Lagrangian Stability and Global Optimality in Nonconvex Quadratic Minimization Over Euclidean Balls and Spheres

where A is an n × n real symmetric matrix, b ∈ IRn, r is a positive number. If A is positive semi-definite then (Q1) is a convex quadratic problem. In general (Q1) is nonconvex. Problem (Q2), whose feasible domain is a sphere, is always nonconvex even if A is positive semi-definite. These are among the few nonconvex optimization problems which possess a complete characterization of their optimal solutions. These problems play an important role in optimization and numerical analysis ([1], [4], [8], [9], [11], [12], [16]). Golub et al. studied (Q2) from both theoretical and computational viewpoints. In [3] the sensitivity of the solutions to primal problem was discussed.

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