A Continuity Result for the Adjusted Normal Cone Operator

Marco Castellani and Massimiliano Giuli ()
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Marco Castellani: Computer Science and Mathematics
Massimiliano Giuli: Computer Science and Mathematics

Journal of Optimization Theory and Applications, 2024, vol. 200, issue 2, No 16, 858-873

Abstract: Abstract The concept of adjusted sublevel set for a quasiconvex function was introduced by Aussel and Hadjisavvas who proved the local existence of a norm-to-weak $$^*$$ ∗ upper semicontinuous base-valued submap of the normal operator associated with the adjusted sublevel set. When the space is finite-dimensional, a globally defined upper semicontinuous base-valued submap is obtained by taking the intersection of the unit sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique does not work in the infinite-dimensional case. We propose a partition of unity technique to overcome this problem in Banach spaces. An application is given to a quasiconvex quasioptimization problem through the use of a new existence result for generalized quasivariational inequalities which is based on the Schauder fixed point theorem.

Keywords: Cone upper semicontinuity; Normal operator; Quasiconvexity; Quasivariational inequality; Quasioptimization; 47H04; 49J40; 49J52; 49J53; 90C26 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-023-02326-w

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