Chapter 3 Quadratic Programming

Optimization I; Chapter 3 76
Theorem 3.10 Further properties of the logarithmic barrier function
Assume that F int ≠ ∅ and that x ∗ is a local solution of (3.62a)-(3.62b) with
multiplier µ ∗ such that the KKT conditions are satisfied.
Suppose further that the LICQ, strict complementarity, and the second order
sufficient optimality conditions fold true at (x ∗ , µ ∗ ). Then there holds:
(i) For sufficiently small β > 0 there exists a strict local minimizer x(β) of
B (β) (x) such that the function x(β) is continuously differentiable in some neighborhood
of x ∗ and x(β) → x ∗ as β → 0.
(ii) For the perturbed complementarity z (β) there holds:
z (β) → µ ∗ as β → 0 . (3.78)
(iii) For sufficiently small β > 0, the Hessian ∇ xx B (β) (x) is positive definite.
Proof. We refer to M.H. Wright; Interior methods for constrained optimization.
Acta Numerica, 341-407, 1992.
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