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Robust constrained best approximation with nonconvex constraints H. Mohebi1

· S. Salkhordeh2

Received: 8 August 2019 / Accepted: 28 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract In this paper, we consider the set D of inequalities with nonconvex constraint functions in the face of data uncertainty. We show under a suitable condition that “perturbation property” of the robust best approximation to any x ∈ Rn from the set K˜ := C¯ ∩ D is characterized by the strong conical hull intersection property (strong CHIP) of C¯ and D. The set C is an open convex subset of Rn and the set D is represented by D := {x ∈ Rn : g j (x, v j ) ≤ 0, ∀ v j ∈ V j , j = 1, 2, . . . , m}, where the functions g j : Rn × V j −→ R, j = 1, 2, . . . , m, are continuously Fréchet differentiable that are not necessarily convex, and v j is the uncertain parameter which belongs to an uncertainty set V j ⊂ Rq j , j = 1, 2, . . . , m. This is done by first proving a dual cone characterization of the robust constraint set D. Finally, following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty under the robust nondegeneracy constraint qualification. Given examples illustrate the nature of our assumptions. Keywords Robust best approximation · Robust nondegeneracy Constraint qualification · Robust Slater’s constraint qualification · Lagrange multipliers · Robust constrained best approximation · Strong conical hull intersection property · Perturbation property Mathematics Subject Classification 41A29 · 41A50 · 90C22 · 90C25

1 Introduction Over the years, studies of characterizing the best approximation to any x ∈ Rn from the set C ∩ {x ∈ Rn : g j (x) ≤ 0, j = 1, 2 . . . , m} commonly assume accurate values for the data

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H. Mohebi [email protected] S. Salkhordeh [email protected]

1

Department of Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, P.O. Box: 76169133, 7616914111 Kerman, Iran

2

Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran

123

Journal of Global Optimization

or parameters in the constraint set D0 := {x ∈ Rn : g j (x) ≤ 0, j = 1, 2, . . . , m}, whenever D0 is a closed convex set and has a convex representation in the sense that g j , j = 1, 2, . . . , m, are convex functions [6,8,9,11,12,17,18]. An effective approach to dealing with the data uncertainty is to treat uncertainty as deterministic and describes it in terms of bounded sets. This approach in optimization is known as robust optimization [2,3] and is a complementary approach to stochastic optimization [21] which describes uncertainty in terms of probability distributions. Various characterizations of the perturbation property have been given by using local constraint qualifications such as the strong conical hull intersection property (strong CHIP) of C and D0 at the best approximation [5,9,11–13,17]. These constraint qualificat