Linear convergence of a type of iterative sequences in nonconvex quadratic programming

Tuan, Hoang Ngoc

doi:10.1016/j.jmaa.2014.10.048

Cited by 10 publications

(5 citation statements)

References 20 publications

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“…We choose a natural number N and define the mesh size . Since the optimal control is assumed to be bang–bang, we identify the discretized control with its piece-wise constant extension: Moreover, we identify the discretized state and costate with its piece-wise linear interpolations and The Euler discretization of ( 1.1 ) is given by where is the cost function defined by Observe that ( ) is a quadratic optimization problem over a polyhedral convex set, where the gradient projection method converges linearly, see e.g., [ 30 ]. This means that for each N , there exists such that In the following examples, we will consider various values of N which suggest that This will confirm the sublinear rate obtained in Theorem 3.2 .…”

Section: Numerical Illustrationsmentioning

confidence: 99%

On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions

Preininger

Vuong

2018

Comput Optim Appl

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We revisit the gradient projection method in the framework of nonlinear optimal control problems with bang-bang solutions. We obtain the strong convergence of the iterative sequence of controls and the corresponding trajectories. Moreover, we establish a convergence rate, depending on a constant appearing in the corresponding switching function and prove that this convergence rate estimate is sharp. Some numerical illustrations are reported confirming the theoretical results.

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Section: Numerical Illustrationsmentioning

confidence: 99%

On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions

Preininger

Vuong

2018

Comput Optim Appl

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“…The first aim of the present paper is to prove that any DCA sequence generated by Algorithm B converges R-linearly to a KKT point. Hence, combining this with Theorem 2.1 from [26], we have a complete solution for the Conjecture in [12, p. 489]. Our result is obtained by applying some arguments of [26] and a new technique in dealing with implicitly defined DCA sequences.…”

Section: Introductionmentioning

confidence: 68%

“…Hence, combining this with Theorem 2.1 from [26], we have a complete solution for the Conjecture in [12, p. 489]. Our result is obtained by applying some arguments of [26] and a new technique in dealing with implicitly defined DCA sequences.…”

Section: Introductionmentioning

confidence: 68%

“…Recently, the Conjecture has been solved in the affirmative for the two-dimensional IQP by Tuan [25]. To solve it in the general case, Tuan [26] has used a local error bound for affine variational inequalities and several specific properties of the KKT point set of the IQP which were obtained by Luo and Tseng [15] (see also Tseng [24] and Luo [14]). The main result of [26] is the following theorem: If the IQP has a nonempty solution set, then every DCA sequence generated by Algorithm A converges R-linearly to a KKT point.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of a Solution Algorithm in Indefinite Quadratic Programming

Cuong¹,

Lim²,

Yen³

2018

Preprint

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It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm R-linearly converges to a Karush-Kuhn-Tucker point, provided that the problem has a solution. Another major result of this paper says that DCA sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably-chosen neighborhood of it. To deal with the implicitly defined iterative sequences, a local error bound for affine variational inequalities and novel techniques are used. Numerical results together with an analysis of the influence of the decomposition parameter, as well as a comparison between the Proximal DC decomposition algorithm and the Projection DC decomposition algorithm, are given in this paper. Our results complement a recent and important paper of Le Thi, Huynh, and Pham Dinh (J. Optim. Theory Appl. 179 (2018), 103-126).

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“…In our second experiment we considered matrices of the form Q μ n as defined in (32). In order to generate hard instances (those which are close to be copositive) we took μ := 1.9.…”

Section: Methodsmentioning

confidence: 99%

The Boosted DC Algorithm for Linearly Constrained DC Programming

Artacho

Campoy

Vuong

2022

Set-Valued Var. Anal

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The Boosted Difference of Convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical Difference of Convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions programs with linear constraints, and prove that every cluster point of the sequence generated by this algorithm is a Karush–Kuhn–Tucker point of the problem if the feasible set has a Slater point. When the objective function is quadratic, we prove that any sequence generated by the algorithm is bounded and R-linearly (geometrically) convergent. Finally, we present some numerical experiments where we compare the performance of DCA and BDCA on some challenging problems: to test the copositivity of a given matrix, to solve one-norm and infinity-norm trust-region subproblems, and to solve piecewise quadratic problems with box constraints. Our numerical results demonstrate that this new extension of BDCA outperforms DCA.

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Linear convergence of a type of iterative sequences in nonconvex quadratic programming

Cited by 10 publications

References 20 publications

On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions

On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions

Convergence of a Solution Algorithm in Indefinite Quadratic Programming

The Boosted DC Algorithm for Linearly Constrained DC Programming

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