Properties of updating methods for the multipliers in augmented Lagrangians

Glad, Torkel

doi:10.1007/bf00933239

Cited by 42 publications

(22 citation statements)

References 17 publications

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“…Since equations (20), (21) are equivalent to the Lagrange system of the sSQP subproblem, (ξ j , η j ) is also the unique stationary point of this subproblem. Sufficient positive definiteness of the matrix (i.e., the fulfillment of (19)) can be achieved by modifying it in the process of its Cholesky factorization [36,Section 3.4], if this is the approach for computing the solution ξ j of the linear system (21). Then η j is given by the explicit formula (20).…”

Section: The Algorithm and Its Convergence Propertiesmentioning

confidence: 99%

“…Proposition 2 Let (ξ 1 , η 1 ) be the unique solution of the system (20), (21) η 2 ) be the unique solution of the system (35), (36).…”

Section: On the Relations Between Algorithm 1 And The Primal-dual Sqpmentioning

confidence: 99%

“…Thus, using sSQP for solving the Aug-L subproblems would seem to be a natural approach, which can be expected to be effective. In fact, the idea of combining some stabilized Newtonian scheme with the Aug-L method goes back at least to [21]; see also [4]. Despite that we use the usual augmented Lagrangian and not the more involved primal-dual augmented Lagrangian as in [17][18][19], it is interesting that in the equality-constrained case there are some close connections between our method and the algorithms in those works; see Section 3 below.…”

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: The Algorithm and Its Convergence Propertiesmentioning

confidence: 99%

“…Proposition 2 Let (ξ 1 , η 1 ) be the unique solution of the system (20), (21) η 2 ) be the unique solution of the system (35), (36).…”

Section: On the Relations Between Algorithm 1 And The Primal-dual Sqpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Combining stabilized SQP with the augmented Lagrangian algorithm

2015

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“…This Newton method was studied, e.g., in [22]. In particular, the multiplier update rule (2.11) corresponds to [22, relation (19)].…”

Section: The Generic Algorithm Given the Current Primal Iterate Xmentioning

confidence: 99%

A Truncated SQP Method Based on Inexact Interior-Point Solutions of Subproblems

Izmailov¹,

Solodov²

2010

SIAM J. Optim.

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Abstract. We consider sequential quadratic programming (SQP) methods applied to optimization problems with nonlinear equality constraints and simple bounds. In particular, we propose and analyze a truncated SQP algorithm in which subproblems are solved approximately by an infeasible predictor-corrector interior-point method, followed by setting to zero some variables and some multipliers so that complementarity conditions for approximate solutions are enforced. Verifiable truncation conditions based on the residual of optimality conditions of subproblems are developed to ensure both global and fast local convergence. Global convergence is established under assumptions that are standard for linesearch SQP with exact solution of subproblems. The local superlinear convergence rate is shown under the weakest assumptions that guarantee this property for pure SQP with exact solution of subproblems, namely, the strict Mangasarian-Fromovitz constraint qualification and second-order sufficiency. Local convergence results for our truncated method are presented as a special case of the local convergence for a more general perturbed SQP framework, which is of independent interest and is applicable even to some algorithms whose subproblems are not quadratic programs. For example, the framework can also be used to derive sharp local convergence results for linearly constrained Lagrangian methods. Preliminary numerical results confirm that it can be indeed beneficial to solve subproblems approximately, especially on early iterations. 1. Introduction. In this paper, we are concerned with truncated and perturbed versions of sequential quadratic programming (SQP) methods [5,27] for constrained optimization. We refer to an algorithm as a truncated SQP method if the solver for the SQP subproblem may be terminated early, producing an inexact solution. This is, in fact, a special case of the more general class of perturbed SQP methods, which we define as any approach where the iterates can be viewed, perhaps a posteriori, as approximate solutions to relevant SQP subproblems. We thus regard truncation as a special way of inducing perturbations. Our local convergence analysis allows other forms of perturbations as well and is also applicable to some methods that are not modifications of SQP as such. One example is the linearly constrained (augmented) Lagrangian method [42,39,18,33].We shall consider problems with equality constraints and simple bounds:

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“…Local g-superlinear convergence in (x, A) of the SQP augmented Lagrangian BFGS and DFP secant methods was established by Han [15], Tapia [19] and Glad [14].…”

Section: Introductionmentioning

confidence: 99%

On Secant Updates for Use in General Constrained Optimization

Tapia¹

1988

Mathematics of Computation

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Abstract.In this paper we present two new classes of successive quadratic programming (SQP) secant methods for the equality-constrained optimization problem. One class of methods uses the SQP augmented Lagrangian formulation, while the other class uses the SQP Lagrangian formulation.We demonstrate, under the standard assumptions, that in both cases the BFGS and DFP versions of the algorithm are locally q-superlinearly convergent. To our knowledge this is the first time that either local or (j-superlinear convergence has been established for an SQP Lagrangian secant method which uses either the BFGS or DFP updating philosophy and assumes no more than the standard assumptions. Since the standard assumptions do not require positive definiteness of the Hessian of the Lagrangian at the solution, it is no surprise that our BFGS and DFP updates possess the hereditary positive definiteness property only on a proper subspace.

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Properties of updating methods for the multipliers in augmented Lagrangians

Cited by 42 publications

References 17 publications

Combining stabilized SQP with the augmented Lagrangian algorithm

Combining stabilized SQP with the augmented Lagrangian algorithm

A Truncated SQP Method Based on Inexact Interior-Point Solutions of Subproblems

On Secant Updates for Use in General Constrained Optimization

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