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The linearized constraints may be inconsistent, for example when $-c(x_k)$ is not in the image of a linear map defined by $A(x_k)$. Even if feasible, the quadratic objective may have negative curvature along its feasible tangent space, making it unbounded below and leaving no local minimizer.

Fletcher's Sℓ1QP replaces the hard linearized constraints by an L1 exact penalty and bounds the step inside a trust region. One usual form isThe zero step is always admissible. The trust region is compact, so a minimum exists even when the model Hessian matrix is indefinite; the constraint penalty measures infeasibility rather than making the subproblem impossible to solve. Adjusting its weight promotes feasibility, while the radius keeps the local model within a controlled neighbourhood. Introducing nonnegative positive- and negative-part slack variables converts the absolute values to an equivalent smooth quadratic program with box constraints.^[ref]

Roger Fletcher's method uses an L1 exact penalty on linearized constraints and a bounded trust region. It therefore admits a model minimizer even when hard linearized constraints are inconsistent or the quadratic model is indefinite.^[ref]