Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Supervised classification and mathematical optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Supervised Classification and Mathematical Optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data

Fundamental remote sensing science research program. Part 1: Status report of the mathematical pattern recognition and image analysis project

The Mathematical Pattern Recognition and Image Analysis (MPRIA) Project is concerned with basic research problems related to the study of the Earth from remotely sensed measurement of its surface characteristics. The program goal is to better understand how to analyze the digital image that represents the spatial, spectral, and temporal arrangement of these measurements for purposing of making selected inference about the Earth

Exploratory Analysis of Functional Data via Clustering and Optimal Segmentation

We propose in this paper an exploratory analysis algorithm for functional data. The method partitions a set of functions into $K$ clusters and represents each cluster by a simple prototype (e.g., piecewise constant). The total number of segments in the prototypes, $P$, is chosen by the user and optimally distributed among the clusters via two dynamic programming algorithms. The practical relevance of the method is shown on two real world datasets

Optimisation based approaches for machine learning

Machine learning has attracted a lot of attention in recent years and it has become an integral part of many commercial and research projects, with a wide range of applications. With current developments in technology, more data is generated and stored than ever before. Identifying patterns, trends and anomalies in these datasets and summarising them with simple quantitative models is a vital task. This thesis focuses on the development of machine learning algorithms based on mathematical programming for datasets that are relatively small in size. The first topic of this doctoral thesis is piecewise regression, where a dataset is partitioned into multiple regions and a regression model is fitted to each one. This work uses an existing algorithm from the literature and extends the mathematical formulation in order to include information criteria. The inclusion of such criteria targets to deal with overfitting, which is a common problem in supervised learning tasks, by finding a balance between predictive performance and model complexity. The improvement in overall performance is demonstrated by testing and comparing the proposed method with various algorithms from the literature on various regression datasets. Extending the topic of regression, a decision tree regressor is also proposed. Decision trees are powerful and easy to understand structures that can be used both for regression and classification. In this work, an optimisation model is used for the binary splitting of nodes. A statistical test is introduced to check whether the partitioning of nodes is statistically meaningful and as a result control the tree generation process. Additionally, a novel mathematical formulation is proposed to perform feature selection and ultimately identify the appropriate variable to be selected for the splitting of nodes. The performance of the proposed algorithm is once again compared with a number of literature algorithms and it is shown that the introduction of the variable selection model is useful for reducing the training time of the algorithm without major sacrifices in performance. Lastly, a novel decision tree classifier is proposed. This algorithm is based on a mathematical formulation that identifies the optimal splitting variable and break value, applies a linear transformation to the data and then assigns them to a class while minimising the number of misclassified samples. The introduction of the linear transformation step reduces the dimensionality of the examined dataset down to a single variable, aiding the classification accuracy of the algorithm for more complex datasets. Popular classifiers from the literature have been used to compare the accuracy of the proposed algorithm on both synthetic and publicly available classification datasets