ON INTERIOR LOGARITHMIC SMOOTHING AND STRONGLY STABLE STATIONARY POINTS

作     者：Jongen, Hubertus Th. Rueckmann, Jan-J. 

作者机构：Rhein Westfal TH Aachen Lehrstuhl Math C D-52056 Aachen Germany Univ Birmingham Sch Math Birmingham B15 2TT W Midlands England 

出 版 物：《SIAM JOURNAL ON OPTIMIZATION》 (工业与应用数学会最优化杂志)

年 卷 期：2010年第20卷第5期

页      面：2137-2156页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：logarithmic barrier function interior point approximation smoothing strongly stable stationary point degenerate solution of interior-point methods stationary index 

摘      要：The paper deals with a nonlinear programming problem (P) and, by using a logarithmic barrier function, a parametric family of interior point approximations M(gamma) of its feasible set M, where M(gamma) is described by a single smooth inequality constraint. Assuming that a stationary point (x) over bar of (P) under consideration is strongly stable, it is shown that for all sufficiently small gamma 0 there exists locally around (x) over bar a uniquely determined stationary point x(gamma) of (P(gamma)), where (P(gamma)) is obtained from (P) by substituting M by M(gamma). In particular, x(gamma) is strongly stable, even nondegenerate, and it has the same stationary index as (x) over bar. Furthermore, it turns out that x(gamma) and its uniquely determined Lagrange multiplier mu(gamma) form a solution pair of a corresponding interior-point problem, where (x(gamma), mu(gamma)) depends continuously differentiable (under linear independence constraint qualification (LICQ)) or continuous (under Mangasarian-Fromovitz constraint qualification (MFCQ)) on the parameter. and x(gamma) converges to (x) over bar as gamma - 0. The stationary point (x) over bar might be degenerate and, a priori, no strict complementarity is assumed. Finally, a globalization of this one-to-one correspondence between the stationary points of (P) and (P(gamma)) as well as some further topological properties of M and M. are discussed.