Local Structure of Feasible Sets in Nonlinear Programming. Part I. Regularity.

Nonlinear programming is concerned with the problem of minimizing a function, often fairly smooth, over a set the so-called feasible set described by nonlinear inequality and equality constraints, as well as perhaps some bounds or other restrictions on the variables. Problems of nonlinear programming arise in statistics, in chemical engineering, in economics, and in many other areas. Certain basic conditions, called optimality conditions, must be satisfied by any candidate for a solution of such a problem, provided that the constraints satisfy a reasonable regularity condition. These conditions describe the relationship of the derivatives of the function being minimized to the derivatives of the constraint functions and the set over which the minimization is being done. They form the basis for most numerical algorithms for solving such problems. Given a feasible point for a nonlinear programming problem, the authors investigate the structure of the feasible set near that point. Under the constraint qualification called regularity, it is shown how to compute the tangent cone to the feasible set, and to produce feasible arcs with prescribed first and second derivatives. In order to carry out these constructions, a particular way of representing the feasible set as a system of equations with constrained variables is particularly useful. Also given are fairly short proofs of the first-order and second-order necessary optimality conditions in very general forms, using the arc constructions mentioned above.