Generating Optimal Robust Continuous Piecewise Linear Regression with Outliers Through Combinatorial Benders Decomposition

Using piecewise linear (PWL) functions to model discrete data has applications for example in healthcare, engineering and pattern recognition. Recently, mixed-integer linear programming (MILP) approaches have been used to optimally fit continuous PWL functions. We extend these formulations to allow for outliers. The resulting MILP models rely on binary variables and big-M constructs to model … Read more

A robust method based on LOVO functions for solving least squares problems

The robust adjustment of nonlinear models to data is considered in this paper. When data comes from real experiments, it is possible that measurement errors cause the appearance of discrepant values, which should be ignored when adjusting models to them. This work presents a Lower Order-value Optimization (LOVO) version of the Levenberg-Marquardt algorithm, which is … Read more

Outlier detection in time series via mixed-integer conic quadratic optimization

We consider the problem of estimating the true values of a Wiener process given noisy observations corrupted by outliers. The problem considered is closely related to the Trimmed Least Squares estimation problem, a robust estimation procedure well-studied from a statistical standpoint but poorly understood from an optimization perspective. In this paper we show how to … Read more

Size Matters: Cardinality-Constrained Clustering and Outlier Detection via Conic Optimization

Plain vanilla K-means clustering is prone to produce unbalanced clusters and suffers from outlier sensitivity. To mitigate both shortcomings, we formulate a joint outlier-detection and clustering problem, which assigns a prescribed number of datapoints to an auxiliary outlier cluster and performs cardinality-constrained K-means clustering on the residual dataset. We cast this problem as a mixed-integer … Read more

Fast Algorithms for the Minimum Volume Estimator

The MVE estimator is an important tool in robust regression and outlier detection in statistics. We develop fast and efficient algorithms for the MVE estimator problem and discuss how they can be implemented efficiently. The novelty of our approach stems from the recent developments in the first-order algorithms for solving the related Minimum Volume Enclosing … Read more