Nonlinear Integer Programming

Abstract-The anticipated explosion in the total data traffic load will impose to mobile network operators (MNOs) the necessity to densify their networks to provide coverage. At the same time, since MNOs plan their networks according to their high-peak traffic load, base station underutilization during the low traffic hours raises the issue of unnecessary power consumption and excessive cost. In the present paper, we plan to study the energy and cost efficiency of a heterogeneous network (HetNet) that is a cooperation result of many MNOs. Each MNO is owner of a HetNet, composed of eNodeBs (eNBs) and small cells (SCs) and they cooperate by sharing their infrastructure and by switching off a part of it. BS type and traffic load constitute switching off criteria and a roaming cost based user association scheme is used to roam traffic to neighbouring BSs. We assess the cost alterations created by the possible MNO coalitions and we propose a bankruptcy game to allocate the obtained cost to the cooperative MNOs and to motivate thus them to maintain their sharing agreement instead of following a non-cooperative tactic. The bankruptcy game uses Shapley Value to portray each MNO's contribution to cost savings. The MNOs' satisfaction from their payoffs (i.e., the allocated cost) and the overall fairness of the method are evaluated. According to the extracted results, the proposed switching off scheme achieves significant improvement of energy efficiency for the studied network, while the proposed bankruptcy game achieves a balanced and satisfactory cost allocation for different MNO traffic loads.

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“…The problem of eq. (11) is an NP-hard, non-linear integer problem [29], [30]. This is due to the fact that the solution of problem in eq.…”

Section: A Energy Efficiency Problem Formulationmentioning

confidence: 99%

Evaluating Cost Allocation Imposed by Cooperative Switching Off in Multioperator Shared HetNets

Oikonomakou

Antonopoulos

Alonso

et al. 2017

IEEE Trans. Veh. Technol.

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“…This begins to pose severe challenges for high-dimensional optimization problems (such as loan portfolio selection), which will be discussed in more detail below. Moreover, for portfolio optimization, one has a nonlinear integer program, which makes the problem even more challenging (Hemmecke, Koppe, Lee & Weismantel 2010).…”

Section: Sources Of Computational Advantages Of Aopmentioning

confidence: 99%

Large-Scale Loan Portfolio Selection

2015

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We consider the problem of optimally selecting a large portfolio of risky loans, such as mortgages, credit cards, auto loans, student loans, or business loans. Examples include loan portfolios held by financial institutions and fixed-income investors as well as pools of loans backing mortgage-and asset-backed securities. The size of these portfolios can range from the thousands to even hundreds of thousands. Optimal portfolio selection requires the solution of a high-dimensional nonlinear integer program and is extremely computationally challenging. For larger portfolios, this optimization problem is intractable. We propose an approximate optimization approach that yields an asymptotically optimal portfolio for a broad class of data-driven models of loan delinquency and prepayment. We prove that the asymptotically optimal portfolio converges to the optimal portfolio as the portfolio size grows large. Numerical case studies using actual loan data demonstrate its computational efficiency. The asymptotically optimal portfolio's computational cost does not increase with the size of the portfolio. It is typically many orders of magnitude faster than nonlinear integer program solvers while also being highly accurate even for moderate-sized portfolios. AbstractWe consider the problem of optimally selecting a large portfolio of risky loans, such as mortgages, credit cards, auto loans, student loans, or business loans. Examples include loan portfolios held by financial institutions and fixed-income investors as well as pools of loans backing mortgage-and asset-backed securities. The size of these portfolios can range from the thousands to even hundreds of thousands. Optimal portfolio selection requires the solution of a high-dimensional nonlinear integer program and is extremely computationally challenging. For larger portfolios, this optimization problem is intractable. We propose an approximate optimization approach that yields an asymptotically optimal portfolio for a broad class of data-driven models of loan delinquency and prepayment. We prove that the asymptotically optimal portfolio converges to the optimal portfolio as the portfolio size grows large. Numerical case studies using actual loan data demonstrate its computational efficiency. The asymptotically optimal portfolio's computational cost does not increase with the size of the portfolio. It is typically many orders of magnitude faster than nonlinear integer program solvers while also being highly accurate even for moderate-sized portfolios.

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“…In general, solving IQPs is intractable [16]. But our problem has a special structure: we can show that the matrix corresponding to the system (2)-(4) is totally unimodular, and that the matrix corresponding to the quadratic term of the optimization function (1) is principally unimodular [12].…”

Section: Solving Cardinality Constraintsmentioning

confidence: 99%

Scalable test data generation from multidimensional models

Torlak

2012

Proceedings of the ACM SIGSOFT 20th International Symposium on the Foundations of Software Engineering

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Multidimensional data models form the core of modern decision support software. The need for this kind of software is significant, and it continues to grow with the size and variety of datasets being collected today. Yet real multidimensional instances are often unavailable for testing and benchmarking, and existing data generators can only produce a limited class of such structures. In this paper, we present a new framework for scalable generation of test data from a rich class of multidimensional models. The framework provides a small, expressive language for specifying such models, and a novel solver for generating sample data from them. While the satisfiability problem for the language is NP-hard, we identify a polynomially solvable fragment that captures most practical modeling patterns. Given a model and, optionally, a statistical specification of the desired test dataset, the solver detects and instantiates a maximal subset of the model within this fragment, generating data that exhibits the desired statistical properties. We use our framework to generate a variety of high-quality test datasets from real industrial models, which cannot be correctly instantiated by existing data generators, or as effectively solved by general-purpose constraint solvers.

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Nonlinear Integer Programming

Cited by 137 publications

References 92 publications

Evaluating Cost Allocation Imposed by Cooperative Switching Off in Multioperator Shared HetNets

Evaluating Cost Allocation Imposed by Cooperative Switching Off in Multioperator Shared HetNets

Large-Scale Loan Portfolio Selection

Scalable test data generation from multidimensional models

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