Local Search for Nonsmooth DC Optimization with DC Equality and Inequality Constraints

Strekalovsky, Alexander S.

doi:10.1007/978-3-030-34910-3_7

Cited by 17 publications

(24 citation statements)

References 41 publications

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“…The key observation for this paper is that as is well known (see e.g., [8]), one can propagate the absolute value operation according to the identity…”

Section: Propagating Bounds And/or Radiimentioning

confidence: 99%

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Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Griewank

Walther

2020

Algorithms

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For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f and a concave part f ^ , including a pair of generalized gradients g ∈ R n ∋ g ^ . The latter satisfy strict chain rules and can be computed in the reverse mode of algorithmic differentiation, at a small multiple of the cost of evaluating f itself. It is shown how f and f ^ can be expressed as a single maximum and a single minimum of affine functions, respectively. The two subgradients g and − g ^ are then used to drive DCA algorithms, where the (convex) inner problem can be solved in finitely many steps, e.g., by a Simplex variant or the true steepest descent method. Using a reflection technique to update the gradients of the concave part, one can ensure finite convergence to a local minimizer of f, provided the Linear Independence Kink Qualification holds. For piecewise smooth objectives the approach can be used as an inner method for successive piecewise linearization.

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“…The key observation for this paper is that as is well known (see e.g., [8]), one can propagate the absolute value operation according to the identity…”

Section: Propagating Bounds And/or Radiimentioning

confidence: 99%

“…The first equation in Equation (8) means that for all quantities u that are affine functions of the independent variables x the corresponding radius δu is zero so that q u " u " p u until we reach the first absolute value. Notice that δv does indeed grow additively for the subtraction just like for the addition.…”

Section: Lemmamentioning

confidence: 99%

Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Griewank

Walther

2020

Algorithms

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“…Finally, let us note that very little research on DC optimization problems with DC equality constraints exists. To the best of the author's knowledge, only in the recent papers by Strekalovsky [44,45] optimization methods for general DC optimization problems with equality and inequality constraints have been considered.…”

Section: Introductionmentioning

confidence: 99%

“…The second method presented in this paper is based on the general steering exact penalty methodology developed for sequential linear/quadratic programming methods for nonlinear programming problems by Byrd et al in [6,7]. A DCA-type method using steering exact penalty rules for constrained nonsmooth DC optimization problems was first presented by Strekalovsky in [45]. However, both the description of this method and its convergence analysis in [45] contain several inaccuracies (see Remark 7 below for more details).…”

Section: Introductionmentioning

confidence: 99%

“…A DCA-type method using steering exact penalty rules for constrained nonsmooth DC optimization problems was first presented by Strekalovsky in [45]. However, both the description of this method and its convergence analysis in [45] contain several inaccuracies (see Remark 7 below for more details). In particular, the case when a point computed by the method is critical for the penalty term (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Steering exact penalty DCA for nonsmooth DC optimization problems with equality and inequality constraints

Долгополик¹

2021

Preprint

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We propose and study two DC (difference of convex functions) algorithms based on exact penalty functions for solving nonsmooth DC optimization problems with nonsmooth DC equality and inequality constraints. Both methods employ adaptive penalty updating strategies to improve their performance. The first method is based on exact penalty functions with individual penalty parameter for each constraint (i.e. multidimensional penalty parameter) and utilizes a primal-dual approach to penalty updates. The second method is based on the so-called steering exact penalty methodology and relies on solving some auxiliary convex subproblems to determine a suitable value of the penalty parameter. We present a detailed convergence analysis of both methods and give several simple numerical examples highlighting peculiarites of two different penalty updating strategies studied in this paper.

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Global and Local Search Methods for D.C. Constrained Problems

Strekalovsky

2020

Mathematical Optimization Theory and Operations Research

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Local Search for Nonsmooth DC Optimization with DC Equality and Inequality Constraints

Cited by 17 publications

References 41 publications

Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions

Steering exact penalty DCA for nonsmooth DC optimization problems with equality and inequality constraints

Global and Local Search Methods for D.C. Constrained Problems

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