Primal Superlinear Convergence of Sqp Methods in Piecewise Linear-Quadratic Composite Optimization

Local Convergence Analysis of Augmented Lagrangian Methods for Piecewise Linear-Quadratic Composite Optimization Problems

“…When g is the indicator function of a polyhedral convex set and Θ = IR n , one can observe that this SOSC boils down to that of classical nonlinear programming problems. To demonstrate the importance of this condition in the local convergence analysis of the ALM for (1.1), we recall below a result, obtained recently in [26], showing that it yields an error bound estimate for the KKT system (2.9), which is central to our local convergence analysis in this paper. To this end, recall that, given a constant r > 0, the proximal mapping of a convex function g : IR m → IR is defined by prox rg (x) := argmin…”

“…Note that the SOSC (2.11) is strictly stronger than the error bound estimate (2.14); see [26] for more details. Indeed, it was shown in [26,Theorem 3.6] that the latter error bound estimate is equivalent to the fact that the Lagrange multiplier λ therein is noncritical in sense of [26,Definition 3.1].…”

“…Note that the SOSC (2.11) is strictly stronger than the error bound estimate (2.14); see [26] for more details. Indeed, it was shown in [26,Theorem 3.6] that the latter error bound estimate is equivalent to the fact that the Lagrange multiplier λ therein is noncritical in sense of [26,Definition 3.1]. While this may suggest that the SOSC (2.11) can be replaced with the latter noncriticality assumption to ensure the error bound estimate (2.14), it is not clear whether this can be done for all other results, established in this paper.…”

“…For such a convex function, it is known that for any x ∈ IR m we have prox rg (x) = I + r∂g −1 (x) and ∇e r g(x) = rI + (∂g) −1 −1 (x), (2.13) where I stands for the m × m identity matrix; see [24,Proposition 12.19] and [24,Theorem 2.26] for more details. We are now in a position to state the aforementioned result for the KKT system (2.9), which comes from combining [26,Theorem 3.6] and [26, Proposition 3.9] with [26,Remark 3.8].…”

“…Using the latter quadratic growth condition, we show that the partially augmented problem (1.3) is always solvable and its optimal solution mapping is uniformly isolated calm provided that the penalty parameter ρ therein is sufficiently large. Appealing to an error bound estimate for the KKT system of (1.1), established recently in [26], to an error bound for consecutive iterates of the ALM, obtained in Theorem 5.5, and employing an iterative framework, proposed by Fisher in [5,Theorem 1], to deal with generalized equations with non-isolated solutions, we justify the linear convergence of the primal-dual sequence of the ALM for the composite problem (1.1) when the penalty parameter ρ is sufficiently large.…”