A note on the convergence of an augmented Lagrangian algorithm to second-order stationary points

Andreani, Roberto; Secchin, Leonardo D.

doi:10.5540/03.2018.006.01.0303

Cited by 3 publications

(4 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Indeed, [53] gives an explicit example illustrating that accumulation points of trajectories generated by barrier algorithms will converge to stationary points satisfying the weak second-order necessary condition, but not the strong version. Later, Andreani and Secchin [10] made a small modification in Gould and Toint's counterexample to come to the same conclusion for augmented Lagrangian-type algorithms. Hence, the most we can expect from our method is that it generates points that approximately satisfy the weak second-order necessary optimality conditions.…”

mentioning

confidence: 80%

See 1 more Smart Citation

Hessian barrier algorithms for non-convex conic optimization

Dvurechensky,

Staudigl

2024

Math. Program.

View full text Add to dashboard Cite

A key problem in mathematical imaging, signal processing and computational statistics is the minimization of non-convex objective functions that may be non-differentiable at the relative boundary of the feasible set. This paper proposes a new family of first- and second-order interior-point methods for non-convex optimization problems with linear and conic constraints, combining logarithmically homogeneous barriers with quadratic and cubic regularization respectively. Our approach is based on a potential-reduction mechanism and, under the Lipschitz continuity of the corresponding derivative with respect to the local barrier-induced norm, attains a suitably defined class of approximate first- or second-order KKT points with worst-case iteration complexity $$O(\varepsilon ^{-2})$$ O ( ε - 2 ) (first-order) and $$O(\varepsilon ^{-3/2})$$ O ( ε - 3 / 2 ) (second-order), respectively. Based on these findings, we develop new path-following schemes attaining the same complexity, modulo adjusting constants. These complexity bounds are known to be optimal in the unconstrained case, and our work shows that they are upper bounds in the case with complicated constraints as well. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained non-convex optimization problems.

show abstract

mentioning

confidence: 80%

“…Therefore, conditions (10) and ( 11) both hold. We now check for the complementarity condition ( 14).…”

Section: Per-iteration Analysis and A Bound For The Number Of Iterationsmentioning

confidence: 96%

Hessian barrier algorithms for non-convex conic optimization

Dvurechensky,

Staudigl

2024

Math. Program.

View full text Add to dashboard Cite

show abstract

“…Assim como em PNL, temos que T (x) ̸ ⊃ T BS lin (x) em geral. Por exemplo, considere as restrições −x 3 1…”

Section: Uma Linearização Adequada Do Conjunto Viávelunclassified

“…É bem conhecido no campo da PNL que a condição necessária de segunda ordem forte não pode ser esperada nos pontos limite de algoritmos práticos [3]. Em vez disso, o conceito adequado neste contexto é a condição necessária de segunda ordem fraca, que consiste em relaxar os requisitos em relação a ∇g em (5).…”

Section: Condições De Otimalidade De 2ª Ordem: Forte E Fracaunclassified

Uma condição de otimalidade de segunda ordem para Problemas com Restrição de Cardinalidade

Medeiros,

Ribeiro,

Sachine

et al. 2023

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

View full text Add to dashboard Cite

Resumo. Neste trabalho propusemos duas novas condições necessárias de segunda ordem para o problema de programação matemática com restrições de cardinalidade (cuja sigla, em inglês, é MPCaC), referidas como MPCaC-SSONC e MPCaC-WSONC, que são baseadas no conceito de primeira ordem de M-estacionariedade. Também discutimos condições de qualificação (CQ) necessárias para que a M-estacionariedade de segunda ordem seja válida nos minimizadores. Também propusemos uma CQ de posto constante relaxado especializada (MPCaC-RCRCQ) para otimalidade de segunda ordem em MPCaC. Por fim comparamos esses resultados com trabalhos anteriores que utilizaram diferentes linearizações do conjunto viável e conceitos de estacionariedade.

show abstract