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Solving equilibrium problems using extended mathematical programming

MATHEMATICAL programming COMPUTATION 2019年第3期11卷 457-501页

作者： Kim, Youngdae Ferris, Michael C. Argonne Natl Lab Math & Comp Sci Div 9700 Cass Ave Lemont IL 60439 USA Univ Wisconsin Dept Comp Sci 1210 West Dayton St Madison WI 53706 USA Univ Wisconsin Wisconsin Inst Discovery 1210 West Dayton St Madison WI 53706 USA

We introduce an extended mathematical programming framework for specifying equilibrium problems and their variational representations, such as generalized Nash equilibrium, multiple optimization problems with equilibrium constraints, and (quasi-) variational inequalities, and computing solutions of them from modeling languages. We define a new set of constructs with which users annotate variables and equations of the model to describe equilibrium and variational problems. Our constructs enable a natural translation of the model from one formulation to another more computationally tractable form without requiring the modeler to supply derivatives. In the context of many independent agents in the equilibrium, we facilitate expression of sophisticated structures such as shared constraints and additional constraints on their solutions. We define shared variables and demonstrate their uses for sparse reformulation, economic equilibrium problems sharing economic states, mixed pricing behavior of agents, and so on. We give some equilibrium and variational examples from the literature and describe how to formulate them using our framework. Experimental results comparing performance of various complementarity formulations for shared variables are provided. Our framework has been implemented and is available within GAMS/EMP.

关键词： equilibrium programming Nash equilibrium problems Quasi-variational inequalities

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Antisymmetric Extremal Mapping and Linear Dynamics

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2025年第3期65卷 468-485页

作者： Antipin, A. S. Khoroshilova, E. V. Russian Acad Sci Comp Ctr Fed Res Ctr Comp Sci & Control Moscow 119333 Russia Lomonosov Moscow State Univ Fac Computat Math & Cybernet Moscow 119991 Russia

An optimal control problem is considered on a fixed time interval. Choosing a control generates a phase trajectory of this problem. The left end of the trajectory is fixed, while a finite-dimensional problem of calculating a fixed point of an extremal mapping is set up at the right end. In the optimal situation, the right end of the phase trajectory coincides with the fixed point of the mapping. In other words, the task is, by choosing a suitable control, to construct a phase trajectory in a Hilbert space that leaves the initial position at the left end of the time interval and arrives at the fixed point of the extremal mapping at the right end of the time interval. To solve the problem within the framework of the Lagrangian formalism, we propose a new approach based on saddle point sufficient optimality conditions. An iterative computational process of saddle point gradient type is investigated. The process is proved to converge strongly in phase and dual trajectories, as well as in terminal variables of the finite-dimensional boundary value problem of linear programming, and to converge weakly in controls. The emphasis is placed on the fact that only proof-based computational techniques transform a mathematical model into a tool for making a guaranteed decision.

关键词： optimal control Lagrangian formalism duality equilibrium programming saddle point methods convergence proof-based computations

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Shooting method for solving equilibrium programming problems

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2013年第12期53卷 1819-1824页

作者： Budak, B. A. Moscow MV Lomonosov State Univ Fac Computat Math & Cybernet Moscow 119991 Russia

A new iterative method is proposed for solving equilibrium programming problems. The sequence of points it generates is proved to converge weakly to the solution set of the equilibrium problem under study. If the initial point has at least one projection onto the solution set of the equilibrium problem, the sequence generated by the method is shown to converge strongly to the set of these projections. The partial gradient of the initial data is assumed to be invertible and strictly monotone, which differs from the classical skew-symmetry condition.

关键词： equilibrium programming invertible and strictly monotone gradient shooting method strong convergence weak convergence skew-symmetry

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Simulating the Evolution of Business Models for Electricity Recharging Infrastructure Development by 2030: A Case Study for Greece

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ENERGIES 2021年第9期14卷 2345页

作者： Statharas, Stergios Moysoglou, Yannis Siskos, Pelopidas Capros, Pantelis Natl Tech Univ Athens E3MLab Sch Elect & Comp Engn 9 Iroon Polytech St Athens 15773 Greece E3 Modelling SA Panormou 70-72 Athens 11524 Greece

It is widely accepted that the market uptake of electric vehicles is essential for the decarbonisation of transport. However, scaling up the roll out of electric vehicles (EV) is challenging considering the lack of charging infrastructure. The latter is, currently, developing in an uneven way across the EU countries. A charging infrastructure with wide coverage addresses range limitations but requires high investment with uncertain returns during the early years of deployment. The aim of this paper is to assess how different policy options affect EV penetration and the involvement of private sector in infrastructure deployment. We propose a mathematical programming model of the decision problem and the interaction between the actors of EV charging ecosystem and apply it to the case of Greece from the time period until 2030. Greece represents a typical example of a country with ambitious targets for EV penetration by 2030 (10% of the total stock) with limited effort made until now. The results indicate that it is challenging to engage private investors in the early years, even using subsidies;thus, publicly financed infrastructure deployment is important for the first years. In the mid-term, subsidization on the costs of charging points is necessary to positively influence the uptake of private investments. These are mainly attracted from 2025 onwards, after a critical mass of EVs and infrastructure has been deployed.

关键词： electric vehicles electricity recharging infrastructure business models equilibrium programming Greek EV mobility 2030 private investments in infrastructure

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Investigating the Ability of Demand Shifting to Mitigate Electricity Producers' Market Power

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IEEE TRANSACTIONS ON POWER SYSTEMS 2018年第4期33卷 3800-3811页

作者： Ye, Yujian Papadaskalopoulos, Dimitrios Strbac, Goran Imperial Coll London Dept Elect & Elect Engn London SW7 2AZ England

Previous work on the role of the demand side in imperfect electricity markets has demonstrated that its self-price elasticity reduces electricity producers' ability to exercise market power. However, the concept of self-price elasticity cannot accurately capture consumers' flexibility, as the latter mainly involves shifting of loads' operation in time. This paper provides for the first time theoretical and quantitative analysis of the beneficial impact of demand shifting (DS) in mitigating market power by the generation side. Quantitative analysis is supported by a multiperiod equilibrium programming model of the imperfect electricity market, accounting for the time-coupling operational constraints of DS as well as network constraints. The decision making process of each strategic producer is modeled through a bi-level optimization problem, which is solved after converting it to a Mathematical Program with equilibrium Constraints (MPEC) and linearizing the latter through suitable techniques. The oligopolistic market equilibria resulting from the interaction of multiple independent producers are determined by employing an iterative diagonalization method. Case studies on a test market reflecting the general generation and demand characteristics of the GB system quantitatively demonstrate the benefits of DS in mitigating producers' market power, by employing relevant indexes from the literature.

关键词： Bi-level optimization demand shifting electricity markets equilibrium programming market power

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Multi-Period and Multi-Spatial equilibrium Analysis in Imperfect Electricity Markets: A Novel Multi-Agent Deep Reinforcement Learning Approach

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IEEE ACCESS 2019年 7卷 130515-130529页

作者： Ye, Yujian Qiu, Dawei Li, Jing Strbac, Goran Fetch AI Cambridge CB4 0WS England Imperial Coll London Fac Engn Dept Elect & Elect Engn London SW7 2AZ England

Previously works on analysing imperfect electricity markets have employed conventional game-theoretic approaches. However, such approaches necessitate that each strategic market player has full knowledge of the operating parameters and the strategies of its rivals as well as the computational algorithm of the market clearing process. This unrealistic assumption, along with the modeling and computational complexities, renders such approaches less applicable for conducting practical multi-period and multispatial equilibrium analysis. This paper proposes a novel multi-agent deep reinforcement learning (MA-DRL) based methodology, combining multi-agent intelligence, the deep policy gradient (DPG) method, and an innovative long short term memory (LSTM) based representation network for optimizing the offering strategies of multiple self-interested generation companies (GENCOs) as well as exploring the market outcome stemming from their interactions. The proposed approach is tailored to align with the nature of the examined problem by posing it, for the first time, in multi-dimensional continuous state and action spaces, enabling GENCOs to receive accurate feedback regarding the impact of their offering strategies on the market clearing outcome, and devise more profitable bidding decisions by exploiting the entire action domain, and thereby facilitates more accurate equilibrium analysis. The proposed LSTM-based representation network extracts discriminative features which further improves the learning performance and thus promises more profitable offerings strategies for each GENCO. Case studies demonstrate that the proposed method i) achieves a significantly higher profit than state-of-the-art RL methods for a single GENCO's optimal offering strategy problem and ii) outperforms the state-of-the-art equilibrium programming models in efficiently identifying an imperfect market equilibrium with / without network congestion. Quantitative economic analysis is carried out on the

关键词： Deep neural networks deep reinforcement learning electricity markets equilibrium programming imperfect competition multi-agent intelligence strategic offering

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Exploring the Role of Demand Shifting in Oligopolistic Electricity Markets

Exploring the Role of Demand Shifting in Oligopolistic Elect...

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2017 IEEE Power & Energy Society General Meeting

作者： Papadaskalopoulos, Dimitrios Ye, Yujian Strbac, Goran Imperial Coll London Dept Elect & Elect Engn London England

ISBN: (纸本)9781538622124

Previous work has demonstrated that the price elasticity of the demand side reduces electricity producers' ability to exercise market power. However, price elasticity cannot capture alone consumers' flexibility, as the latter mainly involves shifting of loads' operation in time. This paper provides for the first time qualitative and quantitative analysis of the value of demand shifting in mitigating market power by the generation side. An equilibrium programming model of the oligopolistic market setting is developed, taking into account the inter-temporal characteristics of demand shifting. The decision making process of each strategic producer is modelled through a bi-level optimization problem, which is solved after transforming it to a Mathematical Program with equilibrium Constraints (MPEC). The market equilibria resulting from the interaction of multiple independent producers are determined by employing an iterative diagonalization method. Case studies on a test market with day-ahead horizon and hourly resolution quantitatively demonstrate the benefits of demand shifting in limiting generation market power, by employing relevant indexes from the literature.

关键词： Bi-level optimization demand shifting electricity markets equilibrium programming market power mathematical program with equilibrium constraints

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Variable-Metric Continuous Proximal Methods

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Computational Mathematics and Modeling 2015年第1期26卷 87-106页

作者： Budak, B.A. Lomonosov Moscow State University Faculty of Computation Mathematics and Cybernetics Moscow Russian Federation

The article considers a continuous extra-proximal method for equilibrium-programming problems and proves the convergence of its trajectory to one of the solutions. A regularized analogue is constructed under classical assumptions regarding errors in input data. Its convergence to the normal solution is proved. © 2014, Springer Science+Business Media New York.

关键词： equilibrium programming extra-proximal method

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Entropy Methods for equilibrium programming with Its Application in Traffic

Entropy Methods for Equilibrium Programming with Its Applica...

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第八届运筹学及其应用国际研讨会

作者： Baibai Fu School of Architecture and Urban Planning Shandong Jianzhu University State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University

In this paper,we first develop the concept of the maximum entropy function for equilibrium programming by employing prior distribution information and Kullback entropy. It plays a key role in entropy function method for solving equilibrium *** analyze and discuss the relation with the model transformation equivalent applied in mixed traffic assignment problems,and finally we give an optimal model and effective algorithm for mixed traffic assignment problems.

关键词： Operations research equilibrium programming entropy function mixed traffic assignment.

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A regularized Newton method for solving equilibrium programming problems with an inexactly specified set

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Computational Mathematics and Mathematical Physics 2007年第1期47卷 19-31页

作者： Antipin, A.S. Stukalov, A.S. Dorodnicyn Computing Center Russian Academy of Sciences Moscow 119991 ul. Vavilova 40 Russian Federation Faculty of Computational Mathematics and Cybernetics Moscow State University Leninskie Gory Moscow 119992 Russian Federation

Unstable equilibrium problems are examined in which the objective function and the set where the equilibrium point is sought are specified inexactly. A regularized Newton method, combined with penalty functions, is proposed for solving such problems, and its convergence is analyzed. A regularizing operator is constructed. © Nauka/Interperiodica 2007.

关键词： equilibrium programming Newton method Penalty functions Regularization

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