We consider global convergence properties of the augmented Lagrangian methods on problems with degenerate constraints, with a special emphasis on mathematical programs with complementarity constraints (MPCC). In the general case, we show convergence to stationary points of the problem under an error bound condition for the feasible set (which is weaker than constraint qualifications), assuming ...

We study subdifferential conditions of the calmness property for multifunctions representing convex constraint systems in a Banach space. Extending earlier work in finite dimensions [R. Henrion and J. Outrata, J. Math. Anal. Appl., 258 (2001), pp. 110–130], we show that, in contrast to the stronger Aubin property of a multifunction (or metric regularity of its inverse), calmness can be ensured ...

We give an alternative formulation for the so-called closed cone constraint qualification (CCCQ) related to a convex optimization problem in Banach spaces recently introduced in the literature. This new formulation allows to prove in a simple way that (CCCQ) is weaker than some generalized interiorpoint constraint qualifications given in the past. By means of some insights from the theory of co...

We consider a (finite or infinite) family of closed convex sets with nonempty intersection in a normed space. A property relating their epigraphs with their intersection’s epigraph is studied, and its relations to other constraint qualifications (such as the linear regularity, the strong CHIP, and Jameson’s (G)-property) are established. With suitable continuity assumption we show how this prop...

This paper is based on the author’s thesis, “On duality theorems for quasiconvex programming”. In this paper, we investigate duality theorems for quasiconvex programming as generalizations of results in convex programming, and consists of three topics. The first topic is about quasiconjugates and polar sets. The second is about three types of set containment characterizations. The third is abou...

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constrai...

The constant rank constraint qualification, introduced by Janin in 1984 for nonlinear programming, has been extensively used sensitivity analysis, global convergence of first- and second-order algorithms, computing the directional derivative value function. In this paper we discuss naive extensions rank-type qualifications to cone programming semidefinite which are based on Approximate-Karush–K...

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min-max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min-max duality, and show some examples at which such duality result...

We investigate ill-posed semidefinite programming problems for which Slater’s constraint qualifications fail, and propose a new reliable termination criterium dealing with such problems. This criterium is scale-independent and provides verified forward error bounds for the true optimal value, where all rounding errors due to floating point arithmetic are taken into account. It is based on a bou...