Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification. (English) Zbl 0920.90119

Mangasarian, O. L.

Math. Program. 83, No. 2 (A), 187-194 (1998).

Summary: A global error bound is given on the distance between an arbitrary point in the \(n\)-dimensional real space \(\mathbb{R}^n\) and its projection on a nonempty convex set determined by \(m\) convex, possibly nondifferentiable, inequalities. The bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong Slater constraint qualification (CQ) which implies the ordinary Slater CQ. A very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary Slater CQ is given first and then used to derive the principal global error bound.

MSC:

90C25

Convex programming

Keywords:

convex inequalities; global error bound; nondifferentiable, inequalities; strong Slater constraint qualification