Duality And A Class Of Smoothing-regularization Methods To Mathematical Programming With Vanishing Constraints

Mathematical programs with vanishing constraints(MPVC for short)are a kind of constrained optimization problems which are difficult to solve directly by the classical optimization methods.It has a wide applications such as optimal topological design,robot motion planning,electric power economic scheduling and nonlinear optimal control.The main contents of this paper are as follows:First of all,we study the duality of mathematical programming with vanishing constraints.We mainly give the improved models of Wolfe,Mond-Weir duality that proposed by S.K.Mishra,Vinay Singh,Vivek Laha et al,which makes the calculation of the index set not be involved.At the same time,the corresponding duality theorems are given,and an example is given to explain the validity of the dual models.Secondly,we study a class of smooth regularization methods for solving the mathematical programming with vanishing constraints.These methods include the smoothing method proposed by Kanzow et al in 2013.At the same time,we prove that MFCQ holds at the feasible points of smoothing-regularization model under the VC-MFCQ condition which is weaker than VC-LICQ condition that been used by Kanzow et al.Also under the mild conditions,the convergence analysis is given in this paper.Numerical experiments are given in the end.