Best Subset Selection via a Modern Optimization Lens

In the last twenty-five years (1990-2014), algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than {\texttt {Lasso}} and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.Comment: This is a revised version (May, 2015) of the first submission in June 201

The Discrete Dantzig Selector: Estimating Sparse Linear Models via Mixed Integer Linear Optimization

We propose a novel high-dimensional linear regression estimator: the Discrete Dantzig Selector, which minimizes the number of nonzero regression coefficients subject to a budget on the maximal absolute correlation between the features and residuals. Motivated by the significant advances in integer optimization over the past 10-15 years, we present a Mixed Integer Linear Optimization (MILO) approach to obtain certifiably optimal global solutions to this nonconvex optimization problem. The current state of algorithmics in integer optimization makes our proposal substantially more computationally attractive than the least squares subset selection framework based on integer quadratic optimization, recently proposed in [8] and the continuous nonconvex quadratic optimization framework of [33]. We propose new discrete first-order methods, which when paired with state-of-the-art MILO solvers, lead to good solutions for the Discrete Dantzig Selector problem for a given computational budget. We illustrate that our integrated approach provides globally optimal solutions in significantly shorter computation times, when compared to off-the-shelf MILO solvers. We demonstrate both theoretically and empirically that in a wide range of regimes the statistical properties of the Discrete Dantzig Selector are superior to those of popular $\ell_{1}$-based approaches. We illustrate that our approach can handle problem instances with p = 10,000 features with certifiable optimality making it a highly scalable combinatorial variable selection approach in sparse linear modeling

Automated multigravity assist trajectory planning with a modified ant colony algorithm

The paper presents an approach to transcribe a multigravity assist trajectory design problem into an integrated planning and scheduling problem. A modified Ant Colony Optimization (ACO) algorithm is then used to generate optimal plans corresponding to optimal sequences of gravity assists and deep space manoeuvers to reach a given destination. The modified Ant Colony Algorithm is based on a hybridization between standard ACO paradigms and a tabu-based heuristic. The scheduling algorithm is integrated into the trajectory model to provide a fast time-allocation of the events along the trajectory. The approach demonstrated to be very effective on a number of real trajectory design problems

Machine Learning Playground

Machine learning is a science that “learns” about the data by finding unique patterns and relations in the data. There are a lot of libraries or tools available for processing machine learning datasets. You can upload your dataset in seconds and quickly start using these tools to get prediction results in a few minutes. However, generating an optimal model is a time consuming and tedious task. The tunable parameters (hyper-parameters) of any machine learning model may greatly affect the accuracy metrics. While most of the tools have models with default parameter setting to provide good results, they can often fail to provide optimal results for reallife datasets. This project will be to develop a GUI application where a user could upload a dataset and dynamically visualize accuracy results based on the selected algorithm and its hyperparameters

The SOS Platform: Designing, Tuning and Statistically Benchmarking Optimisation Algorithms

open access articleWe present Stochastic Optimisation Software (SOS), a Java platform facilitating the algorithmic design process and the evaluation of metaheuristic optimisation algorithms. SOS reduces the burden of coding miscellaneous methods for dealing with several bothersome and time-demanding tasks such as parameter tuning, implementation of comparison algorithms and testbed problems, collecting and processing data to display results, measuring algorithmic overhead, etc. SOS provides numerous off-the-shelf methods including: (1) customised implementations of statistical tests, such as the Wilcoxon rank-sum test and the Holm–Bonferroni procedure, for comparing the performances of optimisation algorithms and automatically generating result tables in PDF and formats; (2) the implementation of an original advanced statistical routine for accurately comparing couples of stochastic optimisation algorithms; (3) the implementation of a novel testbed suite for continuous optimisation, derived from the IEEE CEC 2014 benchmark, allowing for controlled activation of the rotation on each testbed function. Moreover, we briefly comment on the current state of the literature in stochastic optimisation and highlight similarities shared by modern metaheuristics inspired by nature. We argue that the vast majority of these algorithms are simply a reformulation of the same methods and that metaheuristics for optimisation should be simply treated as stochastic processes with less emphasis on the inspiring metaphor behind them