Application of general semi-infinite programming to lapidary cutting problems

Winterfeld, Anton

doi:10.1016/j.ejor.2007.01.057

Cited by 33 publications

(19 citation statements)

References 10 publications

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“…Applications of GSIP are found in many areas, e.g., kinetic model reduction [23], robust optimization [6,24], gemstone cutting [25] and Chebyshev approximation [26]. Among the challenges is that a minimum may not even exist.…”

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confidence: 99%

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Mitsos

Tsoukalas

2014

J Glob Optim

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The algorithm proposed in Mitsos (Optimization 60(10-11):1291(Optimization 60(10-11): -1308(Optimization 60(10-11): , 2011 for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs. No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) deterministic global NLP solver. The lower bounding procedure is based on a discretization of the lower-level host set; the set is populated with Slater points of the lower-level program that result in constraint violations of prior upper-level points visited by the lower bounding procedure. The purpose of the lower bounding procedure is only to generate a certificate of optimality; in trivial cases it can also generate a global solution point. The upper bounding procedure generates candidate optimal points; it is based on an approximation of the feasible set using a discrete restriction of the lower-level feasible set and a restriction of the right-hand side constraints (both lower and upper level). Under relatively mild assumptions, the algorithm is shown to converge finitely to a truly feasible point which is approximately optimal as established from the lower bound. Test cases from the literature are solved and the algorithm is shown to be computationally efficient.

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confidence: 99%

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Mitsos

Tsoukalas

2014

J Glob Optim

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“…Convergence of this method under standard assumptions is shown in [1,4]. In [3], the GSIP problems, obtained from diamond cutting problems, are solved by using an interior-point method developed by Stein [5]. Compared with the method used in [3], the advantage of our method is that at each iteration, only a system of linear equations is solved to get search direction.…”

Section: Discussionmentioning

confidence: 99%

“…In [3], the GSIP problems, obtained from diamond cutting problems, are solved by using an interior-point method developed by Stein [5]. Compared with the method used in [3], the advantage of our method is that at each iteration, only a system of linear equations is solved to get search direction. The numerical examples show that semismooth Newton method works well for diamond cutting problems.…”

Section: Discussionmentioning

confidence: 99%

“…In a general design centering problem, the aim is to maximize some measure (e.g., volume) of a parametrized body B(x) under the constraint that B(x) is contained in a fixed body G. The formulation of a general design centering problem can be written as max x∈R n Vol(B(x)), such that B (x) ⊂ G. Let G = {y ∈ R m | g (y) ≤ 0} be the fixed body called as container, then GSIP formulation of general design centering problem can be written as follows: max x∈R n Vol(B(x)), such that g (y) ≤ 0, for all y ∈ B(x) A good example of design centering problem in three dimensions is the gemstone (diamond) cutting problem. In these problems, the optimal prototype gemstone (diamond being cut) is inscribed into the functional approximation of irregularly shaped rough gemstone [3].…”

Section: Design Centering Problemsmentioning

confidence: 99%

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On an Application of Generalized Semi-Infinite Optimization

Özturan

2016

Acta Phys. Pol. A

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Generalized semi-infinite optimization problems are optimization problems having infinitely many constraints. In addition, the infinite index set depends on the decision variable of optimization. In this study, as an application of generalized semi-infinite optimization problems a type of design centering problems is considered. In a general design centering problem some measure of a parametrized body is maximized under the constraint that parametrized body is inscribed in a fixed body. In this study, diamond cutting problem is considered as a type of design centering problems. Here the aim is to maximize the volume of a round cut diamond from functional approximation of a rough irregularly shaped gemstone. First of all, the problem is converted to a generalized semiinfinite optimization problem, then corresponding first order optimality conditions are obtained. Several numerical examples are presented by solving reformulated Karush-Kuhn-Tucker optimality conditions with semismooth Newton method. The advantage of the method is that a linear system of equations has to be solved in each iteration.

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“…In Winterfeld (2006), the lapidary cutting problem is solved deterministically. The authors formulate the problem as a general semi-infinite problem (GSIP) based on an interiorpoint method developed by Stein (O.Stein, 2006).…”

Section: Related Workmentioning

confidence: 99%

A real-valued genetic algorithm for gemstone cutting

Silva

Ritt

Carvalho

et al. 2012

2012 XXXVIII Conferencia Latinoamericana en Informatica (CLEI)

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In this paper we present a genetic algorithm for solving the gemstone cutting problem. The goal of this problem is to find the largest faceted cut design which fits inside a given rough gemstone. We propose a fast algorithm for finding the largest scaling factor of a faceted cut, once its center and orientation angles are given, as well as a real-valued genetic algorithm for finding the cut having the largest volume. Finally we present experimental results obtained using a set of 50 scanned gemstones and compare the results with similar ones from the literature.

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Application of general semi-infinite programming to lapidary cutting problems

Cited by 33 publications

References 10 publications

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Global optimization of generalized semi-infinite programs via restriction of the right hand side

On an Application of Generalized Semi-Infinite Optimization

A real-valued genetic algorithm for gemstone cutting

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