Active Set and Interior Methods for Nonlinear Optimization

We discuss several fundamental questions concerning the problem of minimizing a nonlinear function subject to a set of inequality constraints. We begin by asking: What makes the problem intrinsically diicult to solve, and which characterizations of the solution make its solution more tractable? This leads to a discussion of two important methods of solution: active set and interior points. We make a critical assessment of the two approaches, and describe the main issues that must be resolved to make them eeective in the solution of very large problems. The most important open problem in nonlinear optimization is the solution of large constrained problems of the form minimize f(x) subject to h(x) = 0 (1) g(x) 0; where the functions f : R n ! R; h : R n ! R m and g : R n ! R t are assumed to be smooth. Assuming that certain regularity assumptions hold, the solution of (1) is characterized by the Karush-Kuhn-Tucker conditions 4]. They state that any solution x must satisfy the system rf(x) + A h (x) h + A g (x) g = 0 (2)