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•

A simple computational procedure for optimization problems with functional inequality constraints

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Kok Lay Teo¹, C. Goh¹•Institutions (1)

National University of Singapore¹

01 Oct 1987-IEEE Transactions on Automatic Control

TL;DR: In this article, a simple computational procedure is given for solving a class of optimization problems, where an objective function is to be minimized subject to conventional inequality constraints as well as to inequality constraints of the functional type.

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Abstract: In this note, a simple computational procedure is given for solving a class of optimization problems, where an objective function is to be minimized subject to conventional inequality constraints as well as to inequality constraints of the functional type.

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55 citations

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

..., Auslender [2], Teboulle [28], Teo and Goh [29], Teo et al....
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...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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Journal Article•DOI•

Penalty and Barrier Methods: A Unified Framework

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A. Auslender

01 May 1999-Siam Journal on Optimization

TL;DR: A generic algorithm, based on the properties of recession functions, is proposed, which encompasses almost all penalty and barrier methods in nonlinear programming and in semidefinite programming, but also generates new types of methods.

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Abstract: It is established that many optimization problems may be formulated in terms of minimizing a function $x\rightarrow f_0 (x) + H_\infty (f_1 (x), f_2 (x),\ldots,f_m (x)) + L_\infty (Ax-b)$, where the $f_i$ are closed functions defined on $\mathbb{R}^N$, and where $H_\infty$ and $L_\infty$ are the recession functions of closed, proper, convex functions $H$ and $L$. $A$ is a linear transformation from $\mathbb{R}^N$ to a finite dimensional vector space $Y$ with $b\in Y$. A generic algorithm, based on the properties of recession functions, is proposed. This algorithm not only encompasses almost all penalty and barrier methods in nonlinear programming and in semidefinite programming, but also generates new types of methods. Primal and dual convergence theorems are given.

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54 citations

"Penalty and Smoothing Methods for C..." refers methods 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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...In Auslender [2], Q is supposed to be compact....
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...Vol. 34, No. 2, May 2009, pp. 303–319 issn 0364-765X eissn 1526-5471 09 3402 0303 informs ® doi 10.1287/moor.1080.0362 ©2009 INFORMS Penalty and Smoothing Methods for Convex Semi-Infinite Programming Alfred Auslender Université de Lyon, CNRS, UMR 5208 Institut Camille Jordan, 69622 Villeurbanne, Cedex, France, and Department of Economics, Ecole Polytechnique, F-91128 Palaiseau, Cedex, France, auslender.alfred@gmail.com Miguel A. Goberna, Marco A. López Department of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain {mgoberna@ua.es, marco.antonio@ua.es} In this paper we consider min-max convex semi-infinite programming....
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...The references Auslender [2], Lin et al. [16], and Teo et al. [30] deal with problems of the first type where F is 1 and %k = 0 ∀k....
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Dissertation•

Algorithmes pour la résolution de problèmes d'optimisation et de minimax

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Bernard Martinet

24 Apr 1972

53 citations

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

...Such a rule has been proposed in Martinet [17], where convergence is proved in the convex case, with the assumption just cited above but imposing the additional one that F is uniformly strictly convex....
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...Furthermore, in Martinet [17] there is no duality analysis as in the following section....
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...With a completely different proof, convergence in Martinet [17] was obtained in the nonconvex case, but under the stronger assumption which imposes that F be level bounded on Q instead on the feasible set Q ∩ D, as in Theorem 3....
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...Concerning ordinary CSIP ( T1 = 1) problems, to the best of our knowledge, Remez-type algorithms coupled with penalty methods have only been introduced by Martinet [17]....
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...The class of penalty functions considered in Martinet [17] consists of continuous functions ! → + such that ! t = 0 if t ≤ 0....
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Journal Article•DOI•

A regularization method for solving the finite convex min-max problem

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Cristina Gígola, Susana Gómez

01 Nov 1990-SIAM Journal on Numerical Analysis

TL;DR: In this article, the finite min-max problem is solved by generating a sequence of differentiable subproblems, where u and v are variables in the dual space and the method will minimize this smooth function until sufficient decrease of $\varphi _v (x)$ is attained and then updates the parameter v.

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Abstract: The finite min-max problem \[ \min {\text{-}} \max (f_i (x),i = 1, \cdots ,m) \] is solved generating a sequence of differentiable subproblems \[ \mathop {\min }\limits_x \varphi _v (x) = \mathop {\min }\limits_x \mathop {{\text{Sup}}}\limits_{v,u \in U} (u^t f(x) - \frac{1}{2}\left\| {u - v} \right\|^2 )\] where u and v are variables in the dual space.The method will minimize this smooth function until sufficient decrease of $\varphi _v (x)$ is attained and then updates the parameter v.Convergence is proved assuming convexity, and numerical results are given.

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45 citations

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

...There are many ways to smooth F k (see in particular Gigola and Gomez [13] and Polak et al....
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...There are many ways to smooth F k (see in particular Gigola and Gomez [13] and Polak et al. [20]), but for the sake of simplicity we consider here only the most important and widely used in different fields in the literature....
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Journal Article•DOI•

Algorithms for finite and semi-infinite min-max-min problems using adaptive smoothing techniques

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Elijah Polak¹, Johannes O. Royset¹•Institutions (1)

University of California, Berkeley¹

01 Dec 2003-Journal of Optimization Theory and Applications

TL;DR: Two implementable algorithms are developed for the solution of finite and semi-infinite min-max-min problems, each of which reduces the potential ill-conditioning due to high smoothing precision parameter values and computational costdue to high levels of discretization.

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Abstract: We develop two implementable algorithms, the first for the solution of finite and the second for the solution of semi-infinite min-max-min problems. A smoothing technique (together with discretization for the semi-infinite case) is used to construct a sequence of approximating finite min-max problems, which are solved with increasing precision. The smoothing and discretization approximations are initially coarse, but are made progressively finer as the number of iterations is increased. This reduces the potential ill-conditioning due to high smoothing precision parameter values and computational cost due to high levels of discretization. The behavior of the algorithms is illustrated with three semi-infinite numerical examples.

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41 citations

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

...In another context they are coupled with adaptive grid methods (see, for example, Kaplan and Tichatschke [14], Polak and Royset [19], and references therein) where the parameters of the procedures of discretization, smoothing, regularization, and penalization are adjusted....
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