Penalty and Smoothing Methods for Convex Semi-Infinite Programming

doi:10.1287/MOOR.1080.0362

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Penalty and Smoothing Methods for Convex Semi-Infinite Programming

Journal Article•DOI•

Penalty and Smoothing Methods for Convex Semi-Infinite Programming

Alfred Auslender¹, Miguel A. Goberna², Marco A. López²•Institutions (2)

University of Lyon¹, University of Alicante²

01 May 2009-Mathematics of Operations Research (INFORMS)-Vol. 34, Iss: 2, pp 303-319

TL;DR: This paper introduces a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods that subsumes well-known classical algorithms, but also provides some new methods with interesting properties.

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Abstract: In this paper we consider min-max convex semi-infinite programming. To solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions.

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Convex Analysisの二,三の進展について

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徹丸山

01 Feb 1977

5,933 citations

Journal Article•DOI•

A New Exchange Method for Convex Semi-Infinite Programming

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Liping Zhang¹, Soon-Yi Wu², Marco A. López•Institutions (2)

Tsinghua University¹, National Cheng Kung University²

01 Aug 2010-Siam Journal on Optimization

TL;DR: A new dropping-rule is introduced in the proposed exchange algorithm, which only keeps those active constraints with positive Lagrange multipliers and exploits the idea of looking for $\eta$-infeasible indices of the lower level problem as the adding-rule in the algorithm.

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Abstract: In this paper we propose a new exchange method for solving convex semi-infinite programming (CSIP) problems. We introduce a new dropping-rule in the proposed exchange algorithm, which only keeps those active constraints with positive Lagrange multipliers. Moreover, we exploit the idea of looking for $\eta$-infeasible indices of the lower level problem as the adding-rule in our algorithm. Hence the algorithm does not require to solve a maximization problem over the index set at each iteration; it only needs to find some points such that a certain computationally-easy criterion is satisfied. Under some reasonable conditions, the new adding-dropping rule guarantees that our algorithm provides an approximate optimal solution for the CSIP problem in a finite number of iterations. In the numerical experiments, we apply the proposed algorithm to solve some test problems from the literature, including some medium-sized problems from complex approximation theory and FIR filter design. We compare our algorithm with an existing central cutting plane algorithm and with the semi-infinite solver fseminf in MATLAB toolbox, and we find that our algorithm solves the CSIP problem much faster. For the FIR filter design problem, we show that our algorithm solves the problem better than some algorithms that were technically established for the problem.

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54 citations

Cites background from "Penalty and Smoothing Methods for C..."

...1 in [1] shows some particular cases where the set Ω0 is easily obtainable....
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Journal Article•DOI•

Recent contributions to linear semi-infinite optimization: an update

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Miguel A. Goberna¹, Marco A. López², Marco A. López¹•Institutions (2)

University of Alicante¹, Australian National University²

02 Aug 2018-Annals of Operations Research

TL;DR: The state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization is reviewed, some numerical approaches to this type of problems are presented, and a selection of recent applications are described.

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Abstract: This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-infinite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of fields. Extensions to related optimization areas, as convex semi-infinite optimization, linear infinite optimization, and multi-objective linear semi-infinite optimization, are also commented.

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33 citations

Journal Article•DOI•

Stationarity and Regularity of Infinite Collections of Sets

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Alexander Y. Kruger¹, Marco A. López², Marco A. López¹•Institutions (2)

Federation University Australia¹, University of Alicante²

05 Apr 2012-Journal of Optimization Theory and Applications

TL;DR: Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces.

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Abstract: This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Frechet normals to infinite intersections of sets in Asplund spaces.

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31 citations

Journal Article•DOI•

Recent contributions to linear semi-infinite optimization

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Miguel A. Goberna¹, Marco A. López¹, Marco A. López²•Institutions (2)

University of Alicante¹, Australian National University²

29 Aug 2017-A Quarterly Journal of Operations Research

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25 citations

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Journal Article•DOI•

Combined entropic regularization and path-following method for solving finite convex min-max problems subject to infinitely many linear constraints

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Ruey-Lin Sheu¹, Soon-Yi Wu¹•Institutions (1)

National Cheng Kung University¹

01 Apr 1999-Journal of Optimization Theory and Applications

TL;DR: In this paper, the authors studied the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints, and proposed a path-following algorithm with a convergence proof.

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Abstract: In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications.

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4 citations

"Penalty and Smoothing Methods for C..." refers background in this paper

...It is well known that this function is convex (sum of log-convex functions) and that we have the uniform estimate (see, for example, Sheu and Wu [27])...
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...[21], by Sheu and Wu [27] for finite min-max problems subject to infinitely many linear constraints and, more recently, by Sheu and Lin [26] for continuous min-max problems, motivated by the global approach of Fang and Wu [12] using an integral analog....
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Proceedings Article•DOI•

Nonlinear perturbations for linear semi-infinite optimization problems

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Marc Teboulle¹•Institutions (1)

University of Baltimore¹

05 Dec 1990

TL;DR: In this article, a perturbation method for solving semi-infinite optimization problems is introduced, which uses the continuous structure of the problem rather than an a priori discretization of the constraint set.

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Abstract: A perturbation method for solving semi-infinite optimization problems is introduced. The approach is to use the continuous structure of the problem rather than an a priori discretization of the constraint set. A duality theory for infinite-dimensional convex programs is used to construct a nonlinear dual problem which is a finite-dimensional unconstrained concave problem. This induced dual problem penalizes the classical semi-infinite problem. This formulation lends itself to computing a solution of the dual by Newton's type method and allows for solving both the primal and dual problems. Implementation of a primal-dual algorithm, the connection with interior point methods, and further results are briefly discussed. >

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3 citations

"Penalty and Smoothing Methods for C..." refers background in this paper

...This kind of integral methods has been studied by many researchers (see, e.g., Auslender [2], Teboulle [28], Teo and Goh [29], Teo et al. [30], Lin et al. [16], Schattler [25], Polak et al. [20], Fang and Wu [12]) and has the advantage of avoiding nonconvex global optimization in Step 2 of Remez-type methods, via integrals which convexify the approximated functions....
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...We recall that the asymptotic function f of f is defined through the relation epi f = epi f As a straightforward consequence, we get (cf. Auslender and Teboulle [4, Theorem 2.5.1]) f d = inf { lim inf k→+ f kx k k k → + xk → d } (7) where k ⊂ and xk ⊂ n. Note that f is positively homogeneous; that is, f d = f d ∀d ∀ > 0 (8) Remark 2.1....
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..., Auslender [2], Teboulle [28], Teo and Goh [29], Teo et al....
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...This requires, as for Remez-type algorithms, an analysis more subtle than usual, which is built on the use of the theory of recession functions developed in Auslender and Teboulle [4]....
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...More precisely, this smoothing gives F k x = log ∑ t∈T k1 exp f t x p p with p = log T k1 2 (2) This type of smoothing has been proposed by many authors for solving convex finite min-max problems, in particular by Bertsekas [7], Ben-Tal and Teboulle [6], Alvarez [1], and Nesterov [18]....
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Journal Article•DOI•

Méthodes et théorèmes de dualité

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A. Auslender

01 Jan 1970

TL;DR: In this article, the authors present a legal opinion on the use of commercial or impression systématique for copyright violation in the context of the Série rouge agreement, with conditions générales d'utilisation.

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Abstract: © AFCET, 1970, tous droits réservés. L’accès aux archives de la revue « Revue française d’informatique et de recherche opérationnelle. Série rouge » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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2 citations

"Penalty and Smoothing Methods for C..." refers background or methods or result in this paper

...The references Auslender [2], Lin et al....
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...Without this regularization (%k = 0 ∀k , this unified framework contains, in particular, the classical penalty and smoothing methods introduced in Auslender [2], Fang and Wu [12], Lin et al....
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...In Auslender [2], ! = !4, while in Lin et al....
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...In Auslender [2], Q is supposed to be compact....
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..., Auslender [2], Teboulle [28], Teo and Goh [29], Teo et al....
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