Principal component analysis (PCA) is one of the most widely used dimensionality reduction tools in scientific data analysis. The PCA direction, given by the leading eigenvector of a covariance matrix, is a linear com...
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Principal component analysis (PCA) is one of the most widely used dimensionality reduction tools in scientific data analysis. The PCA direction, given by the leading eigenvector of a covariance matrix, is a linear combination of all features with nonzero loadings;this impedes interpretability. Sparse principal component analysis (SPCA) is a framework that enhances interpretability by incorporating an additional sparsity requirement in the feature weights (factor loadings) while finding a direction that explains the maximal variation in the data. However, unlike PCA, the optimization problem associated with the SPCA problem is NP-hard-Most conventionalmethods for solving SPCA are heuristics with no guarantees, such as certificates of optimality on the solution quality via associated dual bounds. Dual bounds are available via standard semidefinite programming (SDP).based relaxations, which may not be tight, and the SDPs are difficult to scale using off-the-shelf solvers. In this paper, we present a convex integer programming (IP) framework to derive dual bounds. At the heart of our approach is the so-called 1-relaxation of SPCA. Although the 1-relaxation leads to convex optimization problems for .0-sparse linear regressions and relatives, it results in a nonconvex optimization problem for the PCA problem. We first show that the 1-relaxation gives a tight multiplicative bound on SPCA. Then, we show how to use standard integer programming techniques to further relax the 1-relaxation into a convex IP for which there are good commercial solvers. We present worst-case results on the quality of the dual bound provided by the convex *** empirically observe that the dual bounds are significantly better than the worst-case performance and are superior to the SDP bounds on some real-life instances. Moreover, solving the convex IP model using commercial IP solvers appears to scalemuch better that solving the SDP-relaxation using commercial solvers. To the best of our knowledge
In this paper, we consider a scheduling issue for parcel delivery and pickup services by a truck-drone last-mile delivery system. We are given a single carrier truck and multiple identical drones to serve a finite set...
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ISBN:
(数字)9798350373332
ISBN:
(纸本)9798350373349
In this paper, we consider a scheduling issue for parcel delivery and pickup services by a truck-drone last-mile delivery system. We are given a single carrier truck and multiple identical drones to serve a finite set of customers. The single carrier truck plays the role of a mobile depot for the drones, and it visits several stops in a given order of them. The given order of stops constitutes a fixed truck route, and the carrier truck is allowed to launch (resp., to retrieve) a drone to (resp., from) a customer at a stop. For each sortie of a drone, the launching stop and the retrieving stop must be the same. The load capacity of a drone is limited to one, and for a sortie of a drone at a stop, there are three options: (i) only deliver a parcel to a customer, (ii) only pick up a parcel from a customer, and (iii) deliver a parcel to a customer, and then pick up a parcel from a customer. The scheduling problem asks to find an assignment of customers to drones with sortie options and a choice of a truck stop for each sortie in the assignment. The objective is to minimize the total duration of the carrier truck over all the stops, which is equivalent to minimizing the makespan. In this paper, we first propose an extended integer program for the case with both delivery and pickup services from an existing one for the case with delivery service only. We also conduct numerical experiments to demonstrate the solution quality of the proposed integer program by utilizing an integer programming solver, and report the results.
We study the problem of learning directed acyclic graphs from continuous observational data, generated according to a linear Gaussian structural equation model. State-of-the-art structure learning methods for this set...
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Evacuation planning is a complex process that requires detailed planning with multiple stages and phases. The main objective of this report is to propose an application of operational research in an evacuation plannin...
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This paper addresses the problem of complete coverage task allocation for a heterogeneous robot team operating in a mapped workarea like an indoor building. The robots in the team have varying execution speeds, and th...
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integer programming (IP) is an essential class of combinatorial optimization problems (COPs). Its inherent NP-hardness has fostered considerable efforts towards the development of heuristic strategies. An emerging app...
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Mixed-integer linear programming (MILP) is at the core of many advanced algorithms for solving fundamental problems in combinatorial optimization. The complexity of solving MILPs directly correlates with their support...
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The underwater data center is a project to install Internet facilities such as servers in submarine containers with cooling function, and sink the data center into the submarine to solve many shortcomings of land-base...
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As an auxiliary space in engineering projects, the process of parking lot design is often under-appreciated, which is highly repetitive, time-consuming, and leaves little space for creativity. Thus, optimizing existin...
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Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. As opposed to binary matrix factorization which uses standard arithmetic, BMF uses the Boolean ...
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