Robust linear programming discrimination of two linearly inseparable sets

By Kristin P. Bennett, Olvi L. Mangasarian. 1992

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Mathematical optimization in classification and regression trees (2021, background): “In Bennett (1992), the hyperplane that minimizes the average distance from the misclassified individuals to the hyperplane is modeled as an LP problem, while in Bennett and Blue (1996), for a fixed topology of the tree and fixed predictions at the leaf nodes, the problem of finding the optimal obliq”
Forward stability and model path selection (2021, neutral): “2 Breast Cancer Identification We now consider a dataset describing cell nuclei images taken from 699 breast cancer patients (Bennett and Mangasarian, 1992).”
Robustness and Regularization of Support Vector Machines (2008, background): “When the samples are not separable, a penalty term that approximates the total training error is considered (Bennett and Mangasarian, 1992; Cortes and Vapnik, 1995).”
Kernel methods in machine learning (2007, neutral): “Bennett and Mangasarian [17] and Cortes and Vapnik [34] impose a linear penalty on the violation of the large-margin constraints to obtain”
An asynchronous mini-batch algorithm for regularized stochastic optimization (2015, neutral): “Examples include Tikhonov and elastic net regularization, Lasso, sparse logistic regression and support vector machines [1]–[5].”
Optimal Decision Trees (1996, methodology): “MSMT used a linear program [6] to recursively construct the decisions.”
Bundle Methods for Regularized Risk Minimization (2010, background): “A case in point are the papers introducing (binary) hinge loss (Bennett and Mangasarian, 1992; Cortes and Vapnik, 1995) and structured loss (Taskar et al.”
Breast Cancer Diagnosis and Prognosis Via Linear Programming (1995, methodology): “The classification procedure used to separate benign from malignant samples is a variant on the Mul- tisurface Method (MSM) [10] [11] known as MSM-Tree (MSM-T) [1] [2].”
Learning from Sparse Data by Exploiting Monotonicity Constraints (2005, methodology): “We have chosen five data sets from the UCI ML repository: auto-mpg[21], haberman[14], pima-indian-diabetes[22], breast-cancer-wisconsin[4], and car[6, 29].”
Identification of Hybrid Systems: A Tutorial (2007, methodology): “The originalRobust Linear Programming(RLP) method proposed in [7] is a particular case of (29), while theSupport Vector Machines (SVM) method [19] solves a quadratic program under the same constraints as (29).”
SSVM: A Smooth Support Vector Machine for Classification (2001, methodology): “For moderate sized datasets, SSVM gave the highest tenfold cross validation correctness on six publicly available datasets as well as the fastest times when compared with four other methods given in [2, 4].”
Machine Learning via Polyhedral Concave Minimization (1996, methodology): “Thus in the exceptional case when the convex hulls of A and B do not intersect, a single linear program [2] will generate a plane P that strictly separates the sets A and B as follows: Aw >= e + e; Bw <= e e (2) Our concern here is with the usually occurring case when no plane P exists satisfying (2”
Inferring qualitative relations in genetic networks and metabolic pathways (2000, background): “As previously mentioned, robust LP (Bennett and Mangasarian, 1992) might be useful for handling noisy data.”
On the Approximability of Minimizing Nonzero Variables or Unsatisfied Relations in Linear Systems (1998, background): “During the last decade many mathematical programming formulations have been studied to design linear discriminant classi ers (see [49, 12] as well as the included references).”
Pattern Recognition and Machine Learning (2006, unknown): “, N , with one slack variable for each training data point (Bennett, 1992; Cortes and Vapnik, 1995).”
Advances in kernel methods: support vector learning (1999, methodology): “To make MSM more tolerant of noise, Bennett and Mangasarian proposed theRobust Linear Programming method (RLP) (Bennett and Mangasarian, 1992)using the following linear program:minw; ; lXi=1 i isubject to yi(w xi ) + i 1 i 0 i = 1; ::; ` (19.3)Robust LinearProgrammingMethod where i > 0 is the xed…”