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A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

Gabriel Haeser (ghaeser***at***ime.usp.br)

Abstract: We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives a new second-order optimality condition without constraint qualifications stronger than previous ones associated with global convergence of algorithms. We prove that second-order variants of augmented Lagrangian and interior point methods generate sequences satisfying our optimality condition. We present also a companion minimal constraint qualification, weaker than the ones known for second-order methods, that ensures usual global convergence results to a classical second-order stationary point. Finally, our optimality condition naturally suggests definition of second-order stationarity suitable for the computation of iteration complexity bounds and for the definition of stopping criteria.

Keywords: second-order optimality conditions, complementarity, global convergence, constraint qualifications.

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )


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Entry Submitted: 10/28/2016
Entry Accepted: 10/28/2016
Entry Last Modified: 04/11/2018

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