct a regularization function,which does not depend on any information of the feasible region.With the special structure of the constructed regularization function,we construct a neurodynamic algorithm,and prove the global existence,uniqueness and “slow solution” property of the state solution.Moreover,the state solution is proved to be convergent to the feasible region in finite time and to the optimal solution set of the related optimization problem subsequently.In particular,the convergence to an exact optimal solution of the state under some conditions is also discussed.4.For a class of nonsmooth and nonconvex point clouds matching problems,we construct two improved optimal transport models,and design the solving algorithms.The traditional optimal transport model is lack of robustness for the affine transformation and even nonlinear transformation between image point sets in handling with point clouds matching problems.In order to improve the matching accuracy of point clouds affected by complex deformation or noise,combining with the optimal transport theory,we induce an improved optimal transport model by using the orthogonal matrix and diagonal matrix as variables.And then,combining with parallel technique,we construct a solving algorithm for this improved optimal transport model.Furthermore,in order to deal with point clouds matching in more complex environments(such as with outliers or occlusion),we design a corresponding regularization term,and then propose a relaxed and regularized optimal transport model for this class of point clouds matching problems.Moreover,the solving algorithms are designed.