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Journal of Convex Analysis 28 (2021), No. 3, 983--1014
Copyright Heldermann Verlag 2021

Projecting onto Intersections of Halfspaces and Hyperplanes

Hui Ouyang
Dept. of Mathematics, University of British Columbia, Kelowna B.C., Canada

It is well-known that the sequence of iterations of the composition of projections onto closed affine subspaces converges linearly to the projection onto the intersection of the affine subspaces when the sum of the corresponding linear subspaces is closed. Inspired by this, we systematically study the relation between the projection onto intersection of halfspaces and hyperplanes, and the composition of projections onto halfspaces and hyperplanes. In addition, as by-products, we provide the Karush-Kuhn-Tucker conditions for characterizing the optimal solution of convex optimization with finitely many equality and inequality constraints in Hilbert spaces and construct an explicit formula for the projection onto the intersection of hyperplane and halfspace.

Keywords: Projection, halfspace, hyperplane, best approximation mapping, linear convergence, Karush-Kuhn-Tucker conditions, convex optimization.

MSC: 47N10, 41A50 , 65K10; 65K05, 90C25, 90C90.

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