Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

Boţ, Radu Ioan; Csetnek, Ernö Robert; Nimana, Nimit

doi:10.1007/s11590-017-1158-1

Cited by 13 publications

(5 citation statements)

References 34 publications

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“…Let us also notice that for there exists such that , hence, for every it holds For situations where () is satisfied we refer the reader to [5,8,9,11].…”

Section: The General Monotone Inclusion Problemmentioning

confidence: 99%

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A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

Boţ

Nguyen

2018

Optimization

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We investigate a forward–backward splitting algorithm of penalty type with inertial effects for finding the zeros of the sum of a maximally monotone operator and a cocoercive one and the convex normal cone to the set of zeroes of an another cocoercive operator. Weak ergodic convergence is obtained for the iterates, provided that a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions, strong convergence for the sequence of generated iterates is proved. As a particular instance we consider a convex bilevel minimization problem including the sum of a non-smooth and a smooth function in the upper level and another smooth function in the lower level. We show that in this context weak non-ergodic and strong convergence can be also achieved under inf-compactness assumptions for the involved functions.

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“…Let us also notice that for there exists such that , hence, for every it holds For situations where () is satisfied we refer the reader to [5,8,9,11].…”

Section: The General Monotone Inclusion Problemmentioning

confidence: 99%

“…Moreover, as emphasized in [19], see also [20], algorithms with inertial effects may detect optimal solutions of minimization problems which cannot be found by their non-inertial variants. In the last years, a huge interest in inertial algorithms can be noticed (see, for instance, [8,9,15,17,20–32]).…”

Section: Introductionmentioning

confidence: 99%

A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

Boţ

Nguyen

2018

Optimization

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“…Subsequently, there are many authors who are interested in studying the inertial-type algorithm. We refer interested readers to [23][24][25][26][27][28][29][30][31] for more information. In 2015, Combettes and Yamada [32] presented a new Mann algorithm combining error term for solving a common fixed point of averaged nonexpansive mappings in a Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems

Artsawang

Ungchittrakool

2020

Symmetry

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In this article, we establish a new Mann-type method combining both inertial terms and errors to find a fixed point of a nonexpansive mapping in a Hilbert space. We show strong convergence of the iterate under some appropriate assumptions in order to find a solution to an investigative fixed point problem. For the virtue of the main theorem, it can be applied to an approximately zero point of the sum of three monotone operators. We compare the convergent performance of our proposed method, the Mann-type algorithm without both inertial terms and errors, and the Halpern-type algorithm in convex minimization problem with the constraint of a non-zero asymmetric linear transformation. Finally, we illustrate the functionality of the algorithm through numerical experiments addressing image restoration problems.

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“…Another approach to motivate problem (2) in the context of nonautonomous multiscaled differential inclusion is due to [1]. We refer the reader to [2,3,13,14,18,23,27] for a rich literature devoted to problem (2). Assume that the solution set of the problem (2) is nonempty and some qualification conditions hold, for instance,…”

Section: Introductionmentioning

confidence: 99%

Generalized forward–backward splitting with penalization for monotone inclusion problems

Nimana

Petrot

2018

J Glob Optim

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We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided that the condition corresponding to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a largescale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.Keywords Monotone operator · monotone inclusion · hierarchical optimization · forward-backward algorithm Mathematics Subject Classification (2010) 47H05 · 65K05 · 65K10 · 90C25

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Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

Cited by 13 publications

References 34 publications

A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

Inertial Mann-Type Algorithm for a Nonexpansive Mapping to Solve Monotone Inclusion and Image Restoration Problems

Generalized forward–backward splitting with penalization for monotone inclusion problems

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