We survey recent work on machine learning (ML) techniques for selecting cutting planes (or cuts) in mixed-integer linear programming (MILP). Despite the availability of various classes of cuts, the task of choosing a ...
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ISBN:
(纸本)9781956792034
We survey recent work on machine learning (ML) techniques for selecting cutting planes (or cuts) in mixed-integer linear programming (MILP). Despite the availability of various classes of cuts, the task of choosing a set of cuts to add to the linear programming (LP) relaxation at a given node of the branch-and-bound (B&B) tree has defied both formal and heuristic solutions to date. ML offers a promising approach for improving the cut selection process by using data to identify promising cuts that accelerate the solution of MILP instances. This paper presents an overview of the topic, highlighting recent advances in the literature, common approaches to data collection, evaluation, and ML model architectures. We analyze the empirical results in the literature in an attempt to quantify the progress that has been made and conclude by suggesting avenues for future research.
Productivity in open pit operations in the mining industry is conditioned by the manual assignment of trucks by the dispatcher, who does not have the ability to find the optimal policy by himself, having many variable...
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We study a mutually enriching connection between response time analysis in real-time systems and the mixing set problem. Thereby generalizing over known results we present a new approach to the computation of response...
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ISBN:
(纸本)9798400717246
We study a mutually enriching connection between response time analysis in real-time systems and the mixing set problem. Thereby generalizing over known results we present a new approach to the computation of response times in fixed-priority uniprocessor realtime scheduling. We even allow that the tasks are delayed by some period-constrained release jitter. By studying a dual problem formulation of the decision problem as an integer linear program we show that worst-case response times can be computed by algorithmically exploiting a conditional reduction to an instance of the mixing set problem. In the important case of harmonic periods our new technique admits a near-quadratic algorithm to the exact computation of worst-case response times. We show that generally, a smaller utilization leads to more efficient algorithms even in fixed-priority scheduling. Worst-case response times can be understood as least fixed points to non-trivial fixed point equations and as such, our approach may also be used to solve suitable fixed point problems. Furthermore, we show that our technique can be reversed to solve the mixing set problem by computing worst-case response times to associated real-time scheduling task systems. Finally, we also apply our optimization technique to solve 4-block integer programs with simple objective functions.
In a seminal paper, Kannan and Lovasz (1988) considered a quantity mu(KL)(Lambda, K) which denotes the best volume-based lower bound on the covering radius mu(Lambda, K) of a convex body K with respect to a lattice La...
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ISBN:
(纸本)9798350318944
In a seminal paper, Kannan and Lovasz (1988) considered a quantity mu(KL)(Lambda, K) which denotes the best volume-based lower bound on the covering radius mu(Lambda, K) of a convex body K with respect to a lattice Lambda. Kannan and Lovasz proved that mu(Lambda, K)=*** KL(Lambda, K) and the Subspace Flatness Conjecture by Dadush (2012) claims a O(log(2n)) factor suffices, which would match the lower bound from the work of Kannan and Lovasz. We settle this conjecture up to a constant in the exponent by proving that mu(Lambda, K) <= O(log(3)(2n)).mu(KL)(Lambda, K). Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012,2019), we obtain a (log(2n))(O(n))-time randomized algorithm to solve integer programs in n variables. Another implication of our main result is a near-optimal flatness constant of O(n log(3)(2n)).
Neural networks are known to be vulnerable to adversarial attacks, which are small, imperceptible perturbations that can significantly alter the network's output. Conversely, there may exist large, meaningful pert...
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Neural networks are known to be vulnerable to adversarial attacks, which are small, imperceptible perturbations that can significantly alter the network's output. Conversely, there may exist large, meaningful perturbations that do not affect the network's decision (excessive invariance). In our research, we investigate this latter phenomenon in two contexts: (a) discrete-time dynamical system identification, and (b) the calibration of a neural network's output to that of another network. We examine noninvertibility through the lens of mathematical optimization, where the global solution measures the "safety" of the network predictions by their distance from the non-invertibility boundary. We formulate mixed-integer programs (MIPs) for ReLU networks and L-p norms (p = 1;2, infinity) that apply to neural network approximators of dynamical systems. We also discuss how our findings can be useful for invertibility certification in transformations between neural networks, e.g. between different levels of network pruning.
This paper develops an integer programming approach to two-sided many-to-one matching by investigating stable integral matchings of a fictitious market where each worker is divisible. We show that a stable matching ex...
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This paper develops an integer programming approach to two-sided many-to-one matching by investigating stable integral matchings of a fictitious market where each worker is divisible. We show that a stable matching exists in a discrete matching market when the firms' preference profile satisfies a total unimodularity condition that is compatible with various forms of complementarities. We provide a class of firms' preference profiles that satisfy this condition.
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.
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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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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