Journal ArticleDOI

# Penalty and Smoothing Methods for Convex Semi-Infinite Programming

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.

AbstractIn 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.

Topics: Penalty method (66%), Convex analysis (60%), Smoothing (57%), Duality (optimization) (54%)

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##### Citations
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01 Feb 1977

5,933 citations

Journal ArticleDOI
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.
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.

49 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 ArticleDOI
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.
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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Journal ArticleDOI
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.
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.

23 citations

Journal ArticleDOI
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.
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.

21 citations

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01 Feb 1977

5,933 citations

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

• ...We recall here some basic notions about asymptotic cones and functions (for more details see, for instance, the books of Auslender and Teboulle [4], Rockafellar [24])....

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• ...We recall here some basic notions about asymptotic cones and functions (for more details see, for instance, the books of Auslender and Teboulle [4] and of Rockafellar [24])....

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Journal ArticleDOI
TL;DR: A new approach for constructing efficient schemes for non-smooth convex optimization is proposed, based on a special smoothing technique, which can be applied to functions with explicit max-structure, and can be considered as an alternative to black-box minimization.
Abstract: In this paper we propose a new approach for constructing efficient schemes for non-smooth convex optimization. It is based on a special smoothing technique, which can be applied to functions with explicit max-structure. Our approach can be considered as an alternative to black-box minimization. From the viewpoint of efficiency estimates, we manage to improve the traditional bounds on the number of iterations of the gradient schemes from ** keeping basically the complexity of each iteration unchanged.

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Book
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TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
Abstract: Basic notation.- Introduction.- Background material.- Optimality conditions.- Basic perturbation theory.- Second order analysis of the optimal value and optimal solutions.- Optimal Control.- References.

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

• ...Applied to LSIP, especially Cheney and Goldstein [10] and Kelley [15] turn out to be identical or mere modifications of the dual simplex method discussed above, so that they have similar properties and drawbacks....

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• ...Supposing that F is 1 (as is generally the case in ordinary CSIP), we can use cutting-plane methods of Cheney and Goldstein [10], Kelley [15], Veinott [31], or Elzinga and Moore [11], and their variants (see, e.g., Reemtsen and Görner [22] for more references)....

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• ...To avoid slow convergence, constraint dropping rules are again given under some conditions as strict convexity on F for Cheney and Goldstein [10] and Kelley [15]....

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• ...Supposing that F is 1 (as is generally the case in ordinary CSIP), we can use cutting-plane methods of Cheney and Goldstein [10], Kelley [15], Veinott [31], or Elzinga and Moore [11], and their variants (see, e....

[...]

Journal ArticleDOI
TL;DR: A class of parametric smooth functions that approximate the fundamental plus function, (x)+=max{0, x}, by twice integrating a probability density function leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs).
Abstract: We propose a class of parametric smooth functions that approximate the fundamental plus function, (x)+=max{0, x}, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equations as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter α. Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter α, generate an interior path, which is different from the central path for interior point method. Computational results for 52 test problems compare favorably with these for another Newton-based method. The smooth technique is capable of solving efficiently the test problems solved by Dirkse and Ferris [6], Harker and Xiao [11] and Pang & Gabriel [28].

441 citations

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

• ...In [9], Chen and Mangasarian provided a systematic way to generate elements of 1....

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