Nonlinear Programs with Positively Bounded Jacobians

In [3], it is constructively demonstrated that if M is a square (not necessarily symmetric) matrix all of whose principal minors are positive, the quadratic program\[ (A1) {minimize} z^T ( {Mz + q} ) {subject to} Mz + q 0, z 0, \] has an optimal solution satisfying the equation\[ ({A2}) z^T ( {Mz + q} ) = 0. \]A different prooff is offered here. The analysis is then extended to programs of the form \[ ({A3}) {minimize} z^T W( z ) {subject to} W( z ) 0, z 0, \] where W is a continuously differentiable mapping of real N-space into itself. The condition used to insure the existence of an optimal solution to (A3) is positive boundedness of the Jacobian matrix of the mapping W. Definition: A differentiable mapping W:RN → RN has a positively bounded Jacobian matrix, Jw ( z ), if there exists a real number δ such that 0 < δ < 1 and such that for every z ∈ RN each principal minor of JW ( z ) lies between δ and δ - 1. Mappings of the form $W( z ) = Mz + q$ have positively bounded Jacobian matrices if and only if M has positive principal minors, hence the programs (Al) are subsumed by (A3). Elementary examples show that it is not enough to assume in the general case, (A3), that JW ( z ) has positive principal minors for all z. A consequence of the main result is the Minimax Theorem: If K( x,y ) is a twice continuously differentiable real-valued function on Rⁿ × Rᵐ, and if [ ∇ ₓ K( x,y ), - ∇ y K( x,y ) ] has a positively bounded Jacobian matrix, then\[ {max }y 0 {min }x 0 K( {x,y} ) = {min }x 0 {max }y 0 K( {x,y} ). \]