A note on the second-order convergence of optimization algorithms using barrier functions

It has long been known that barrier algorithms for constrained optimization can produce a sequence of iterates converging to a critical point satisfying weak second-order necessary optimality conditions, when their inner iterations ensures that second-order necessary conditions hold at each barrier minimizer. We show that, despite this, strong second-order necessary conditions may fail to be attained at the limit, even if the barrier minimizers satisfy second-order sufficient optimality conditions. Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 OQX, England, EU Email : n.gould@rl.ac.uk Current reports available by anonymous ftp from joyous-gard.cc.rl.ac.uk (internet 130.246.9.91) in the directory “pub/reports” . Department of Mathematics, FacultCs Universitaires ND de la Paix, 61, rue de Bruxelies, B-5000 Namur, Belgium, EU Email : pht@math.fundp.ac.be Current reports available by anonymous ftp from thales.math.fundp.ac.be (internet 138.48.20.102) in the directory “pub/reports”. Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxon OX11 OQX October 10, 1997.

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