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Generalized Nash equilibrium problems with mixed-integer variables

MATHEMATICAL PROGRAMMING 2025年第1-2期209卷 231-277页

作者： Harks, Tobias Schwarz, Julian Univ Passau Fac Comp Sci & Math D-94032 Passau Germany

We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido-Isoda function. To any given instance of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

关键词： Generalized Nash equilibrium problem mixed-integer nonlinear problem Non-convex games

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Coordination of semi-actuated signals on arterials

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JOURNAL OF ADVANCED TRANSPORTATION 2015年第2期49卷 228-246页

作者： Zhang, Lihui Lou, Yingyan Dalian Univ Technol Sch Transportat & Logist Dalian 116024 Peoples R China Arizona State Univ Ira A Fulton Sch Engn Sch Sustainable Engn & Built Environm Tempe AZ 85287 USA

This paper formulates an integrated mixed-integer nonlinear programming model for coordinating semi-actuated signals on arterials, with the vehicle actuation logic explicitly modeled and embedded. The decision variables are the common cycle length;offset points;phase sequences;green splits of the major phases;minimum green, unit green extension and maximum green of the minor phases which are optimized simultaneously. Two algorithms are employed to search for the optimal signal plans: one is the traditional genetic algorithm (GA), and the other is a GA-based heuristic to save computation effort in earlier search stages. Numerical example is carried out to demonstrate the proposed model and algorithms. Copyright (c) 2013 John Wiley & Sons, Ltd.

关键词： semi-actuated signals coordination mixed-integer nonlinear problem genetic algorithm

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Algorithms for generalized potential games with mixed-integer variables

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2017年第3期68卷 689-717页

作者： Sagratella, Simone Sapienza Univ Rome Dept Comp Control & Management Engn Antonio Ruber Via Ariosto 25 I-00185 Rome Italy

We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss-Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches.

关键词： Generalized Nash equilibrium problem Generalized potential game mixed-integer nonlinear problem Parametric optimization

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Joint optimal resource allocation schemes for downlink cooperative cellular networks over orthogonal frequency division multiplexing carriers

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IET COMMUNICATIONS 2020年第10期14卷 1560-1570页

作者： Kaveh, Fatemeh Soleimanpour, Mohadese Talebi, Siamak Shahid Bahonar Univ Kerman Dept Elect Engn Kerman Iran Higher Educ Complex Bam Dept Mech Engn Bam Iran

This study concentrates on the study of a downlink cooperative cellular (DCC) network in which an access point is located in the centre of each cell and two relay stations are installed in the border between both sectors of a cell to obtain higher cell-edge throughput. To optimise the network efficiency, a joint optimal subcarrier and power allocation scheme is proposed which initially derives the highest achievable rate for a single-legacy user (LU), then, through further evaluation and analysis, this is generalised to the case of a multiple-LU environment. Next, to maximise the overall throughput and ensure fairness, a multi-objective lexicographical optimisation problem (MLOP) is formulated to maximise the minimum LUs rate while improving the overall throughput. As the proposed MLOP is a mixed-integer non-linear problem, finding optimal solutions is impractical and in general, there is no standard method for optimally obtaining a solution. Hence, two efficient algorithms are introduced in this study. Simulation results confirm that the proposed algorithms achieve a fair and near-optimal performance for the multi-LU DCC network based on two relay channels, which makes them desirable in practice compared to some of the sub-optimal well-known algorithms investigated in this study.

关键词： OFDM modulation resource allocation integer programming linear programming minimax techniques cellular radio nonlinear programming near-optimal performance multiLU DCC network relay channels joint optimal resource allocation schemes downlink cooperative cellular network access point relay stations cell-edge throughput network efficiency joint optimal subcarrier power allocation scheme highest achievable rate single-legacy user multiple-LU environment multiobjective lexicographical optimisation problem MLOP minimum LUs rate mixed-integer nonlinear problem fair performance

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Computing equilibria of Cournot oligopoly models with mixed-integer quantities

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2017年第3期86卷 549-565页

作者： Sagratella, Simone Sapienza Univ Rome Dept Comp Control & Management Engn Antonio Ruber Via Ariosto 25 I-00185 Rome Italy

We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called "2-groups partitionable" Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method.

关键词： Nash equilibrium problem mixed-integer nonlinear problem Cournot oligopoly Numerical solution

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On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

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OPTIMIZATION 2019年第1期68卷 197-226页

作者： Sagratella, Simone Sapienza Univ Rome Dept Comp Control & Management Engn Antonio Ruber Via Ariosto 25 I-00185 Rome Italy

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.

关键词： Generalized Nash equilibrium problem mixed-integer nonlinear problem enumerative method equilibrium selection

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Coordinated operation strategies for natural gas and power systems in presence of gas-related flexibilities

IET ENERGY SYSTEMS INTEGRATION

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IET ENERGY SYSTEMS INTEGRATION 2019年第1期1卷 3-13页

作者： Ameli, Hossein Qadrdan, Meysam Strbac, Goran Imperial Coll London Control & Power Grp London SW7 2AZ England Cardiff Univ Inst Energy Cardiff CF24 3AA Wales

A detailed investigation of the interaction between natural gas and power systems is necessary, due to the increasing interdependency of these vectors, especially in the context of renewable generations integration growth into the grid. In this study, an outer approximation with an equality relaxation decomposition method is proposed to solve a mixed-integer non-linear problem representing the operation of coupled natural gas and power systems. The proposed coupled modelling of natural gas and power systems is compared to decoupled operational modelling. It is demonstrated that operating gas and electricity as a coupled system resulted in about 7% of operational cost savings. In addition, the value of gas-related flexibility options, including flexible gas compressors, flexible gas generation plants, and gas interconnections, to the operation of natural gas and power systems is quantified for a 2030 GB energy system. It is shown that if the natural gas and power systems are flexible enough, the operation of the systems in the decoupled approach is almost the same as the coupled model and therefore there is no need to reform the current energy market framework to make the systems fully coupled.

关键词： power markets integer programming nonlinear programming natural gas technology power system economics approximation theory power systems gas-related flexibility options flexible gas compressors flexible gas generation plants gas-related flexibilities coupled natural gas coordinated operation strategies renewable generations integration growth outer approximation equality relaxation decomposition method mixed-integer nonlinear problem operational cost savings energy market framework

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Online Optimal Power Control Of Offshore Oil-platform Power Systems Based On Interior Point And Fast Branch-Bound Methods 12

Online Optimal Power Control Of Offshore Oil-platform Power ...

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12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)

作者： Zhong, Jianliang Sun, Yingyun Xie, Xiaorong Wang, Jianfeng Wei, Che North China Elect Power Univ State Key Lab Alternate Elect Power Syst Renewabl Beijing 102206 Peoples R China Tsinghua Univ State Key Lab Power Syst Dept Elect Engn Beijing 100084 Peoples R China China Natl Offshore Oil Corp Res Inst Beijing 100028 Peoples R China

ISBN: (纸本)9781467376822

Offshore oil-platform power systems are important infrastructure for the exploitation of maritime oil and gas. However, their current energy management systems, with relatively simple control schemes and low-level automation, can hardly operate the systems in a secure and economic manner to match the rapid progress of offshore oil and gas exploitation. To address this issue, an online multi-objective optimal power control strategy is proposed and implemented based on the existing Supervisory Control And Data Acquisition (SCADA) system. By incorporating network loss, gas consumption and voltage deviation into a synthesized objective function and taking the various operational constraints into account, the power-control task is formulated into a mixed-integer nonlinear optimized problem. And then an efficient procedure that combines the interior point and the fast branch-bound methods are developed to solve the problem. Consequently the optimal control strategy can be obtained online and either system security or operational efficiency is achieved. The developed control system has been put into practical use in the Weixinan offshore oil-platform power system (China's first of its kind). Field test results show that it can coordinate active and reactive power for online optimal control as per the changes of operating conditions. The improved safety, efficiency and power quality of the power systems will definitely promote exploitation of offshore oil and gas in the respects of both security and efficiency.

关键词： offshore oil-platform power systems optimal power control mixed-integer nonlinear problem interior point fast branch-bound method

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A progressive method to solve large-scale AC Optimal Power Flow with discrete variables and control of the feasibility

A progressive method to solve large-scale AC Optimal Power F...

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2014 Power Systems Computation Conference (PSCC)

作者： Ruiz, Manuel Maeght, Jean Marie, Alexandre Panciatici, Patrick Renaud, Arnaud Artelys 12 Rue 4 Septembre F-75002 Paris France RTE F-78005 Versailles France

ISBN: (纸本)9788393580132

This paper introduces a study on power system networks which aims to produce a dynamic simulation based security assesment taking into account uncertainties. An extended Optimal Power Flow without any guarantee on feasibility leads to the resolution of a mixed-integer nonlinear problem (MINLP). Solving a (MINLP) is very challenging and even harder when the problem is not convex and could be unfeasible. A custom filtering method which tries to explain infeasibilities and uses the nonlinear solver KNITRO to reformulate discrete variables into nonlinear constraints is proposed. Preliminary results on the French and European power systems network demonstrate the interest of this approach on real-life cases.

关键词： load flow power system security power system simulation European power system French power system MINLP discrete variable dynamic simulation feasibility control filtering method large-scale AC optimal power flow mixed-integer nonlinear problem nonlinear constraint nonlinear solver KNITRO power system network security assesment Artificial neural networks Couplers Extremities Optimization Power transformers Production Security discrete transforms nonlinear constraint Power system simulation Dynamic simulation Power system security ELECTRIC TRANSFORMERS couplers load flow Extremities Artificial neural networks system network optimal load flow Security

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Techno-Economic Optimisation And Life Cycle Assessment Of Alternative Technologies For Converting Municipal Solid Waste Into Chemicals And Electricity In The Uk

Techno-Economic Optimisation And Life Cycle Assessment Of Al...

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作者： Nutchapon Chiarasumran Loughborough University

学位级别：博士

This research aims to develop a systematic life cycle assessment approach for alternative technologies for converting UK's municipal solid waste (MSW) to energy and chemicals. The study focuses on seven scenarios, which are combustion, gasification, methanol synthesis, indirect dimethyl ether (DME) synthesis, direct DME synthesis, solid oxide fuel cell (SOFC) and methanation. The LCA model is developed based on one ton of MSW where three waste compositions are considered. The environmental impacts are evaluated by SimaPro software using IMPACT 2002+ method and Ecoinvent 3 as database. This work does not consider on the MSW collection and transportation, the plant construction and uncertainty of the MSW. The syngas composition employed in this study is predicted from a non-stoichiometric equilibrium gasification model using Lagrange multiplier and Gibb's free energy minimisation approach, which is simulated in MATLAB software. Moreover, a dynamic kinetic model based on the reactions in the gasification is also developed and simulated. The model output is compared to the experimental results and the parameters enhanced using two estimability analysis approaches. The environmental impact results show that the SOFC process has the least impact on most end-point categories for the base case of the waste composition (MSW 1) because it can generate the highest amount of electricity. The other two waste compositions are also compared and found that in case of the second MSW composition (MSW 2), SOFC is in the most environmentally friendly based on the end-point impact categories. For the third composition (MSW 3), the direct DME synthesis becomes the top technology in the human health, whereas SOFC is still the top in the climate change and resource categories. The methanation always has the lowest ecosystem quality impact. A sensitivity analysis is performed by increasing the total yields for methanol, indirect and direct DME synthesis. It appears that when the total yield

关键词： Chemical Engineering not elsewhere classified municipal solid waste (MSW) Waste-to-energy estimability Techno-economic Assessment Multi-objective optimisation life cycle assessment studies mixed-integer nonlinear problem waste utilization

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