Academic literature on the topic 'McCormick envelope'
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Journal articles on the topic "McCormick envelope"
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Liu, Jingyao, Guangsheng Feng, Jiayu Sun, Liying Zheng, and Huiqiang Wang. "QoE-Oriented Cooperative Broadcast Optimization for Vehicular Video Streaming." Wireless Communications and Mobile Computing 2021 (December 23, 2021): 1–22. http://dx.doi.org/10.1155/2021/8653083.
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The popularity of online vehicular video has caused enormous information requests in Internet of vehicles (IoV), which brings huge challenges to cellular networks. To alleviate the pressure of base station (BS), Roadside Units (RSUs) and vehicle peers are introduced to collaboratively provide broadcast services to vehicle requesters where vehicles act as both service providers and service requesters. In this paper, we propose an efficient framework leveraging scalable video coding (SVC) technique to improve quality of experience (QoE) from two perspectives: (1) maximizing the data volume received by all requesters and (2) determining buffer action based on playback fluency and average playback quality. For (1), potential providers cooperate to determine the precached video content and delivery policy with the consideration of vehicular mobility and wireless channel status. If one provider fails, other sources will complement to provide requested content delivery. Therefore, their cooperation can improve the QoE and enhance the service reliability. For (2), according to buffer occupancy status, vehicle requesters manage buffer action whether to buffer new segments or upgrade the enhancement level of unplayed segment. Furthermore, the optimization of the data volume is formulated as an integer nonlinear programming (INLP) problem, which can be converted into some linear integer programming subproblems through McCormick envelope method and Lagrange relaxation. Numerical simulation results show that our algorithm is effective in improving total data throughput and QoE.
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Sharifzadeh, H. "An Enhanced McCormick Envelopes to Represent Kron's Loss Formula." International Journal of Engineering 36, no. 3 (2023): 585–93. http://dx.doi.org/10.5829/ije.2023.36.03c.19.
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Rana, Zaid Ashraf, Cheng Seong Khor, and Haslinda Zabiri. "Computational Experience with Piecewise Linear Relaxations for Petroleum Refinery Planning." Processes 9, no. 9 (September 9, 2021): 1624. http://dx.doi.org/10.3390/pr9091624.
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Refinery planning optimization is a challenging problem as regards handling the nonconvex bilinearity, mainly due to pooling operations in processes such as crude oil distillation and product blending. This work investigated the performance of several representative piecewise linear (or piecewise affine) relaxation schemes (referred to as McCormick, bm, nf5, and nf6t) and de (which is a new approach proposed based on eigenvector decomposition) that mainly give rise to mixed-integer optimization programs to convexify a bilinear term using predetermined univariate partitioning for instances of uniform and non-uniform partition sizes. The computational results showed that applying these schemes improves the relaxation tightness compared to only applying convex and concave envelopes as estimators. Uniform partition sizes typically perform better in terms of relaxation solution quality and convergence behavior. It was also seen that there is a limit on the number of partitions that contribute to relaxation tightness, which does not necessarily correspond to a larger number of partitions, while a direct relationship between relaxation size and tightness does not always hold for non-uniform partition sizes.
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Ur Rehman, Habib, Sajjad Ali Haider, Syed Rameez Naqvi, Muhammad Naeem, Kyung-Sup Kwak, and S. M. Riazul Islam. "Environment Friendly Energy Cooperation in Neighboring Buildings: A Transformed Linearization Approach." Energies 15, no. 3 (February 4, 2022): 1160. http://dx.doi.org/10.3390/en15031160.
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Energy consumption in residential, commercial and industrial buildings is one of the major contributors to global warming. Due to the increase in the latter, and growing global energy crisis, more attention is being paid to renewable energy resources (RES). The use of innovative concepts in existing buildings is gaining popularity to provide reduction in energy requirements for electricity, heating and cooling. In this paper, an electricity, heating and cooling cooperation mechanism among neighboring buildings with RES is proposed. It relies on adjusting the RES tariff with a mutual agreement between the neighboring buildings, with an aim to minimize the operational costs. For this purpose, a mathematical model is developed for joint energy cooperation, where surplus energy in one of the buildings is shared with others, thereby reducing dependency on the grid. The optimization structure of the environment friendly energy cooperation is nonlinear, which is linearized using the McCormick envelopes. A scenario for the city of Islamabad, Pakistan, is considered by utilizing its environmental data obtained from public domain websites. The simulation results show more than twenty percent energy cost savings with the proposed cooperation model.
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Zhao, Xiao, Xuhui Xia, Lei Wang, and Guodong Yu. "Risk-Averse Facility Location for Green Closed-Loop Supply Chain Networks Design under Uncertainty." Sustainability 10, no. 11 (November 6, 2018): 4072. http://dx.doi.org/10.3390/su10114072.
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With the increasing attention given to environmentalism, designing a green closed-loop supply chain network has been recognized as an important issue. In this paper, we consider the facility location problem, in order to reduce the total costs and CO2 emissions under an uncertain demand and emission rate. Particularly, we are more interested in the risk-averse method for providing more reliable solutions. To do this, we employ a coherent risk measure, conditional value-at-risk, to represent the underlying risk of uncertain demand and CO2 emission rate. The resulting optimization problem is a 0-1 mixed integer bi-objective programming, which is challenging to solve. We develop an improved reformulation-linearization technique, based on decomposed piecewise McCormick envelopes, to generate lower bounds efficiently. We show that the proposed risk-averse model can generate a more reliable solution than the risk-neutral model, both in reducing penalty costs and CO2 emissions. Moreover, the proposed algorithm outperforms and classic reformulation-linearization technique in convergence rate and gaps. Numerical experiments based on random data and a ‘real’ case are performed to demonstrate the performance of the proposed model and algorithm.
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Schweidtmann, Artur M., Dominik Bongartz, Daniel Grothe, Tim Kerkenhoff, Xiaopeng Lin, Jaromił Najman, and Alexander Mitsos. "Deterministic global optimization with Gaussian processes embedded." Mathematical Programming Computation 13, no. 3 (June 25, 2021): 553–81. http://dx.doi.org/10.1007/s12532-021-00204-y.
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AbstractGaussian processes (Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the free variables and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the “MeLOn—MachineLearning Models for Optimization” toolbox (https://git.rwth-aachen.de/avt.svt/public/MeLOn).
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Basto-Gil, Jerson Daniel, Angel David Maldonado-Cardenas, and Oscar Danilo Montoya. "Optimal Selection and Integration of Batteries and Renewable Generators in DC Distribution Systems through a Mixed-Integer Convex Formulation." Electronics 11, no. 19 (September 30, 2022): 3139. http://dx.doi.org/10.3390/electronics11193139.
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The problem concerning the optimal placement and sizing of renewable energy resources and battery energy storage systems in electrical DC distribution networks is addressed in this research by proposing a new mathematical formulation. The exact mixed-integer nonlinear programming (MINLP) model is transformed into a mixed-integer convex model using McCormick envelopes regarding the product between two positive variables. Convex theory allows ensuring that the global optimum is found due to the linear equivalent structure of the solution space and the quadratic structure of the objective function when all the binary variables are defined. Numerical results in the 21-bus system demonstrate the effectiveness and robustness of the proposed solution methodology when compared to the solution reached by solving the exact MINLP model. Numerical results showed that the simultaneous allocation of batteries and renewable energy resources allows for the best improvements in the daily operating costs, i.e., about 53.29% with respect to the benchmark case of the 21-bus grid, followed by the scenario where the renewable energy resources are reallocated while considering a fixed location for the batteries, with an improvement of 43.33%. In addition, the main result is that the difference between the exact modeling and the proposed formulation regarding the final objective function was less than 3.90% for all the simulation cases, which demonstrated the effectiveness of the proposed approach for operating distributed energy resources in monopolar DC networks.
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Kleinert, Thomas, Martine Labbé, Fränk Plein, and Martin Schmidt. "Closing the gap in linear bilevel optimization: a new valid primal-dual inequality." Optimization Letters, November 11, 2020. http://dx.doi.org/10.1007/s11590-020-01660-6.
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Abstract Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.
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Paláncz, Béla, and Lajos Völgyesi. "A Numeric-Symbolic Solution of GNSS Phase Ambiguity." Periodica Polytechnica Civil Engineering, February 3, 2020. http://dx.doi.org/10.3311/ppci.15092.
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Solution of the Global Navigation Satellite Systems (GNSS) phase ambiguity is considered as a global quadratic mixed integer programming task, which can be transformed into a pure integer problem with a given digit of accuracy. In this paper, three alter-native algorithms are suggested. Two of them are based on local and global linearization via McCormic Envelopes, respectively. These algorithms can be effective in case of simple configuration and relatively modest number of satellites. The third method is a locally nonlinear, iterative algorithm handling the problem as {-1, 0, 1} programming and also lets compute the next best integer solution easily. However, it should keep in mind that the algorithm is a heuristic one, which does not guarantee to find the global integer optimum always exactly. The procedure is very powerful utilizing the ability of the numeric-symbolic abilities of a computer algebraic system, like Wolfram Mathematica and it is properly fast for minimum 4 satellites with normal configuration, which means the Geometric Dilution of Precision (GDOP) should be between 1 and 8. Wolfram Alpha and Wolfram Clouds Apps give possibility to run the suggested code even via cell phones. All of these algorithms are illustrated with numerical examples. The result of the third one was successfully compared with the LAMBDA method, in case of ten satellites sending signals on two carrier frequencies (L1 and L2) with weighting matrix used to weight the GNSS observation and computed as the inverse of the corresponding covariance matrix.
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Mahmoodian, Vahid, Iman Dayarian, Payman Ghasemi Saghand, Yu Zhang, and Hadi Charkhgard. "A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs." INFORMS Journal on Computing, January 4, 2022. http://dx.doi.org/10.1287/ijoc.2021.1097.
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This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.
Dissertations / Theses on the topic "McCormick envelope"
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Silva, Mauro Viegas da. "Desenvolvimento de um modelo de programação convexa para o problema de fluxo de potência ótimo /." Ilha Solteira, 2018. http://hdl.handle.net/11449/180413.
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Orientador: José Roberto Sanches MantovaniResumo: Neste trabalho, o modelo matemático do problema de fluxo de potência ótimo básico não linear é analisado e manipulado algebricamente para obter um modelo de programação convexa, do tipo cônico de segunda ordem. O conceito de envelopes convexos é apresentado para tratar a não linearidade e não convexidade da restrição trigonométrica inversa que surge ao escrever o modelo de FPO como um modelo cônico. Aplicando duas proposições apresentadas neste trabalho a restrição trigonométrica é resolvida em um pré-processamento por um solver de otimalidade local, neste caso o KNITRO, que enumera todas as possibilidades dos pontos de KKT para obter os envelopes convexos e tornar o modelo de FPO totalmente convexo. O modelo é implementado no AMPL e é resolvido com solvers de otimalidade global com sistemas testes da literatura, nesta tese usam-se os sistemas testes IEEE 14, 30, 57 e 118 barras. Os resultados obtidos são validados comparando-os com resultados fornecidos pelo Matpower, que é um simulador para FPO. Como contribuição desta tese, o modelo convexo de FPO obtido é utilizado como exemplo de aplicações no problema de despacho ótimo de potência ativa e reativa, considerando competições via programação binível. São apresentados dois modelos biníveis e dois modelos uníveis. O modelo iterativo convexo utiliza-se do modelo proposto de FPO convexo e as não linearidades são convexificadas fazendo uso dos envelopes de McCormick. O conceito de dualidade forte é empregado afim de obter um mod... (Resumo completo, clicar acesso eletrônico abaixo)
Abstract: In this work, the basic nonlinear mathematical model for the optimal power flow (OPF) problem is analyzed and manipulated algebraically in order to obtain a second-order conic convex programming model. The concept of convex envelopes is presented to deal with the nonlinearity and nonconvexity of the inverse trigonometric constraint that arises when transforming the nonconvex OPF model into an equivalent conic model. By applying two propositions presented in this work, the trigonometric constraint is solved in a pre-processing stage by a local optimization solver, in this case, the KNITRO solver, which considers all the possibilities of the KKT points to obtain the convex envelopes and find a completely convex OPF model, is used. The model is implemented in AMPL and is solved via global optimization solvers while to show the effectiveness of the model several IEEE systems such as the IEEE 14-, 30-, 57-, and 118-bus systems are used. The obtained results are validated by comparing them with the results provided by Matpower, which is an OPF solver. As a contribution of this thesis, the obtained convex OPF model is used as an application in the active and reactive optimal power dispatch problem, considering competition via bilevel programming. Two bilevel models and two single-level models are presented. The convex iterative model uses the proposed convex OPF model, and the nonlinearities are convexified using McCormick envelopes. The concept of strong duality is employed to obta... (Complete abstract click electronic access below)
Doutor
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Gupte, Akshay. "Mixed integer bilinear programming with applications to the pooling problem." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/45761.
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Solution methodologies for mixed integer bilinear problems (MIBLP) are studied in this dissertation. This problem class is motivated using the pooling problem, a multicommodity network flow problem that typically arises in chemical engineering applications. Stronger than previously known results are provided to compare the strengths of polyhedral relaxations of the pooling problem. A novel single node flow relaxation, defined by a bilinear equality constraint and flow balance, is proposed for the pooling problem. Linear valid inequalities in the original space of variables are derived using a well-known technique called lifting. Mixed integer linear (MILP) formulations are proposed for generating feasible solutions to the pooling problem. Some of these MILP models arise from variable discretizations while others possess a network flow interpretation. The effectiveness of these MILP models is empirically validated on a library of medium and large-scale instances. General MIBLPs, not necessarily pooling problems, are solved using extended MILP reformulations. The reformulation is obtained by writing binary representation for each general integer variable. Facet-defining inequalities are provided for the reformulation of each bilinear term. New valid inequalities are also proposed for bilinear terms with a nontrivial upper bound. The proposed reformulation and cutting planes are compared against a global solver on five different classes of MIBLPs.
Book chapters on the topic "McCormick envelope"
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Koch, Christof. "Bursting Cells." In Biophysics of Computation. Oxford University Press, 1998. http://dx.doi.org/10.1093/oso/9780195104912.003.0022.
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Some neurons throughout the animal kingdom respond to an intracellular current injection or to an appropriate sensory stimulus with a stereotypical sequence of two to five fast spikes riding upon a slow depolarizing envelope. The entire event, termed a burst, is over within 10-40 msec and is usually terminated by a profound afterhyperpolarization (ΑΗΡ). Such bursting cells are not a random feature of a certain fraction of all cells but can be identified with specific neuronal subpopulations. What are the mechanisms generating this intrinsic firing pattern and what is its meaning? Bursting cells can easily be distinguished from a cell firing at a high maintained frequency by the fact that bursts will persist even at a low firing frequency. As illustrated by the thalamic relay cell of Fig. 9.4, some cells can switch between a mode in which they predominantly respond to stimuli via single, isolated spikes and one in which bursts are common. Because we believe that bursting constitutes a special manner of signaling important information, we devote a single, albeit small chapter to this topic. In the following, we describe a unique class of cells that frequently signal with bursts, and we touch upon the possible biophysical mechanisms that give rise to bursting. We finish this excursion by focussing on a functional study of bursting cells in the electric fish and speculate about the functional relevance of burst firing. Neocortical cells are frequently classified according to their response to sustained current injections. While these distinctions are not all or none, there is broad agreement for three classes: regular spiking, fast spiking, and intrinsically bursting neurons (Connors, Gutnick, and Prince, 1982; McCormick et al., 1985; Connors and Gutnick, 1990; Agmon and Connors, 1992; Baranyi, Szente, and Woody, 1993; Nuńez, Amzica, and Steriade, 1993; Gutnick and Crill, 1995; Gray and McCormick, 1996). Additional cell classes have been identified (e.g., the chattering cells that fire bursts of spikes with interburst intervals ranging from 15 to 50 msec; Gray and McCormick, 1996), but whether or not they occur widely has not yet been settled. The cells of interest to us are the intrinsically bursting cells.
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Bynum, Michael, Anya Castillo, Jean-Paul Watson, and Carl D. Laird. "Strengthened SOCP Relaxations for ACOPF with McCormick Envelopes and Bounds Tightening." In 13th International Symposium on Process Systems Engineering (PSE 2018), 1555–60. Elsevier, 2018. http://dx.doi.org/10.1016/b978-0-444-64241-7.50254-8.
Full textConference papers on the topic "McCormick envelope"
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Lin, Xuan, Min Sung Ahn, and Dennis Hong. "Designing Multi-Stage Coupled Convex Programming with Data-Driven McCormick Envelope Relaxations for Motion Planning." In 2021 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2021. http://dx.doi.org/10.1109/icra48506.2021.9561418.
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Tang, Xueyong, Xiaocong Sun, Xia Yan, Sheng Wang, Yu Zhang, Changzheng Shao, and Yi Ding. "Linearized Modeling of Integrated Electricity and District Heating Systems with VF-VT Strategy Based on McCormick Envelopes." In 2020 IEEE Sustainable Power and Energy Conference (iSPEC). IEEE, 2020. http://dx.doi.org/10.1109/ispec50848.2020.9351222.
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Manarelli Gaspar, João Paulo, and Leonardo Nepomuceno. "Modelo de Leilão Multi-período de Sistemas Hidrotérmicos para Mercados do Dia Seguinte." In Congresso Brasileiro de Automática - 2020. sbabra, 2020. http://dx.doi.org/10.48011/asba.v2i1.1406.
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Este artigo propõe um modelo de leilão para mercados do dia seguinte de sistemas hidrotérmicos que utiliza a linearização da função de produção hidráulica por meio de envelopes de McCormick. Além disso, o artigo também propõe a linearização das cotas de montante, a partir de linearização local em torno do ponto de operação, a linearização das cotas de jusantes por meio de aproximantes lineares, e linearizações das funções de potência e vazão turbinada máximas. O modelo proposto busca a otimização da função de bem estar social, que corresponde aos excedentes de produção e de consumo e representa as principais restrições físicas de unidades geradoras termelétricas e hidrelétricas. Os resultados mostram que os despachos de geração e consumo, bem como as políticas de operação hidráulica são consistentes.
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Zhang, Youzhi, Bo An, and V. S. Subrahmanian. "Correlation-Based Algorithm for Team-Maxmin Equilibrium in Multiplayer Extensive-Form Games." In Thirty-First International Joint Conference on Artificial Intelligence {IJCAI-22}. California: International Joint Conferences on Artificial Intelligence Organization, 2022. http://dx.doi.org/10.24963/ijcai.2022/86.
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Efficient algorithms computing a Nash equilibrium have been successfully applied to large zero- sum two-player extensive-form games (e.g., poker). However, in multiplayer games, computing a Nash equilibrium is generally hard, and the equilibria are not exchangeable, which makes players face the problem of selecting one of many different Nash equilibria. In this paper, we focus on an alternative solution concept in zero-sum multiplayer extensive-form games called Team-Maxmin Equilibrium (TME). It is a Nash equilibrium that maximizes each team member’s utility. As TME is unique in general, it avoids the equilibrium selection problem. However, it is still difficult (FNP- hard) to find a TME. Computing it can be formulated as a non-convex program, but existing algorithms are capable of solving this program for only very small games. In this paper, we first refine the complexity result for computing a TME by using a correlation plan to show that a TME can be found in polynomial time in a specific class of games according to our boundary for complexity. Second, we propose an efficient correlation-based algorithm to solve the non-convex program for TME in games not belonging to this class. The algorithm combines two special correlation plans based on McCormick envelopes for convex relaxation and von Stengel-Forges polytope for correlated equilibria. We show that restricting the feasible solution space to von Stengel-Forges polytope will strictly reduce the feasible solution space after convex re- laxation of nonlinear terms. Finally, experiments show that our algorithm is about four orders of magnitude faster than the prior state of the art and can solve many previously unsolvable games.