Practical Augmented Lagrangian Methods for Constrained Optimization

Birgin, Ernesto G.; Martnez, Jos Mario

doi:10.1137/1.9781611973365

Cited by 288 publications

(366 citation statements)

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“…We note that the theoretical possibility of convergence to infeasible points is something standard for augmented Lagrangian methods [5], and thus carries over also to related techniques, including what is presented here. Specifically, when all iterations from some point on are of Aug-L type, every primal accumulation point is stationary for the infeasibility minimizing problem minimize h(x) 2 + max{0, g(x)} 2 , x ∈ R n , and if this point is feasible and satisfies some weak constraints qualifications, then it is necessarily stationary in the original problem (1).…”

Section: ⊓ ⊔mentioning

confidence: 89%

“…In the case of a linesearch iteration, if it gives an acceptable approximate stationary point of the augmented Lagrangian, i.e., (13) below does not hold, we accept this point and update the dual iterates, the penalty parameter, and the stationarity tolerance the same way as the usual Aug-L methods [5] do, and proceed to the next outer iteration. Otherwise, we consider the linesearch step as an inner iteration within the process of solving the current Aug-L subproblem (i.e., within minimizing the augmented Lagrangian for fixed dual variables and fixed penalty parameter).…”

Section: The Algorithm and Its Convergence Propertiesmentioning

confidence: 99%

“…Our algorithm uses the boundsλ min <λ max andμ max > 0 to safeguard the dual iterates, as does the ALGENCAN solver [1], for example (see [2,Algorithm 3.1] and also [5]). …”

Section: The Algorithm and Its Convergence Propertiesmentioning

confidence: 99%

“…However, any subsequence with this property must be generated by Aug-L iterations (at least from some point on), and in particular, the iterative process then reduces to the augmented Lagrangian algorithm. More specifically, to the version used in the ALGENCAN [1] ( [2, Algorithm 3.1]; see also [5]). Therefore, the tendency of Algorithm 1 to converge to infeasible points cannot be stronger than that of ALGENCAN.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…Essentially, we combine sSQP with the usual augmented Lagrangian algorithm (Aug-L) [4,5]. Roughly speaking, we employ sSQP as inner iterations for solving the subproblems of the outer algorithm (in this sense, this is the common feature with [11,[17][18][19]).…”

Section: Introductionmentioning

confidence: 99%

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Combining stabilized SQP with the augmented Lagrangian algorithm

2015

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For an optimization problem with general equality and inequality constraints, we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type for approximately solving subproblems of the augmented Lagrangian method. The motivation is to take advantage of the well-known robust behavior of the augmented Lagrangian algorithm, including on problems with degenerate constraints, and at the same time try to reduce the overall algorithm locally to sSQP (which gives fast local convergence rate under weak assumptions). Specifically, the algorithm first verifies whether the primal-dual sSQP step (with unit stepsize) makes good progress towards decreasing the violation of optimality conditions for the original problem, and if so, makes this step. Otherwise, the primal part of the sSQP direction is used for linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate fixed for the time being. The overall algorithm has reasonable global convergence guarantees, and inherits strong local convergence rate properties of sSQP under the same weak assumptions. Numerical results on degenerate problems and comparisons with some alternatives are reported.

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Section: ⊓ ⊔mentioning

confidence: 89%

Section: The Algorithm and Its Convergence Propertiesmentioning

confidence: 99%

“…Our algorithm uses the boundsλ min <λ max andμ max > 0 to safeguard the dual iterates, as does the ALGENCAN solver [1], for example (see [2,Algorithm 3.1] and also [5]). …”

Section: The Algorithm and Its Convergence Propertiesmentioning

confidence: 99%

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Combining stabilized SQP with the augmented Lagrangian algorithm

2015

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Decomposition of control and optimization problems by network structure: Concepts, methods, and inspirations from biology

2019

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Topology optimization of continuum structures with stress constraints and uncertainties in loading

Silva

Beck

Cardoso

2017

Numerical Meth Engineering

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SummaryTopology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first-order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient-based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.

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Practical Augmented Lagrangian Methods for Constrained Optimization

Cited by 288 publications

References 0 publications

Combining stabilized SQP with the augmented Lagrangian algorithm

Combining stabilized SQP with the augmented Lagrangian algorithm

Decomposition of control and optimization problems by network structure: Concepts, methods, and inspirations from biology

Topology optimization of continuum structures with stress constraints and uncertainties in loading

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