A heuristic algorithm for optimal fleet composition with vehicle routing considerations

This paper proposes a fast heuristic algorithm for solving a combined optimal fleet composition and multi-period vehicle routing problem. The aim of the problem is to determine an optimal fleet mix, together with the corresponding vehicle routes, to minimize total cost subject to various customer delivery requirements and vehicle capacity constraints. The total cost includes not only the fixed, variable, and transportation costs associated with operating the fleet, but also the hiring costs incurred whenever vehicle requirements exceed fleet capacity. Although the problem under consideration can be formulated as a mixed-integer linear program (MILP), the MILP formulation for realistic problem instances is too large to solve using standard commercial solvers such as the IBM ILOG CPLEX optimization tool. Our proposed heuristic decomposes the problem into two tractable stages: in the first (outer) stage, the vehicle routes are optimized using cross entropy; in the second (inner) stage, the optimal fleet mix corresponding to a fixed set of routes is determined using dynamic programming and golden section search. Numerical results show that this heuristic approach generates high-quality solutions and significantly outperforms CPLEX in terms of computational speed

Learning Sparse Classifiers: Continuous and Mixed Integer Optimization Perspectives

We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to solve (to optimality) $\ell_0$-regularized regression problems at scales much larger than what was conventionally considered possible. Despite their usefulness, MIP-based global optimization approaches are significantly slower compared to the relatively mature algorithms for $\ell_1$-regularization and heuristics for nonconvex regularized problems. We aim to bridge this gap in computation times by developing new MIP-based algorithms for $\ell_0$-regularized classification. We propose two classes of scalable algorithms: an exact algorithm that can handle $p\approx 50,000$ features in a few minutes, and approximate algorithms that can address instances with $p\approx 10^6$ in times comparable to the fast $\ell_1$-based algorithms. Our exact algorithm is based on the novel idea of \textsl{integrality generation}, which solves the original problem (with $p$ binary variables) via a sequence of mixed integer programs that involve a small number of binary variables. Our approximate algorithms are based on coordinate descent and local combinatorial search. In addition, we present new estimation error bounds for a class of $\ell_0$-regularized estimators. Experiments on real and synthetic data demonstrate that our approach leads to models with considerably improved statistical performance (especially, variable selection) when compared to competing methods.Comment: To appear in JML

On Optimally Partitioning Variable-Byte Codes

The ubiquitous Variable-Byte encoding is one of the fastest compressed representation for integer sequences. However, its compression ratio is usually not competitive with other more sophisticated encoders, especially when the integers to be compressed are small that is the typical case for inverted indexes. This paper shows that the compression ratio of Variable-Byte can be improved by 2x by adopting a partitioned representation of the inverted lists. This makes Variable-Byte surprisingly competitive in space with the best bit-aligned encoders, hence disproving the folklore belief that Variable-Byte is space-inefficient for inverted index compression. Despite the significant space savings, we show that our optimization almost comes for free, given that: we introduce an optimal partitioning algorithm that does not affect indexing time because of its linear-time complexity; we show that the query processing speed of Variable-Byte is preserved, with an extensive experimental analysis and comparison with several other state-of-the-art encoders.Comment: Published in IEEE Transactions on Knowledge and Data Engineering (TKDE), 15 April 201