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STABILIZED SEQUENTIAL QUADRATIC PROGRAMMING FOR OPTIMIZATION AND A STABILIZED NEWTON-TYPE METHOD FOR VARIATIONAL PROBLEMS WITHOUT CONSTRAINT QUALIFICATIONS
, 2007
"... The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence ..."
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Cited by 24 (14 self)
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The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the secondorder sufficient condition for optimality (SOSC) and the Mangasarian-Fromovitz constraint qualification, or under the strong second-order sufficient condition for optimality (in that case, without constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming SOSC only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush-Kuhn-Tucker systems for variational problems.
On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers
- MATHEMATICAL PROGRAMMING
, 2009
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Examples of dual behaviour of Newton-type methods on optimization problems with degenerate constraints
- Computational Optimization and Applications
"... discuss possible scenarios of behaviour of the dual part of sequences generated by primal-dual Newton-type methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so- ..."
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Cited by 16 (10 self)
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discuss possible scenarios of behaviour of the dual part of sequences generated by primal-dual Newton-type methods when applied to optimization problems with nonunique multipliers associated to a solution. Those scenarios are: (a) failure of convergence of the dual sequence; (b) convergence to a so-called critical multiplier (which, in particular, violates some second-order sufficient conditions for optimality), the latter appearing to be a typical scenario when critical multipliers exist; (c) convergence to a noncritical multiplier. The case of mathematical programs with complementarity constraints is also discussed. We illustrate those scenarios with examples, and discuss consequences for the speed of convergence. We also put together a collection of examples of optimization problems with constraints violating some standard constraint qualifications, intended for preliminary testing of existing algorithms on degenerate problems, or for developing special new algorithms designed to deal with constraints degeneracy. Keywords Degenerate constraints · Second-order sufficiency · Newton method · SQP
SHARP PRIMAL SUPERLINEAR CONVERGENCE RESULTS FOR SOME NEWTONIAN METHODS FOR CONSTRAINED OPTIMIZATION
, 2009
"... As is well known, superlinear or quadratic convergence of the primal-dual sequence generated by an optimization algorithm does not, in general, imply superlinear convergence of the primal part. Primal convergence, however, is often of particular interest. For the sequential quadratic programming (SQ ..."
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Cited by 9 (8 self)
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As is well known, superlinear or quadratic convergence of the primal-dual sequence generated by an optimization algorithm does not, in general, imply superlinear convergence of the primal part. Primal convergence, however, is often of particular interest. For the sequential quadratic programming (SQP) algorithm, local primal-dual quadratic convergence can be established under the assumptions of uniqueness of the Lagrange multiplier associated to the solution and the second-order sufficient condition. At the same time, previous primal superlinear convergence results for SQP required to strengthen the first assumption to the linear independence constraint qualification. In this paper, we show that this strengthening of assumptions is actually not necessary. Specifically, we show that once primal-dual convergence is assumed or already established, for primal superlinear rate one only needs a certain error bound estimate. This error bound holds, for example, under the second-order sufficient condition, which is needed for primal-dual local analysis in any case. Moreover, in some situations even second-order sufficiency can be relaxed to the weaker assumption that the multiplier in question is noncritical. Our study is performed for a rather general perturbed SQP framework, which covers in addition to SQP and quasi-Newton SQP some other algorithms as well. For example, as a by-product,
A NOTE ON UPPER LIPSCHITZ STABILITY, ERROR BOUNDS, AND CRITICAL MULTIPLIERS FOR LIPSCHITZ-CONTINUOUS KKT SYSTEMS
, 2012
"... We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qual ..."
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Cited by 8 (5 self)
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We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.
Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints
, 2009
"... We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismo ..."
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Cited by 7 (5 self)
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We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismooth and BD-regular at the solution under reasonable assumptions. Thus, fast local convergence can be obtained by applying the semismooth Newton method. Moreover, it turns out that the squared residual of the Lagrange system is continuously differentiable (even though the system itself is not), which opens the way for a natural globalization of the local algorithm.
Stabilized SQP revisited
- MATH. PROGRAM., SER. A
, 2010
"... The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key ..."
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Cited by 3 (2 self)
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The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91–124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47–73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover,
Local convergence of the method of multipliers for variational and optimization problems under the sole noncriticality assumption. August 2013 (Revised January 2014). Available at http://pages.cs.wisc.edu/˜solodov/solodov.html
"... ABSTRACT We present local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange mult ..."
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Cited by 3 (2 self)
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ABSTRACT We present local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local superlinear convergence is established under the appropriate control of the penalty parameter values. For optimization problems, we demonstrate in addition local linear convergence for sufficiently large fixed penalty parameters. Both exact and inexact versions of the method are considered. Contributions with respect to previous state-of-the-art analyses for equality-constrained problems consist in the extension to the variational setting, in using the weaker noncriticality assumption instead of the usual second-order sufficient optimality condition, and in relaxing the smoothness requirements on the problem data. In the context of optimization problems, this gives the first local convergence results for the augmented Lagrangian method under the assumptions that do not include any constraint qualifications and are weaker than the second-order sufficient optimality condition. We also show that the analysis under the noncriticality assumption cannot be extended to the case with inequality constraints, unless the strict complementarity condition is added (this, however, still gives a new result).
An active-set Newton method for mathematical programs with complementarity constraints
- SIAM J. on Optimization
"... Abstract. For a mathematical program with complementarity constraints (MPCC), we propose an active-set Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCC-LICQ) and the standard second-order sufficient condition (SOSC ..."
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Cited by 3 (2 self)
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Abstract. For a mathematical program with complementarity constraints (MPCC), we propose an active-set Newton method, which has the property of local quadratic convergence under the MPCC linear independence constraint qualification (MPCC-LICQ) and the standard second-order sufficient condition (SOSC) for optimality. Under MPCC-LICQ, this SOSC is equivalent to the piecewise SOSC on branches of MPCC, which is weaker than the special MPCC-SOSC often employed in the literature. The piecewise SOSC is also more natural than MPCC-SOSC because, unlike the latter, it has an appropriate second-order necessary condition as its counterpart. In particular, our assumptions for local quadratic convergence are weaker than those required by standard SQP when applied to MPCC and are equivalent to assumptions required by piecewise SQP for MPCC. Moreover, each iteration of our method consists of solving a linear system of equations instead of a quadratic program. Some globalization issues of the local scheme are also discussed, and illustrative examples and numerical experiments are presented.
