How to solve a design centering problem

Harwood, Stuart M.; Barton, Paul I.

doi:10.1007/s00186-017-0591-3

Cited by 17 publications

(11 citation statements)

References 62 publications

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“…For a detailed discussion of the reformulation of design centering problems as semiinfinite ones we refer to Stein (2006) and the references therein. Different solution techniques are discussed in Harwood and Barton (2017). One interesting application of design centering in the context of semi-infinite optimization is the maximal material usage in gemstone cutting.…”

Section: Design Centeringmentioning

confidence: 99%

A transformation-based discretization method for solving general semi-infinite optimization problems

2020

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Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the $$\mathbf {x}$$ x -dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In this paper we assume that such a transformation exists globally. However, this approach may destroy convexity in the lower level, which is very important for numerical methods. We present in this paper a solution approach for GSIP problems, which cleverly combines the above mentioned two techniques. It is shown that the convergence results for discretization methods in the case of SIP problems can be transferred to our transformation-based discretization method under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on several examples, including a problem of volume-maximal inscription of multiple variable bodies into a larger fixed body, which has never before been considered as a GSIP test problem.

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Section: Design Centeringmentioning

confidence: 99%

A transformation-based discretization method for solving general semi-infinite optimization problems

2020

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“…The trained MLP provides a computationally cheap surrogate, transforming the DS samples into a suitable form for exploitation. It may be used readily to predict the feasibility probability for any design parameters d ∈ K. It may also be embedded into a designcentering optimization problem 26 for finding a subset of points, e.g. in the form of a box or an ellipsoid, with feasibility probability above a desired reliability value.…”

Section: Exploitation Of the Resultsmentioning

confidence: 99%

“…Another drawback is that design-centering problems give rise to complex mathematical programs with either robust (semi-infinite) or chance constraints that are computationally hard to tackle rigorously. 25,26 By contrast, sampling algorithms discretize the process parameter range and return a subset of the samples that satisfy all of the CQA limits and other constraints up to the desired reliability value. Exhaustive sampling may be achieved via a fine uniform gridding or (quasi) Monte Carlo sampling.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Approach to Probabilistic Design Space Characterization: A Nested Sampling Strategy

Kusumo

Gomoescu

Paulen

et al. 2019

Ind. Eng. Chem. Res.

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Quality by design in pharmaceutical manufacturing hinges on computational methods and tools that are capable of accurate quantitative prediction of the design space. This paper investigates Bayesian approaches to design space characterization, which determine a feasibility probability that can be used as a measure of reliability and risk by the practitioner. An adaptation of nested sampling-a Monte Carlo technique introduced to compute Bayesian evidence-is presented. The nested sampling algorithm maintains a given set of live points through regions with increasing probability feasibility until reaching a desired reliability level. It furthermore leverages efficient strategies 1 arXiv:2008.05917v1 [math.OC] 13 Aug 2020 from Bayesian statistics for generating replacement proposals during the search. Features and advantages of this algorithm are demonstrated by means of a simple numerical example and two industrial case studies. It is shown that nested sampling can outperform conventional Monte Carlo sampling and be competitive with flexibility-based optimization techniques in low-dimensional design space problems. Practical aspects of exploiting the sampled design space to reconstruct a feasibility probability map using machine learning techniques are also discussed and illustrated. Finally, the effectiveness of nested sampling is demonstrated on a higher-dimensional problem, in the presence of a complex dynamic model and significant model uncertainty.

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“…More details about design-centering problems can for example be found in [22] and [5]. In this easy case the problem can be solved analytically and its solution is given by:…”

Section: Numerical Examplementioning

confidence: 99%

An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence

Seidel¹,

Küfer²

2019

Preprint

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Semi-infinite programming can be used to model a large variety of complex optimization problems. The simple description of such problems comes at a price: semi-infinite problems are often harder to solve than finite nonlinear problems. In this paper we combine a classical adaptive discretization method developed by Blankenship and Falk in [2] and techniques regarding a semi-infinite optimization problem as a bi-level optimization problem. We develop a new adaptive discretization method which combines the advantages of both techniques and exhibits a quadratic rate of convergence. We further show that a limit of the iterates is a stationary point, if the iterates are stationary points of the approximate problems.

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How to solve a design centering problem

Cited by 17 publications

References 62 publications

A transformation-based discretization method for solving general semi-infinite optimization problems

A transformation-based discretization method for solving general semi-infinite optimization problems

Bayesian Approach to Probabilistic Design Space Characterization: A Nested Sampling Strategy

An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence

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