An augmented Lagrangian approach with a variable transformation in nonlinear programming

作者：Zhang Liwei^*; Yang Xiaoqi

来源：Nonlinear Analysis-Theory Methods & Applications, 2008, 69(7): 2095-2113.

DOI：10.1016/j.na.2007.07.048

• augmented Lagrangian
• duality
• exact penalty representation
• tangent cone
• normal cone
• subderivative
• subdifferential

Tangent cone and (regular) normal cone of a closed set under ail invertible variable transformation around a given point are investigated, which lead to the concepts of theta(-1)-tangent cone of a set and theta(-1)-subderivative of a function. When the notion of theta(-1)-subderivative is applied to perturbation functions, a class of augmented Lagrangians involving an invertible mapping of perturbation variables are obtained, in which dualizing parameterization and augmenting functions are not necessarily convex in -turbation variables. A necessar and sufficient condition for the exact penalty representation under the proposed all-ruented Lagrangian scheme is obtained. For an augmenting function with an Euclidean norm, a sufficient condition (resp., a sufficient and necessary condition) for an arbitrary vector (resp., 0) to Support an exact penalty representation is given in terms of theta(-1)subderivatives. An example of the variable transformation applied to constrained optimization problems is given, which yields several exact penalization results in the literature.

• 出版日期2008-10-1
• 单位大连理工大学; 香港理工大学; 沈阳航空航天大学