P-th Power Lagrangian For Nonconvex Constrained Optimization Problem

Nonconvex constrained optimization problem is an optimization problem that has been widely used in economy and management.The classical Lagrangian function plays an important role in solving the problem of convex constrained optimization.However,the existence of a zero duality gap can not be guaranteed for the nonconvex constrained optimization problem.Therefore,the original Lagrangian duality theory can not be established in the nonconvex constrained optimization problem.In order to overcome this problem,some scholars use the original problem of convex method to eliminate the duality gap.In this paper,we study an equivalent p-th power reformulation for a class of nonconvex constrained optimization problems.Meanwhile,we consider a class of p-th power Lagrangian.First,the existence of local saddle points and global saddle points is discussed under some weaker conditions.Then,we propose an algorithm based on a class of p-th power Lagrangian and analyze its global convergence properties.At last,we propose an algorithm based on a large family of p-th power Lagrangian and analyze its global convergence properties taking into account the possible infeasibility of the problem.This paper is divided into four chapters:In chapter one,we mainly introduce the research background and research situation.In chapter two,we introduce an equivalent p-th power reformulation for a class of nonconvex constrained optimization problems.Under second-order sufficient conditions,it is proved that the p-th power Lagrangian admits a local saddle point,without requiring the gradients of active constraints satisfying linearly independent.In addition,the existence of a global saddle point is then obtained.Specially,we do not require the compactness of the feasible set and the uniqueness of global solution of the original problem.Later,an illustrative example is presented for the theorem of local saddle points and global saddle points.In chapter three,we propose an algorithm based on a class of p-th power Lagrangian and analyze its global convergence properties.Finally,experiments showing how the new algorithms and results are related to practical computations will given.In chapter four,we propose an algorithm based on a large family of p-th power Lagrangian and analyze its global convergence properties taking into account the possible infeasibility of the problem.We illustrate,by means of numerical experiments,that our algorithm is reliable for the p-th power Lagrangian proposed in the literature.