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equilibrium problems with equilibrium constraints Analysis for Power Control and User Scheduling in NOMA Networks

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2020年第5期69卷 5467-5480页

作者： Zhang, Huaqing Di, Boya Chang, Zheng Liu, Xuefang Song, Lingyang Han, Zhu Google LLC Mountain View CA 94043 USA Imperial Coll London Dept Comp London SW7 2BU England Univ Jyvaskyla Fac Informat Technol FIN-40014 Jyvaskyla Finland Xidian Univ Sch Telecommun Engn Xian 710071 Peoples R China Peking Univ Sch Elect Engn & Comp Sci Beijing 100871 Peoples R China Univ Houston Houston TX 77004 USA Kyung Hee Univ Dept Comp Sci & Engn Seoul 446701 South Korea

With limited wireless resources but increasing demands for wireless applications, non-orthogonal multiple access (NOMA) has been considered as a promising technology for future wireless networks. Nevertheless, it remains challenging on motivating the mobile users (MU) to join NOMA networks, scheduling MUs on the same sub-bands, reducing interference among the MUs and setting optimal transmit power in the NOMA networks. In this paper, first we consider the quantization error due to the limitation of receivers' equipments and the decoding error during the successive interference cancellation (SIC) procedure. Second, we model the resource allocation problems as the four-level equilibrium problems with equilibrium constraints (EPEC), where the wireless operator, on behalf of each MU, first predicts the utilities and assign the MU as a NOMA user (NMU) or a traditional MU (TMU). For the NMUs, the interference penalty pricing strategy of other NMUs are proposed. Based on the announced prices of all the NMUs, the sub-channel allocation is managed in either distributed or collaborative ways. According to the announced interference prices of other NMUs and the coalition results, the optimal transmit power to gain the maximum utility is considered. Through extensive simulation analysis, the superior performance obtained by using the proposed scheme is demonstrated.

关键词： equilibrium problems with equilibrium constraints game theory NOMA successive interference cancellation resource allocation

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equilibrium problems with equilibrium constraints via multiobjective optimization

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OPTIMIZATION METHODS & SOFTWARE 2004年第5期19卷 479-492页

作者： Mordukhovich, BS Wayne State Univ Dept Math Detroit MI 48202 USA

This article concerns a new class of optimization-related problems called equilibrium problems with equilibrium constraints (SPEC). One may treat them as two-level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of mathematical programs with equilibrium constraints (MPEC). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This article presents necessary optimality conditions for EPECs in finite-dimensional spaces based on advanced generalized differential tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration.

关键词： equilibrium problems with equilibrium constraints multiobjective optimization necessary optimality conditions variational analysis generalized differentiation

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An evolutionary algorithm based on Nash Dominance for equilibrium problems with equilibrium constraints

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APPLIED SOFT COMPUTING 2012年第1期12卷 161-173页

作者： Koh, Andrew Univ Leeds Inst Transport Studies Leeds LS2 9JT W Yorkshire England

This paper introduces an evolutionary algorithm for the solution of a class of hierarchical ("leader-follower") games known as equilibrium problems with equilibrium constraints (EPECs). In one manifestation of such games, players at the upper level who assume the role of leaders, are assumed to act non cooperatively to maximize individual payoffs. At the same time, each leader's payoffs are constrained not only by their competitor's actions but also by the behaviour of the followers at the lower level which manifests in the form of an equilibrium constraint. By a redefinition of the selection criteria used in evolutionary methods, the paper demonstrates that the solution for such games can be found via a simple modification to a standard evolutionary multiobjective algorithm. We give a proposed algorithm (NDEMO) and illustrate it with numerical examples drawn from both the transportation systems management literature and the electricity generation industry underlying the applicability of NDEMO in multidisciplinary contexts. (C) 2011 Elsevier B. V. All rights reserved.

关键词： Nash equilibrium equilibrium problems with equilibrium constraints Transportation systems management Electricity markets

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REGULARIZED equilibrium problems with equilibrium constraints WITH APPLICATION TO ENERGY MARKETS

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SIAM JOURNAL ON OPTIMIZATION 2023年第3期33卷 1767-1796页

作者： Luna, Juan Pablo Sagastizabal, Claudia Filiberti, Julia Gabriel, Steven A. Solodov, Mikhail V. Univ Fed Rio de Janeiro COPPE Rio De Janeiro Brazil Univ Estadual Campinas IMECC Campinas Brazil Univ Maryland College Pk MD 20742 USA Norwegian Univ Sci & Technol Trondheim Norway IMPA Inst Nacl Matemat Pura & Aplicada BR-22460320 Rio De Janeiro RJ Brazil

equilibrium problems with equilibrium constraints are appropriate modeling formulations in a number of important areas, such as energy markets, transportation planning, and logistics. These models often correspond to bilevel games, in which certain dual variables, representing the equilibrium price, play a fundamental role. We consider multileader single-follower equilibrium problems having a linear program in the lower-level. Because in this setting the lowerlevel response to the leaders' decisions may not be unique, the game formulation becomes ill-posed. We resolve possible ambiguities by considering a sequence of bilevel equilibrium problems, endowed with a special regularization term. We prove convergence of the approximating scheme. Our technique proves useful numerically over several instances related to energy markets. When using PATH to solve the corresponding mixed-complementarity formulations, we exhibit that, in the given context, the regularization approach computes a genuine equilibrium price almost always, while without regularization the outcome is quite the opposite.

关键词： equilibrium problems with equilibrium constraints multileader single-follower games dual-primal regularization energy markets

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Semicontinuity of the solution maps to vector equilibrium problems with equilibrium constraints

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OPTIMIZATION 2022年第3期71卷 737-751页

作者： Lam Quoc Anh Cantho Univ Teacher Coll Dept Math Can Tho Vietnam

In this paper, we are concerned with parametric vector equilibrium problems with equilibrium constraints. We first introduce concepts of generalized semicontinuity and convexity of vector-valued maps in ordered spaces and discuss their properties, as well as their relationships with well-known concepts. Next, employing mentioned concepts of generalized semicontinuity of vector-valued maps, we study sufficient conditions of upper semicontinuity for solution maps to such problems. Then, combining these generalized semicontinuity along with generalized convexity assumptions, sufficient conditions for lower semicontinuity of solution maps to the reference problems are established. Several examples are provided to illustrate the relaxation and essentialness of assumptions. Our results are new, even for special cases of the underlying problems.

关键词： equilibrium problems with equilibrium constraints upper lower semicontinuity level closedness generalized convexity

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ANALYSIS OF M-STATIONARY POINTS TO AN EPEC MODELING OLIGOPOLISTIC COMPETITION IN AN ELECTRICITY SPOT MARKET

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2012年第2期18卷 295-317页

作者： Henrion, Rene Outrata, Jiri Surowiec, Thomas Weierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany Acad Sci Czech Republic Inst Informat Theory & Automat CR-18208 Prague 8 Czech Republic Humboldt Univ D-10099 Berlin Germany

We consider an equilibrium problem with equilibrium constraints (EPEC) arising from the modeling of competition in an electricity spot market (under ISO regulation). For a characterization of equilibrium solutions, so-called M-stationarity conditions are derived. This first requires a structural analysis of the problem, e. g., verifying constraint qualifications. Second, the calmness property of a certain multifunction has to be verified in order to justify using M-stationarity conditions. Third, for stating the stationarity conditions, the coderivative of a normal cone mapping has to be calculated. Finally, the obtained necessary conditions are made fully explicit in terms of the problem data for one typical constellation. A simple two-settlement example serves as an illustration.

关键词： equilibrium problems with equilibrium constraints EPEC M-stationary solutions electricity spot market calmness

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Solving multi-leader-common-follower games

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OPTIMIZATION METHODS & SOFTWARE 2010年第4期25卷 601-623页

作者： Leyffer, Sven Munson, Todd Argonne Natl Lab Math & Comp Sci Div Argonne IL 60439 USA

Multi-leader-common-follower games arise when modelling two or more competitive firms, the leaders, that commit to their decisions prior to another group of competitive firms, the followers, that react to the decisions made by the leaders. These problems lead in a natural way to equilibrium problems with equilibrium constraints (EPECs). We develop a characterization of the solution sets for these problems and examine a variety of nonlinear optimization and nonlinear complementarity formulations of EPECs. We distinguish two broad cases: problems where the leaders can cost-differentiate and problems with price-consistent followers. We demonstrate the practical viability of our approach by solving a range of medium-sized test problems.

关键词： Nash games Stackelberg games nonlinear programming nonlinear complementarity problems mathematical programming with equilibrium constraints equilibrium problems with equilibrium constraints

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Solving quadratic multi-leader-follower games by smoothing the follower's best response

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OPTIMIZATION METHODS & SOFTWARE 2022年第2期37卷 772-799页

作者： Herty, Michael Steffensen, Sonja Thuenen, Anna Rhein Westfal TH Aachen Inst Geometrie & Prakt Math Aachen Germany

We analyse the existence of Nash equilibria for a class of quadratic multi-leader-follower games using the nonsmooth best response function. To overcome the challenge of nonsmoothness, we pursue a smoothing approach resulting in a reformulation as a smooth Nash equilibrium problem. The existence and uniqueness of solutions are proven for all smoothing parameters. Accumulation points of Nash equilibria exist for a decreasing sequence of these smoothing parameters and we show that these candidates fulfil the conditions of S-stationarity and are Nash equilibria to the multi-leader-follower game. Finally, we propose an update on the leader variables for efficient computation and numerically compare nonsmooth Newton and subgradient methods.

关键词： Multi-leader-follower games Nash equilibria game theory equilibrium problems with equilibrium constraints

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COMPETITION BETWEEN CITIES INVESTIGATING THE IMPACTS OF A STOCHASTIC USER equilibrium FRAMEWORK

COMPETITION BETWEEN CITIES INVESTIGATING THE IMPACTS OF A ST...

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17th International Conference of Hong-Kong-Society-for-Transportation-Studies (HKSTS)

作者： Koh, A. Shepherd, S. Watling, D. Univ Leeds Inst Transport Studies Leeds LS2 9JT W Yorkshire England

ISBN: (纸本)9789881581419

Extending our earlier work on the implications of toll competition between cities, this paper studies a variant of an equilibrium Problem with equilibrium constraints (EPEC). Our previous research under a Wardrop User equilibrium route choice setting demonstrated that such competition could potentially result in multiple Nash Equilibria (NE). Hence cities could end up in a Prisoner's Dilemma where they are worse off as a result of being incentivised to engage in the game. Switching to a logit based Stochastic User equilibrium (SUE) framework, this paper investigates the existence of multiple equilibria through simulations. In addition, we are able to implement a gradient based solution algorithm for the EPEC. We show that the issue of multiple NE continues to exist for a wide range of dispersion parameter assumptions studied but that as dispersion increases then the smoothing process of SUE can reduce the problem to having only one NE which can have significant implications for optimal tolls and regulation policy.

关键词： Route Choice Stochastic User equilibrium equilibrium problems with equilibrium constraints Multiple Nash Equilibria Sequential Complementarity Linear Programming

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Cordon toll competition in a network of two cities: Formulation and sensitivity to traveller route and demand responses

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TRANSPORTATION RESEARCH PART B-METHODOLOGICAL 2015年 76卷 93-116页

作者： Watling, D. P. Shepherd, S. P. Koh, A. Univ Leeds Inst Transport Studies Leeds LS2 9JT W Yorkshire England

While there exists extensive literature on the first- and second-best tolling of congested transportation networks, much of it presumes the existence of a single agent responsible for toll-setting. The present paper extends the small but growing body of work studying the impact of several agents independently regulating tolls on different parts of a network. Specifically we consider the problem of a network consisting of two 'cities', each city independently regulated by a city 'authority' able to set a single cordon toll for entry to the city. It is supposed that each authority aims to maximise the social welfare of its own residents, anticipating the impact of its toll on travellers' route and demand decisions, while reacting to the toll level levied by the other authority. In addition, we model the possibility of the cities entering into a 'tax-exporting agreement', in which city A agrees to share with city B the toll revenues it collects from city B residents using city A's network. It is assumed that the sensitivity of travellers, in terms of their route and demand responses, is captured by an elastic demand, Stochastic User equilibrium (SUE) model. Conditions for a Nash equilibrium (NE) between cities are set out as an equilibrium Problem with equilibrium constraints (EPEC). It is shown that weaker, 'local' solutions to the EPEC (which we term LNE for local NE) satisfy a single variational inequality, using the smooth implicit function of the SUE map. Standard variational algorithms may then be used to identify such LNE solutions, allowing NE solutions to be identified from this candidate set;we test the use of a Sequential Linear Complementarity Problem algorithm. Numerical results are reported in which we see that the sensitivity of travellers may affect many factors, including: the number of LNE solutions, the initial conditions for which algorithms might determine such solutions, the gap between LNE and a global regulator solution, and the incentive for ci

关键词： Route choice Stochastic User equilibrium equilibrium problems with equilibrium constraints Multiple Nash Equilibria Complementarity problem

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