This paper concerns the
problem of minimizing a convex function subject to nonnegativity constraints, an
associated nonlinear complementarity problem, and a new approach to solving these
problems. The approach involves solving a sequence of nicer problems which
approximate the given ones better and better, and our focus is on certain natural
“trajectories” of solutions of the approximating problems. Existence, characterization,
and continuous dependence of the solutions is obtained by establishing a complete
analogue of Moreau’s Proximation Theorem. From this analogue also follow two new
facts about the geometric nature of the graphs of subdifferentials in Rn,
as well as new information about monotone conjugacy for coordinatewise
nondecreasing convex functions on the nonnegative orthant. The largest
part of the paper is then devoted to developing a number of rather strong
properties of the solution trajectories, particularly as regards the nature of their
convergence. Perhaps the most striking property is that these trajectories will
locate a maximal strictly complementary solution which, furthermore, can be
arranged to have a certain prescribed strong Pareto optimality property. The
arithmetic-geometric mean inequality enters decisively at several key points, and the
proofs generally rely strongly upon the techniques of finite-dimensional convex
analysis.
