We present a mixed integer programming formulation for the problem of clustering a set of points in Rd with axis-parallel clusters, while allowing to discard a pre-specified number of points, thus declared to be outli...
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We present a mixed integer programming formulation for the problem of clustering a set of points in Rd with axis-parallel clusters, while allowing to discard a pre-specified number of points, thus declared to be outliers. We identify a family of valid inequalities separable in polynomial time, we prove that some inequalities from this family induce facets of the associated polytope, and we show that the dynamic addition of cuts coming from this family is effective in practice.& COPY;2023 Elsevier B.V. All rights reserved.
Binary matrix factorization is an essential tool for identifying discrete patterns in binary data. In this paper, we consider the rank -k binary matrix factorization problem (k- BMF) under Boolean arithmetic: we are g...
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Binary matrix factorization is an essential tool for identifying discrete patterns in binary data. In this paper, we consider the rank -k binary matrix factorization problem (k- BMF) under Boolean arithmetic: we are given an n x m binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension n x k and k x m, respectively, which minimize the distance between X and the Boolean product of A and B in the squared Frobenius distance. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak linear programming (LP) relaxation, whereas the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times the zero is erroneously covered in a rank -k factorization. For one of the exponential size Ips, we describe a computational approach based on column generation. Experimental results on synthetic and real-world data sets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorizations.
Two-stage stochastic mixed-integer programs (SMIPs) with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing a sub...
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Two-stage stochastic mixed-integer programs (SMIPs) with general integer variables in the second-stage are generally difficult to solve. This paper develops the theory of integer set reduction for characterizing a subset of the convex hull of feasible integer points of the second-stage subproblem which can be used for solving the SMIP with pure integer recourse. The basic idea is to use the smallest possible subset of the subproblem feasible integer set to generate a valid inequality like Fenchel decomposition cuts with a goal of reducing computation time. An algorithm for obtaining such a subset based on the solution of the subproblem linear programming relaxation is devised and incorporated into a decomposition method for SMIP. To demonstrate the performance of the new integer set reduction methodology, a computational study based on randomly generated knapsack test instances was performed. The results of the study show that integer set reduction aids in speeding up cut generation, leading to better bounds in solving SMIPs with pure integer recourse than using a direct solver.
Identification of nonlinear dynamical systems using data-driven frameworks facilitates the prediction and control of systems in a range of applications. Identification of a single system from the measurements of the s...
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Identification of nonlinear dynamical systems using data-driven frameworks facilitates the prediction and control of systems in a range of applications. Identification of a single system from the measurements of the system's states leads to the discovery of explicit or implicit models that cannot generalize beyond the system for which the data are provided. By learning the effect of parameters in the system, we propose a generalizable model for the Identification of Parametric forms of dynamical systems using integer programming (IP2). We first build general libraries of basis functions that take into account both states and parameters. Subsequently, leveraging dimension analysis and the assumption of having integer coefficients in the equations, we show that our framework can identify the exact forms of parametric mechanical dynamical systems like an ideal pendulum or an inverted pendulum on a cart. Moreover, by applying object tracking techniques and taking advantage of a sequential filtering scheme, we can identify the state and energy equations of these dynamical systems from videos of the systems, i.e. pixel space noisy data, rather than state-space measurements. The results show that using integer programming makes the proposed framework significantly (more than 40 times in the case of inverted pendulum on a cart) more robust to noise compared to previous optimization models.
The distributionally robust chance constrained integer programming problems are notoriously hard to solve due to the stochastic feature and the discrete nature of integer variables. In this paper, an exact solution me...
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The distributionally robust chance constrained integer programming problems are notoriously hard to solve due to the stochastic feature and the discrete nature of integer variables. In this paper, an exact solution method is developed for linear integer programming problems with the distributionally robust chance constraint. By exploring the geometric properties, some domain cut techniques based on feasible points and infeasible points are derived. Thus cut off some sub-boxes which do not contain any optimal solution and the optimality gap is reduced successively in the solution iterations. Then the optimal solution will be found in a finite number of iterations. Encouraging computational results are also reported in the paper.
Cutting planes for mixed-integer linear programs (MILPs) are typically computed in rounds by iteratively solving optimization problems, the so-called separation. Instead, we reframe the problem of finding good cutting...
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Cutting planes for mixed-integer linear programs (MILPs) are typically computed in rounds by iteratively solving optimization problems, the so-called separation. Instead, we reframe the problem of finding good cutting planes as a continuous optimization problem over weights parametrizing families of valid inequalities. This problem can also be interpreted as optimizing a neural network to solve an optimization problem over subadditive functions, which we call the subadditive primal problem of the MILP. To do so, we propose a concrete two-step algorithm, and demonstrate empirical gains when optimizing generalized Gomory mixed-integer inequalities over various classes of MILPs. Code for reproducing the experiments can be found at https://github .com /dchetelat /subadditive.& COPY;2023 Elsevier B.V. All rights reserved.
Conferences are a key aspect of communicating knowledge, and their schedule plays a vital role in meeting the expectations of participants. Given that many conferences have different constraints and objectives, differ...
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Conferences are a key aspect of communicating knowledge, and their schedule plays a vital role in meeting the expectations of participants. Given that many conferences have different constraints and objectives, different mathematical models and heuristic methods have been designed to address rather specific requirements of the conferences being studied per se. We present a penalty system that allows organisers to set up scheduling preferences for tracks and submissions regarding sessions and rooms, and regarding the utilisation of rooms within sessions. In addition, we also consider hybrid and online conferences where submissions need to be scheduled in appropriate sessions based on timezone information. A generic scheduling tool is presented that schedules tracks into sessions and rooms, and submissions into sessions by minimising the penalties subject to certain hard constraints. Two integer programming models are presented: an exact model and an extended model. Both models were tested on five real instances and on two artificial instances which required the scheduling of several hundreds of time slots. The results showed that the exact model achieved optimal solutions for all instances except for one instance which resulted in 0.001% optimality gap, and the extended model handles more complex and additional constraints for some instances. Overall, this work demonstrates the suitability of the proposed generic approach to optimise schedules for in-person, hybrid, and online conferences.
We consider the problem of mapping a logical quantum circuit onto a given hardware with limited 2-qubit connectivity. We model this problem as an integer linear program, using a network flow formulation with binary va...
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We consider the problem of mapping a logical quantum circuit onto a given hardware with limited 2-qubit connectivity. We model this problem as an integer linear program, using a network flow formulation with binary variables that includes the initial allocation of qubits and their routing. We consider several cost functions: an approximation of the fidelity of the circuit, its total depth, and a measure of cross-talk, all of which can be incorporated in the model. Numerical experiments on synthetic data and different hardware topologies indicate that the error rate and depth can be optimized simultaneously without significant loss. We test our algorithm on a large number of quantum volume circuits, optimizing for error rate and depth;our algorithm significantly reduces the number of CNOTs compared to Qiskit's default transpiler SABRE [19] and produces circuits that, when executed on hardware, exhibit higher fidelity.
Origami architecture (OA) is a fascinating papercraft that involves only a piece of paper with cuts and folds. Interesting geometric structures 'pop up' when the paper is opened. However, manually designing su...
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In this paper, we deal with the unrestricted block relocation problem. We present a new integerprogramming formulation for solving the problem. The initial formulation is improved by tighteningconstraints and a pre-pr...
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In this paper, we deal with the unrestricted block relocation problem. We present a new integerprogramming formulation for solving the problem. The initial formulation is improved by tighteningconstraints and a pre-processing step to fix several variables. We design a exact iterativescheme algorithm based on a fast heuristic for the integer programming formulation (ISA-FH).Computational results show the effectiveness of the improved formulation and algorithm.
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