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A trust-region SQP-filter algorithm of the type introduced by Fletcher and Leyffer [Math. Program., 91 (2002), pp. 239-269] that decomposes the step into its normal and tangential components allows for an approximate solution of the quadratic subproblem and incorporates the safeguarding tests described in Fletcher, Leyffer, and Toint [On the Global Convergence of an SLP-Filter Algorithm, Technical Report 98/13, Department of Mathematics, University of Namur, Namur, Belgium, 1998; On the Global Convergence of a Filter-SQP Algorithm, Technical Report 00/15, Department of Mathematics, University of Namur, Namur, Belgium, 2000] is considered. It is proved that, under reasonable conditions and for every possible choice of the starting point, the sequence of iterates has at least one first-order critical accumulation point.
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