A Complementary Pivoting Approach to Parametric Nonlinear Programming

Abstract

A family of nonlinear programming problems: minimize θ(t, u) subject to g(t, u) ≤ 0 and u ≥ 0 is considered, where t is a scalar parameter changing from 0 to +∞, u an n-dimensional variable vector, θ a continuous real valued mapping defined on the (1 + n)-dimensional space and g a continuous mapping from the (1 + n)-dimensional space into the m-dimensional space. It is assumed that for each fixed t, θ(t, ·) and gj(t, ·) (j = 1, …, m) have partial derivatives ∂θ(t, u)/∂ui and ∂gj(t, u)/∂ui(i = 1, …, n, j = 1, …, m) which are continuous with respect to both t and u. Under a moderate constraint qualification, this paper applies the complementary pivoting theory to the computation of a piecewise linear path in the product space of the parameter t and the variable vector u from which we can approximately know how a Kuhn-Tucker's stationary solution to the problem moves as the parameter t changes from 0 to any given nonnegative number.