Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization

Reyes, Juan C.; Loayza, E.; Merino, Pedro

doi:10.1007/s10589-017-9891-z

Cited by 5 publications

(1 citation statement)

References 25 publications

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“…In practice, the SSN algorithm has been shown to achieve good performance, despite the requirement of solving a large system of equations. Alternatively, if the problem is first discretized, it can be solved by descent methods that will reduce the size of the system, see Reference 20.…”

Section: Numerical Solution and Examplesmentioning

confidence: 99%

Sparse optimal control of Timoshenko's beam using a locking‐free finite element approximation

Hernández,

Merino

2023

Optim Control Appl Methods

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This paper addresses the optimal control problem with sparse controls of a Timoshenko beam, its numerical approximation using the finite element method, and the numerical solution via nonsmooth methods. Incorporating sparsity‐promoting terms in the cost function is practically useful for beam vibration models and results in the localization of the control action that facilitates the placement of actuators or control devices. We consider two types of sparsity‐inducing penalizers: the ‐norm and the ‐penalizer, which measures function support. We analyze discretized problems utilizing linear finite elements with a locking‐free scheme to approximate the states and adjoint states. We confirm that this approximation has the looking‐free property required to achieve a linear convergence linear order of approximation for control case and depending on the set of switching points in the controls. This is similar to the purely ‐norm penalized optimal control, where the order of approximation is independent of the thickness of the beam.

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Section: Numerical Solution and Examplesmentioning

confidence: 99%

Sparse optimal control of Timoshenko's beam using a locking‐free finite element approximation

Hernández,

Merino

2023

Optim Control Appl Methods

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Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite‐dimensional control space

Merino

Nenjer

2020

Optim Control Appl Methods

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Summary In this work, we derive an a priori error estimate of order h2|log(h)| for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box‐constrains on the control are considered and finitely many pointwise state‐constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.

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An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems

Zhang

Sun

et al. 2018

Math. Program.

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The sparse group Lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level. In this paper, we develop an efficient augmented Lagrangian method for large-scale non-overlapping sparse group Lasso problems with each subproblem being solved by a superlinearly convergent inexact semismooth Newton method. Theoretically, we prove that, if the penalty parameter is chosen sufficiently large, the augmented Lagrangian method converges globally at an arbitrarily fast linear rate for the primal iterative sequence, the dual infeasibility, and the duality gap of the primal and dual objective functions. Computationally, we derive explicitly the generalized Jacobian of the proximal mapping associated with the sparse group Lasso regularizer and exploit fully the underlying second order sparsity through the semismooth Newton method. The efficiency and robustness of our proposed algorithm are demonstrated by numerical experiments on both the synthetic and real data sets.

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Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization

Cited by 5 publications

References 25 publications

Sparse optimal control of Timoshenko's beam using a locking‐free finite element approximation

Sparse optimal control of Timoshenko's beam using a locking‐free finite element approximation

Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite‐dimensional control space

An efficient Hessian based algorithm for solving large-scale sparse group Lasso problems

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