Generalized bregman envelopes and proximity operators
- Authors: Burachik, Regina , Dao, Minh , Lindstrom, Scott
- Date: 2021
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 190, no. 3 (2021), p. 744-778
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- Description: Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
The generalized bregman distance
- Authors: Burachik, Regina , Dao, Minh , Lindstrom, Scott
- Date: 2021
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 404-424
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- Description: Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the generalized Bregman distance, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback-Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert \scrW function, whose importance in optimization is of growing interest. © 2021 Society for Industrial and Applied Mathematics
Outer approximation schemes for generalized semi-infinite variational inequality problems
- Authors: Burachik, Regina , Lopes, J.
- Date: 2010
- Type: Text , Journal article
- Relation: Optimization Vol. 59, no. 4 (2010), p. 601-617
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- Description: We introduce and analyse outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems generalized semi-infinite variational inequality problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.
Abstract convexity and augmented Lagrangians
- Authors: Burachik, Regina , Rubinov, Alex
- Date: 2007
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 18, no. 2 (2007), p. 413-436
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- Description: The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semiinfinite optimization. © 2007 Society for Industrial and Applied Mathematics.
- Description: C1
- Description: 2003005362
An update rule and a convergence result for a penalty function method
- Authors: Burachik, Regina , Kaya, Yalcin
- Date: 2007
- Type: Text , Journal article
- Relation: Journal of Industrial & Management Optimization Vol. 3, no. 2 (2007), p. 381-398
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- Description: We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results.
- Description: C1
- Description: 2003004886
On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian
- Authors: Burachik, Regina , Gasimov, Rafail , Ismayilova, Nergiz , Kaya, Yalcin
- Date: 2006
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 34, no. 1 (2006), p. 55-78
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- Description: We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem
- Description: C1
- Description: 2003002552
A dual condition for the convex subdifferential sum formula with applications
- Authors: Burachik, Regina , Jeyakumar, Vaithilingam
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 12, no. 2 (2005), p. 279-290
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- Description: C1
- Description: 2003002555
On the absence of duality gap for Lagrange-type functions
- Authors: Rubinov, Alex , Burachik, Regina
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Industrial and Management Optimization Vol. 1, no. 1 (2005), p. 33-38
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- Description: Given a generic dual program we discuss the absence of duality gap for a family of Lagrange-type functions. We obtain necessary conditions that become sufficient ones under some additional assumptions. We also give examples of Lagrangetype functions for which this sufficient conditions hold.
- Description: C1
- Description: 2003001425