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Technical Report RAL-TR-97-055 A Note on the Second-Order Convergence of Optimization Algorithms Using Barrier Functions N I M Gould and Ph 1 Toint October 1997 COUNCIL FOR THE CENTRAL LABORATORY OF THE RESEARCH COUNCILS 0 Council for the Central Laboratory of the Research Councils 1997 Enquiries about copyright, reproduction and requests for additional copies of this report should be addressed to: The Central Laboratory of the Research Councils Library and Information Services Rutherford Appleton Laboratory C h i I ton Didcot Oxfordshire OX1 1 OQX Tel: 01 235 445384 E- ma i I I i bra r y 8 rl . ac. u k Fox: 01 235 446403 ISSN 1358-6254 Neither the Council nor the Laboratory accept any responsibility for loss or damage arising from the use of information contained in any of their reports o r in any communication about their tests or investigations. RAL-TR-97-055 A note on the second-order convergence of optimization algorithms using barrier functions Nicholas I. M. Gould’2 and Philippe L. T ~ i n t ~ ~ ABSTRACT 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 1) 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 report