57 research outputs found
Proximal-like algorithms for equilibrium seeking in mixed-integer Nash equilibrium problems
We consider potential games with mixed-integer variables, for which we
propose two distributed, proximal-like equilibrium seeking algorithms.
Specifically, we focus on two scenarios: i) the underlying game is generalized
ordinal and the agents update through iterations by choosing an exact optimal
strategy; ii) the game admits an exact potential and the agents adopt
approximated optimal responses. By exploiting the properties of
integer-compatible regularization functions used as penalty terms, we show that
both algorithms converge to either an exact or an -approximate
equilibrium. We corroborate our findings on a numerical instance of a Cournot
oligopoly model
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
Convergence Analysis of the Best Response Algorithm for Time-Varying Games
This paper studies a class of strongly monotone games involving
non-cooperative agents that optimize their own time-varying cost functions. We
assume that the agents can observe other agents' historical actions and choose
actions that best respond to other agents' previous actions; we call this a
best response scheme. We start by analyzing the convergence rate of this best
response scheme for standard time-invariant games. Specifically, we provide a
sufficient condition on the strong monotonicity parameter of the time-invariant
games under which the proposed best response algorithm achieves exponential
convergence to the static Nash equilibrium. We further illustrate that this
best response algorithm may oscillate when the proposed sufficient condition
fails to hold, which indicates that this condition is tight. Next, we analyze
this best response algorithm for time-varying games where the cost functions of
each agent change over time. Under similar conditions as for time-invariant
games, we show that the proposed best response algorithm stays asymptotically
close to the evolving equilibrium. We do so by analyzing both the equilibrium
tracking error and the dynamic regret. Numerical experiments on economic market
problems are presented to validate our analysis
Probably Approximately Correct Nash Equilibrium Learning
We consider a multi-agent noncooperative game with agents' objective
functions being affected by uncertainty. Following a data driven paradigm, we
represent uncertainty by means of scenarios and seek a robust Nash equilibrium
solution. We treat the Nash equilibrium computation problem within the realm of
probably approximately correct (PAC) learning. Building upon recent
developments in scenario-based optimization, we accompany the computed Nash
equilibrium with a priori and a posteriori probabilistic robustness
certificates, providing confidence that the computed equilibrium remains
unaffected (in probabilistic terms) when a new uncertainty realization is
encountered. For a wide class of games, we also show that the computation of
the so called compression set - a key concept in scenario-based optimization -
can be directly obtained as a byproduct of the proposed solution methodology.
Finally, we illustrate how to overcome differentiability issues, arising due to
the introduction of scenarios, and compute a Nash equilibrium solution in a
decentralized manner. We demonstrate the efficacy of the proposed approach on
an electric vehicle charging control problem.Comment: Preprint submitted to IEEE Transactions on Automatic Contro
Study Of Stochastic Market Clearing Problems In Power Systems With High Renewable Integration
Integrating large-scale renewable energy resources into the power grid poses several operational and economic problems due to their inherently stochastic nature. The lack of predictability of renewable outputs deteriorates the power grid’s reliability. The power system operators have recognized this need to account for uncertainty in making operational decisions and forming electricity pricing. In this regard, this dissertation studies three aspects that aid large-scale renewable integration into power systems. 1. We develop a nonparametric change point-based statistical model to generate scenarios that accurately capture the renewable generation stochastic processes; 2. We design new pricing mechanisms derived from alternative stochastic programming formulations of the electricity market clearing problem under uncertainty; 3. We devise a novel approach to coordinate strategic operations of multiple noncooperative system operators.
The current industry practices are based on deterministic models that do not account for the stochasticity of renewable energy. Therefore, the solutions obtained from these deterministic models will not provide accurate measurements. Stochastic programming (SP) can accommodate the stochasticity of renewable energy by considering a set of possible scenarios. However, the reliability of the SP model solution depends on the accuracy of the scenarios. We develop a nonparametric statistical simulation method to develop scenarios for wind generation using wind speed data. In this method, we address the nonstationarity issues that come with wind-speed time-series data using a nonparametric change point detection method. Using this approach, we retain the covariance structure of the original wind-speed time series in all the simulated series.
With an accurate set of scenarios, we develop alternative two-stage SP models for the two-settlement electricity market clearing problem using different representations of the non-anticipativity constraints. Different forms of non-anticipativity constraints reveal different hidden dual information inside the canonical two-stage SP model, which we use to develop new pricing mechanisms. The new pricing mechanisms preserve properties of previously proposed pricing mechanisms, such as revenue adequacy in expectation and cost recovery in expectation. More importantly, our pricing mechanisms can guarantee cost recovery for every scenario. Furthermore, we develop bounds for the price distortion under every scenario instead of the expected distortion bounds. We demonstrate the differences in prices obtained from the alternative mechanisms through numerical experiments.
Finally, we discuss the importance of distributed smart grid operations inside the power grid. We develop an information and electricity exchange system among multiple distribution systems. These distribution systems participate/compete in common markets cohere electricity is exchanged. We develop a standard Nash game treating each distribution system (DS) as an individual player who optimizes their strategies separately. We develop proximal best response (BR) schemes to solve this problem. We present results from numerical experiments conducted on three and six DS settings
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