A Newton’s method for the continuous quadratic knapsack problem

Cominetti, Roberto; Mascarenhas, Walter F.; Silva, Paulo J. S.

doi:10.1007/s12532-014-0066-y

Cited by 30 publications

(41 citation statements)

References 25 publications

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“…The recent method of Cominetti et al [2014] updates the multiplier by taking a semi-smooth Newton step in the direction of λ * :…”

Section: Overview Of Algorithmsmentioning

confidence: 99%

“…These modifications also prevent the iterates from cycling. The Newton algorithm [Cominetti et al 2014] and the variable fixing algorithm [Kiwiel 2008] are closely related. Both algorithms employ the variable fixing operation, and the iterate λ k has the property that λ k ∈ (α k−1 , β k−1 ), where…”

Section: Overview Of Algorithmsmentioning

confidence: 99%

“…For the variable fixing algorithm, this is a consequence of the formula (14)-see Lemma 4.1 in Kiwiel [2008]. For the Newton scheme in Cominetti et al [2014], this property is ensured by replacing the Newton iterate by a secant iterate when the Newton iterate is not contained in (α k−1 , β k−1 ).…”

Section: Overview Of Algorithmsmentioning

confidence: 99%

“…Currently, algorithms that are able to take advantage of a good starting guess include the secant-based algorithm and the semi-smooth Newton method. In Cominetti et al [2014], numerical comparisons were made between algorithms for solving sequences of projection problems arising in Support Vector Machine applications. When hot starts are allowed, the Newton method was shown to be faster than the other methods.…”

Section: Introductionmentioning

confidence: 99%

“…Section 5 presents our heap-based algorithm for solving Problem (3). Finally, Section 6 compares the performance of our hybrid NAPHEAP algorithm to the Newton method from Cominetti et al [2014] using randomly generated test problems. The structure of the dual function is investigated in an effort to understand the implications of numerical results based on randomly generated problems.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

Davis

Hager

Hungerford

2016

ACM Trans. Math. Softw.

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This article considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP has been developed to solve this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in the direction of the solution. The starting guess is computed using the variable fixing method or is supplied by the user. A key innovation in our algorithm is the implementation of a heap data structure for storing the break points of the dual function and computing the solution of the dual problem. Also, a new version of the variable fixing algorithm is developed that is convergent even when the objective Hessian is not strictly positive definite. The hybrid algorithm NAPHEAP that uses a Newton-type method (variable fixing method, secant method, or Newton's method) to bracket a root, followed by a heap-based monotone break point search, can be faster than a Newton-type method by itself, as demonstrated in the numerical experiments.

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“…The recent method of Cominetti et al [2014] updates the multiplier by taking a semi-smooth Newton step in the direction of λ * :…”

Section: Overview Of Algorithmsmentioning

confidence: 99%

Section: Overview Of Algorithmsmentioning

confidence: 99%

Section: Overview Of Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

Davis

Hager

Hungerford

2016

ACM Trans. Math. Softw.

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A Primal-Dual Partial Inverse Algorithm for Constrained Monotone Inclusions: Applications to Stochastic Programming and Mean Field Games

et al. 2023

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In this paper, we provide a generalization of the forward-backward splitting algorithm for minimizing the sum of a proper convex lower semicontinuous function and a differentiable convex function whose gradient satisfies a locally Lipschitztype condition. We prove the convergence of our method and derive a linear convergence rate when the differentiable function is locally strongly convex. We recover classical results in the case when the gradient of the differentiable function is globally Lipschitz continuous and an already known linear convergence rate when the function is globally strongly convex. We apply the algorithm to approximate equilibria of variational mean field game systems with local couplings. Compared with some benchmark algorithms to solve these problems, our numerical tests show similar performances in terms of the number of iterations but an important gain in the required computational time.

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Fast projection onto the simplex and the $$\pmb {l}_\mathbf {1}$$ l 1 ball

2015

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A new algorithm is proposed to project, exactly and in finite time, a vector of arbitrary size onto a simplex or an ℓ 1-norm ball. It can be viewed as a Gauss-Seidel-like variant of Michelot's variable fixing algorithm; that is, the threshold used to fix the variables is updated after each element is read, instead of waiting for a full reading pass over the list of non-fixed elements. This algorithm is empirically demonstrated to be faster than existing methods.

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A Newton’s method for the continuous quadratic knapsack problem

Cited by 30 publications

References 25 publications

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

An Efficient Hybrid Algorithm for the Separable Convex Quadratic Knapsack Problem

A Primal-Dual Partial Inverse Algorithm for Constrained Monotone Inclusions: Applications to Stochastic Programming and Mean Field Games

Fast projection onto the simplex and the $$\pmb {l}_\mathbf {1}$$ l 1 ball

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