Manifold Structure of the Karush–Kuhn–Tucker Stationary Solution Set with Two Parameters

Abstract

This paper deals with a 2-dimensional parameter family of nonlinear programs: minimize $h_0 ( x,t )$ subject to the equality constraints $h_i (x,t) = 0\, (i = 1,\ldots ,l)$ and the inequality constraints $h_j ( x,t ) \leq 0\,( j = l + 1, \ldots ,m )$. Each $h_i \, ( i = 0,1, \ldots ,m )$ is a twice continuously differentiable real-valued map defined on the $( n + 2 )$-dimensional Euclidean space $R^{n + 2} $, where $x \in R^n $ denotes a variable vector and $t \in R^2 $ denotes a 2-dimensional parameter vector. The local properties of the Karush–Kuhn–Tucker stationary solution set, the set $\Sigma $ consisting of all $( x,t )$ such that x is a stationary solution of the program for some t, are studied. In fact, it is shown that if the Mangasarian-Fromovitz constraint qualification and a regular value condition are satisfied, (i) the set $\Sigma $ is a 2-dimensional topological manifold without a boundary, and (ii) the set $\Sigma $ is a generalized creased manifold if, in addition, a constant rank condition holds.