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Two-row and two-column mixed-integer presolve using hashing-based pairing methods

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EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION 2020年第3-4期8卷 205-240页

作者： Gemander, Patrick Chen, Wei-Kun Weninger, Dieter Gottwald, Leona Gleixner, Ambros Martin, Alexander Friedrich Alexander Univ Erlangen Nurnberg Dept Math Cauerstr 11 D-91058 Erlangen Germany Beijing Inst Technol Sch Math & Stat Beijing 100081 Peoples R China Zuse Inst Berlin Dept Math Optimizat Takustr 7 D-14195 Berlin Germany

In state-of-the-art mixed-integer programming solvers, a large array of reduction techniques are applied to simplify the problem and strengthen the model formulation before starting the actual branch-and-cut phase. Despite their mathematical simplicity, these methods can have significant impact on the solvability of a given problem. However, a crucial property for employing presolve techniques successfully is their speed. Hence, most methods inspect constraints or variables individually in order to guarantee linear complexity. In this paper, we present new hashing-based pairing mechanisms that help to overcome known performance limitations of more powerful presolve techniques that consider pairs of rows or columns. Additionally, we develop an enhancement to one of these presolve techniques by exploiting the presence of set-packing structures on binary variables in order to strengthen the resulting reductions without increasing runtime. We analyze the impact of these methods on the MIPLIB 2017 benchmark set based on an implementation in the MIP solver SCIP.

关键词： linear programming mixed-integer linear programming Optimization solver Presolve

Stochastic arbitrage with market index options

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arXiv 2022年

作者： Beare, Brendan K. Seo, Juwon Zheng, Zhongxi School of Economics University of Sydney Australia Department of Economics National University of Singapore Singapore

Opportunities for stochastic arbitrage in an options market arise when it is possible to construct a portfolio of options which provides a positive option premium and which, when combined with a direct investment in the underlying asset, generates a payoff which stochastically dominates the payoff from the direct investment in the underlying asset. We provide linear and mixed-integer linear programs for computing the stochastic arbitrage opportunity providing the maximum option premium to an investor. We apply our programs to 18 years of data on monthly put and call options on the Standard & Poors 500 index, finding no evidence that stochastic arbitrage opportunities are systematically present. A skewed specification of the underlying market return distribution with a constant market risk premium and constant multiplicative variance risk premium is broadly consistent with the pricing of market index options at moderate strikes. Copyright © 2022, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Near-optimal Approaches for Binary-Continuous Sum-of-ratios Optimization

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arXiv 2022年

作者： Pham, Hoang Giang Duong, Ngan Ha Mai, Tien Ta, Thuy Anh School of Computing and Information Systems Singapore Management University Singapore Department of Engineering Computer Science and Mathematics University of L’Aquila Italy Orlab Faculty of Computer Science Phenikaa University Viet Nam

In this paper, we investigate a class of non-convex sum-of-ratios programs relevant to decision-making in key areas such as product assortment and pricing, facility location and cost planning, and security games. These optimization problems, characterized by both continuous and binary decision variables, are highly non-convex and challenging to solve. To the best of our knowledge, no existing methods can efficiently solve these problems to near-optimality with arbitrary precision. To address this challenge, we explore a piecewise linear approximation approach that enables the approximation of complex nonlinear components of the objective function as linear functions. We then demonstrate that the approximated problem can be reformulated as a mixed-integer linear program, a second-order cone program, or a bilinear program, all of which can be solved to optimality using off-the-shelf solvers like CPLEX or GUROBI. Additionally, we provide theoretical bounds on the approximation errors associated with the solutions derived from the approximated problem. We illustrate the applicability of our approach to competitive joint facility location and cost optimization, as well as product assortment and pricing problems. Extensive experiments on instances of varying sizes are conducted to assess the efficiency of our method. Copyright © 2022, The Authors. All rights reserved.

关键词： mixed-integer linear programming

A Nonparametric Approach with Marginals for Modeling Consumer Choice

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arXiv 2022年

作者： Ruan, Yanqiu Li, Xiaobo Murthy, Karthyek Natarajan, Karthik Singapore University of Technology and Design Singapore National University of Singapore Singapore

Given data on the choices made by consumers for different offer sets, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior while being amenable to prescriptive tasks such as pricing and assortment optimization. The marginal distribution model (MDM) is one such model, which requires only the specification of marginal distributions of the random utilities. This paper aims to establish necessary and sufficient conditions for given choice data to be consistent with the MDM hypothesis, inspired by the usefulness of similar characterizations for the random utility model (RUM). This endeavor leads to an exact characterization of the set of choice probabilities that the MDM can represent. Verifying the consistency of choice data with this characterization is equivalent to solving a polynomial-sized linear program. Since the analogous verification task for RUM is computationally intractable and neither of these models subsumes the other, MDM is helpful in striking a balance between tractability and representational power. The characterization is then used with robust optimization for making data-driven sales and revenue predictions for new unseen assortments. When the choice data lacks consistency with the MDM hypothesis, finding the best-fitting MDM choice probabilities reduces to solving a mixed integer convex program. Numerical results using real world data and synthetic data demonstrate that MDM exhibits competitive representational power and prediction performance compared to RUM and parametric models while being significantly faster in computation than RUM. Copyright © 2022, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Flexible Solution Approach for Multistage Transmission Network Expansion Planning with Multiple Generation Scenarios

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JOURNAL OF CONTROL AUTOMATION AND ELECTRICAL SYSTEMS 2020年第3期31卷 705-717页

作者： Freitas, Patricia F. S. Macedo, Leonardo H. Romero, Ruben Sao Paulo State Univ Dept Elect Engn Ave Brasil 56 BR-15385000 Ilha Solteira SP Brazil Sao Paulo Fed Inst Educ Sci & Technol Rua Pedro Cavalo 709Residencial Portal Perola 2 BR-16201407 Birigui SP Brazil

One of the approaches to represent competitive market rules, uncertainties in the generation, and different seasonal climatic conditions in the multistage transmission network expansion planning problem is through scenario-based modeling, by considering several generation scenarios. However, the investment costs of the solutions obtained by this type of approach are much higher when compared to the costs of the expansion plans provided by a single-scenario formulation. To overcome this issue, this work presents a flexible solution approach based on a mixed-integer linear programming formulation, which generates a pool of high-quality solutions with reasonable investment costs and small operating infeasibilities for the problem. The proposed method is evaluated using the IEEE 24-bus system and the southern Brazilian 46-bus system. The results indicate that the proposed approach can provide solutions with acceptable investment costs and small infeasibilities in the operation of the system, which can be easily corrected in the short- and medium-term expansion planning.

关键词： mixed-integer linear programming Multiple generation scenarios Multistage planning Optimization Transmission network expansion planning

Optimizing Connected Components Graph Partitioning with Minimum Size Constraints using integer programming and Spectral Clustering Techniques

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arXiv 2022年

作者： Cordero, Mishelle Miniguano–Trujillo, Andrés Recalde, Diego Torres, Ramiro Vaca, Polo Escuela Politécnica Nacional Quito Ecuador Maxwell Institute for Mathematical Sciences The University of Edinburgh Heriot-Watt University Bayes Centre 47 Potterrow Edinburgh United Kingdom Escuela Politécnica Nacional Department of Mathematics Ladrón de Guevara QuitoE11-253 Ecuador

In this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. mixed integer programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems. Copyright © 2022, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Towards Improving Unit Commitment Economics: An Add-On Tailor for Renewable Energy and Reserve Predictions

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arXiv 2022年

作者： Chen, Xianbang Liu, Yikui Wu, Lei the ECE Department Stevens Institute of Technology HobokenNJ07030 United States the Electrical Engineering College Sichuan University Chengdu610017 China

Generally, day-ahead unit commitment (UC) is conducted in a predict-then-optimize process: it starts by predicting the renewable energy source (RES) availability and system reserve requirements;given the predictions, the UC model is then optimized to determine the economic operation plans. In fact, predictions within the process are raw. In other words, if the predictions are further tailored to assist UC in making the economic operation plans against realizations of the RES and reserve requirements, UC economics will benefit significantly. To this end, this paper presents a cost-oriented tailor of RES-and-reserve predictions for UC, deployed as an add-on to the predict-then-optimize process. The RES-and-reserve tailor is trained by solving a bi-level mixed-integer programming model: the upper level trains the tailor based on its induced operating cost;the lower level, given tailored predictions, mimics the system operation process and feeds the induced operating cost back to the upper level;finally, the upper level evaluates the training quality according to the fed-back cost. Through this training, the tailor learns to customize the raw predictions into cost-oriented predictions. Moreover, the tailor can be embedded into the existing predict-then-optimize process as an add-on, improving the UC economics. Lastly, the presented method is compared to traditional, binary-relaxing, neural network-based, stochastic, and robust methods. © 2022, CC BY.

关键词： mixed-integer linear programming

Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Terms

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arXiv 2022年

作者： Bestuzheva, Ksenia Gleixner, Ambros Achterberg, Tobias AI in Society Science and Technology Zuse Institute Berlin Takustraße 7 Berlin14195 Germany HTW Berlin Berlin10313 Germany Gurobi GmbH Sandstraße 104 Monheim am Rhein 40789 Germany

The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, we present a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs, which is based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Strategies for filtering product relations are discussed and tested. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. This algorithm is applicable to RLT cuts generated for all types of bilinear terms, including but not limited to the detected implicit products. A detailed computational study based on independent implementations in two solvers evaluates the performance impact of the proposed *** Codes 90-08, 90C11, 90C20, 90C26, 90C57 Copyright © 2022, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Optimal energy management strategy in microgrids with mixed energy resources and energy storage system

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IET CYBER-PHYSICAL SYSTEMS: THEORY & APPLICATIONS 2020年第1期5卷 80-84页

作者： Semero, Yordanos Kassa Zhang, Jianhua Zheng, Dehua Trina Solar Energy Shanghai Co Ltd Shanghai 200000 Peoples R China North China Elect Power Univ Sch Elect & Elect Engn Beijing 102206 Peoples R China Goldwind Sci & Technol Co Ltd Beijing 100176 Peoples R China

The continued growth of distributed generation (DG) in the electrical grid has led to the expansion of microgrids. Microgrids contain distributed power generation units, energy storage devices, and controllable loads with the capability to operate in both grid-connected and island modes. The economic operation of a microgrid is achieved through an energy management system that optimally schedules DGs and storage devices and continuously balances supply and demand. In this study, a formulation of optimal unit commitment (UC) and dispatch scheduling of DGs in a grid-connected microgrid system is presented. mixed-integer linear programming is used to implement the optimal UC and dispatch scheduling model. The objective is to minimise the overall operating cost of the system by optimally utilising an energy storage device and a combined heat and power (CHP) generation unit using load and renewable energy generation prediction. Operational constraints such as generation limits of DGs, battery charging/discharging limits and state-of-charge limits are to be satisfied during all intervals of operation. Simulation results indicate that the operational cost of the system is significantly reduced through optimal scheduling of an energy storage system and a CHP unit using the proposed strategy.

关键词： power generation scheduling energy management systems cogeneration power distribution faults renewable energy sources distributed power generation integer programming power generation dispatch linear programming energy storage power grids grid-connected microgrid system mixed-integer linear programming optimal UC scheduling model operating cost energy storage device power generation unit renewable energy generation prediction operational constraints operational cost optimal scheduling energy storage system optimal energy management strategy mixed energy resources continued growth distributed generation electrical grid economic operation energy management system schedules DGs storage devices continuously balances supply optimal unit commitment