On Lipschitz Continuity of Projections onto Polyhedral Moving Sets
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On Lipschitz Continuity of Projections onto Polyhedral Moving Sets · Krzysztof E. Rutkowski2,3
Ewa M. Bednarczuk1
© The Author(s) 2020
Abstract In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left- and right-hand sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied. Keywords Lipschitzness of projection · Relaxed constant rank constraint qualification condition · Lipschitz-likeness · Graphical subdifferential mapping Mathematics Subject Classification 47N10 · 49J52 · 49J53 · 49K40 · 90C31
1 Introduction Continuity of metric projections of a given v¯ onto moving subsets have already been investigated in a number of instances. In the framework of Hilbert spaces, the projection ¯ of v¯ onto closed convex sets C, C , i.e., solutions to optimization problems PC (v)
minimize z − v ¯ subject to z ∈ C,
B
(Proj)
Krzysztof E. Rutkowski [email protected] Ewa M. Bednarczuk [email protected]
1
Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland
2
Faculty of Mathematics and Natural Sciences, School of Exact Sciences, Cardinal Stefan Wyszy´nski University, Warsaw, Poland
3
Systems Research Institute of the Polish Academy of Sciences, Warsaw, Poland
123
Applied Mathematics & Optimization
are unique and Hölder continuous with the exponent 1/2 in the sense that there exists a constant H > 0 with ¯ − PC (v) ¯ ≤ H [dρ (C, C )]1/2 , PC (v) where dρ (·, ·) denotes the bounded Hausdorff distance (see [2] and also [8, Example 1.2]). In the case where the sets, on which we project a given v, ¯ are solution sets to systems of equations and inequalities, the problem Pr oj is a special case of a general parametric problem minimize ϕ0 ( p, x) subject to ϕi ( p, x) = 0 i ∈ I1 , ϕi ( p, x) ≤ 0 i ∈ I2 ,
(Par)
where x ∈ H, p ∈ D ⊂ G, H-Hilbert space, G-metric space, ϕi : D × H → R, i ∈ {0} ∪ I1 ∪ I2 . There exist numerous results concerning continuity of solutions to problem (Par) in finite dimensional settings, see e.g., [6,18,24,27] and the references therein. In a recent paper by Mordukhovich and Nghia [23], in the finite-dimensional setting, the Hölderness and the Lipschitzness of the local minimizers to problem Par with I1 = ∅ are investigated for C 2 functions ϕi , i ∈ I2 , under Mangasarian– Fromowitz (MFCQ) and constant rank (CRCQ) constraints qualifications. Let H be a Hilbert space and let D ⊂ G be a nonempty set of a normed space G, p ∈ D and v ∈ H. We consider the norm topology induced on D by the topology of space G, i.e., U is an open set in D if U = D ∩ U , where U is open in G (see e.g. [11]). We consider the following parametric optimization problem 1 x − v2 , x∈H 2 min
x | gi ( p) = f i ( p), i ∈ I1 , , subject to x ∈ C( p) := x ∈ H
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