Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem

Freund, Robert M.

doi:10.1007/bf01581277

Cited by 25 publications

(19 citation statements)

References 23 publications

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“…We address this problem by making use of extremely fast algorithms based on projections within the interior of polyhedra (much of this work started with Karmarkar 1984). In particular, we draw on the properties of bounding ellipsoids discovered in theorems by Sonnevend (1985aSonnevend ( , 1985b and applied by Freund (1993), Nesterov and Nemirovskii (1994), and Vaidja (1989).…”

Section: Polyhedral Question Design Methodsmentioning

confidence: 99%

Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis

Toubia

Hauser

Simester

2004

Journal of Marketing Research

182

131

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We gratefully acknowledge the contribution of Robert M. Freund who proposed the use of the analytic center and approximating ellipsoids and gave us detailed advice on the application of these methods.This research was supported by the Sloan School of Management and the Center for Innovation in Product Development at M.I.T. This paper may be downloaded from http://mitsloan.mit.edu/vc. That website also contains (1) open source code to implement the methods described in this paper, (2) open source code for the simulations described in this paper, (3) demonstrations of web-based questionnaires based on the methods in this paper, and (4) related papers on web-based interviewing methods. All authors contributed fully and synergistically to this paper. We wish to thank Ray Faith, Aleksas Hauser, Janine Sisk, Limor Weisberg, Toby Woll for the visual design, programming, and project management on the Executive Education Study. This paper has benefited from presentations at the CIPD Spring Research Review, the Epoch Foundation Workshop, the Marketing Science Conferences in Wiesbaden Germany and Alberta Canada, the MIT ILP Symposium on "Managing Corporate Innovation," the MIT Marketing Workshop, the MIT Operations Research Seminar Series, the MSI Young Scholars Conference, the New England Marketing Conference, and Stanford Marketing Workshop, and the UCLA Marketing Seminar Series. Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis AbstractChoice-based conjoint analysis (CBC) is used widely in marketing for product design, segmentation, and marketing strategy. We propose and test a new "polyhedral" question-design method that adapts each respondent's choice sets based on previous answers by that respondent. Individual adaptation appears promising because, as demonstrated in the aggregate customization literature, question design can be improved based on prior estimates of the respondent's partworths -information that is revealed by respondents' answers to prior questions. The otherwise impractical computational problems of individual CBC adaptation become feasible based on recent polyhedral "interior-point" algorithms, which provide the rapid solutions necessary for real-time computation.To identify domains where individual adaptation is promising (and domains where it is not), we evaluate the performance of polyhedral CBC methods with Monte Carlo experiments. We vary magnitude (response accuracy), respondent heterogeneity, estimation method, and question-design method in a We close by describing an empirical application to the design of executive education programs in which 354 web-based respondents answered stated-choice tasks with four service profiles each. The profiles varied on eight multi-level features. With the help of this study a major university is revising its executive education programs with new formats and a new focus.

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Section: Polyhedral Question Design Methodsmentioning

confidence: 99%

Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis

Toubia

Hauser

Simester

2004

Journal of Marketing Research

182

131

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“…We use x 0 to denote the initial point and the symbol '*' is used to denote the center for each iteration. Figure (5) show that the analytic center used 4 cuts against 2 cuts for the p-Center.…”

Section: An Illustrative Examplementioning

confidence: 99%

A weighted projection centering method

Moretti¹

2003

Comput. Appl. Math.

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Abstract. An iterative method for finding the center of a linear programming polytope is presented. The method assumes that we start at a feasible interior point and each iterate is obtained as a convex combination of the orthogonal projection on the half spaces defined by the linear inequalities plus a special projections on the same half spaces.The algorithm is particularly suitable for implementation on computers with parallel processors. We show some examples in two dimensional space to describe geometrically how the method works.Finally, we present computational results on random generated polytopes and linear programming polytopes from NetLib to compare the centering quality of the center using projections and the analytic center approach.Mathematical subject classification: 90C99, 90C90.

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“…Furthermore, the axes of the ellipsoids are well-defined and intuitively capture the concept of an "axis" of a polyhedron. For more details see Vaidja (1989), Freund (1993), Nesterov and Nemirovskii (1994), and Sonnevend (1985aSonnevend ( , 1985b.…”

Section: Interior-point Algorithms and The Analytical Center Of A Polmentioning

confidence: 99%

Fast Polyhedral Adaptive Conjoint Estimation

Toubia¹,

Simester

Hauser³

2001

SSRN Journal

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We gratefully acknowledge the contribution of Robert M. Freund who proposed the use of the analytic center and approximating ellipsoids and gave us detailed advice on the application of these methods. This research was supported by the Sloan School of Management and the Center for Innovation in Product Development at M.I.T. This paper may be downloaded from http://mitsloan.mit.edu/vc. That website also contains (1) open source code to implement the methods described in this paper, (2) open source code for the simulations described in this paper, (3) demonstrations of web-based questionnaires based on the methods in this paper, and (4) related papers on web-based interviewing methods. All authors contributed fully and synergistically to this paper. We wish to thank Limor Weisberg for creating the graphics in this paper. This paper has benefited from presentations at the CIPD Spring Research Review, the Epoch Foundation Workshop, the MIT ILP Symposium on "Managing Corporate Innovation," the MIT Marketing Workshop, the MSI Young Scholars Conference, and the Stanford Marketing Workshop. Fast Polyhedral Adaptive Conjoint Estimation AbstractWeb-based customer panels and web-based multimedia capabilities offer the potential to get information from customers rapidly and iteratively based on virtual product profiles. However, web-based respondents are impatient and wear out more quickly. At the same time, in commercial applications, conjoint analysis is being used to screen large numbers of product features. Both of these trends are leading to a demand for conjoint analysis methods that provide reasonable estimates with fewer questions in problems involving many parameters.In this paper we propose and test new adaptive conjoint analysis methods that attempt to reduce respondent burden while simultaneously improving accuracy. We draw on recent "interior-point" developments in mathematical programming which enable us to quickly select those questions that narrow the range of feasible partworths as fast as possible. We then use recent centrality concepts (the analytic center) to estimate partworths. These methods are efficient, run with no noticeable delay in web-based questionnaires, and have the potential to provide estimates of the partworths with fewer questions than extant methods.After introducing these "polyhedral" algorithms we implement one such algorithm and test it with Monte Carlo simulation against benchmarks such as efficient (fixed) designs and Adaptive Conjoint Analysis (ACA). While no method dominates in all situations, the polyhedral algorithm appears to hold significant potential when (a) profile comparisons are more accurate than the self-explicated importance measures used in ACA, (b) when respondent wear out is a concern, and (c) when the product development and marketing teams wish to screen many features quickly. We also test a hybrid method that combines polyhedral question selection with ACA estimation and show that it, too, has the potential to improve predictions in many contexts. The algorithm w...

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Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem

Cited by 25 publications

References 23 publications

Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis

Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis

A weighted projection centering method

Fast Polyhedral Adaptive Conjoint Estimation

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