An Inertial Tseng’s Type Proximal Algorithm for Nonsmooth and Nonconvex Optimization Problems

Boź, Radu Ioan; Csetnek, Ernö Robert

doi:10.1007/s10957-015-0730-z

Cited by 74 publications

(27 citation statements)

References 34 publications

(49 reference statements)

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“…The first one was often used in the context of Fejer monotonicity techniques for proving convergence results of classical algorithms for convex optimization problems or, more generally, for monotone inclusion problems (see [17]). The second one is easy to verify (see [12]).…”

Section: Preliminariesmentioning

confidence: 97%

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Inertial proximal alternating minimization for nonconvex and nonsmooth problems

Zhang

2017

J Inequal Appl

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In this paper, we study the minimization problem of the type , where f and g are both nonconvex nonsmooth functions, and R is a smooth function we can choose. We present a proximal alternating minimization algorithm with inertial effect. We obtain the convergence by constructing a key function H that guarantees a sufficient decrease property of the iterates. In fact, we prove that if H satisfies the Kurdyka-Lojasiewicz inequality, then every bounded sequence generated by the algorithm converges strongly to a critical point of L.

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Section: Preliminariesmentioning

confidence: 97%

“…The main feature of the inertial proximal algorithm is that the next iterate is defined by using the last two iterates. Recently, there has been an increasing interest in algorithms with inertial effect; see [3–12]. …”

Section: Introductionmentioning

confidence: 99%

Inertial proximal alternating minimization for nonconvex and nonsmooth problems

Zhang

2017

J Inequal Appl

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“…piiq Comparing the assumptions in (iii) and (iv) to the ones in [9], one can notice the presence of the additional condition (2.4c), which is essential in particular when proving the boundedness of the sequence of generated iterates. Notice that in iterative schemes of gradient type, proximal-gradient type or forward-backward-forward type (see [9,11,12]) the boundedness of the iterates follow by combining a descent inequality expressed in terms of the objective function with coercivity assumptions on the later. In our setting this undertaken is less simple, since the descent inequality which we obtain below is in terms of the augmented Lagrangian associated with problem (1.1).…”

Section: A)mentioning

confidence: 99%

A Proximal Minimization Algorithm for Structured Nonconvex and Nonsmooth Problems

Boţ¹,

Csetnek²,

Nguyen³

2019

SIAM J. Optim.

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We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function, each of the nonsmooth summands depending on an independent block variable, and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the boundedness of the generated sequence and prove that any cluster point of the latter is a KKT point of the minimization problem. In the setting of the Kurdyka-Lojasiewicz property we show global convergence, and derive convergence rates for the iterates in terms of the Lojasiewicz exponent.

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“…The resulting iterative schemes share the feature that the next iterate is defined by means of the last two iterates, a fact which induces the inertial effect in the algorithm. Since the works [1,3], one can notice an increasing number of research efforts dedicated to algorithms of inertial type (see [1][2][3]9,[16][17][18][19][24][25][26][27][28][30][31][32]34]). …”

Section: Introductionmentioning

confidence: 99%

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

2017

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We consider the problem of minimizing a smooth convex objective function subject to the set of minima of another differentiable convex function. In order to solve this problem, we propose an algorithm which combines the gradient method with a penalization technique. Moreover, we insert in our algorithm an inertial term, which is able to take advantage of the history of the iterates. We show weak convergence of the generated sequence of iterates to an optimal solution of the optimization problem, provided a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. We also prove convergence for the objective function values to the optimal objective value. The convergence analysis carried out in this paper relies on the celebrated Opial Lemma and generalized Fejér monotonicity techniques. We illustrate the functionality of the method via a numerical experiment addressing image classification via support vector machines.

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An Inertial Tseng’s Type Proximal Algorithm for Nonsmooth and Nonconvex Optimization Problems

Cited by 74 publications

References 34 publications

Inertial proximal alternating minimization for nonconvex and nonsmooth problems

Inertial proximal alternating minimization for nonconvex and nonsmooth problems

A Proximal Minimization Algorithm for Structured Nonconvex and Nonsmooth Problems

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

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