insight - Game theory optimization - # Generalized Nash Equilibrium Problems with Mixed-Integer Variables

Core Concepts

The core message of this paper is to introduce a convexification technique that allows for the characterization of equilibria in generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions, including the important case of games with mixed-integer variables.

Abstract

The paper presents a new approach to characterize equilibria in generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions, including the case of games with mixed-integer variables. The key idea is to reformulate the original non-convex GNEP via a convexification technique using the Nikaido-Isoda function.
The main contributions are:
The authors introduce the concept of convexified instances Iconv for any given instance I of the GNEP. They show that a feasible strategy profile is an equilibrium for the original instance I if and only if it is an equilibrium for any convexified instance Iconv and the convexified cost functions coincide with the initial ones.
The authors develop this convexification approach along three dimensions:
For quasi-linear models, where the convexified instance has a linear cost function and a polyhedral strategy space for each player, the convexification reduces the GNEP to a standard (non-linear) optimization problem.
The authors derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively.
The authors demonstrate the applicability of their 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.

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Key Insights Distilled From

by Tobias Harks... at **arxiv.org** 04-10-2024

Deeper Inquiries

To extend the convexification approach to handle more general classes of non-convex and non-differentiable cost functions and strategy spaces, one can explore techniques such as piecewise linear approximations, reformulations using surrogate functions, or utilizing approximation methods like the convex envelope.
For non-convex cost functions, one approach could involve approximating the non-convex functions with a series of convex functions that closely represent the original function within certain regions. This can be achieved through techniques like piecewise linearization or using convex relaxations. By breaking down the non-convex functions into simpler convex components, the convexification process can still be applied effectively.
Similarly, for non-differentiable cost functions and strategy spaces, one can consider using subgradients or subdifferentials to handle the lack of differentiability. By incorporating subgradients into the convexification process, it becomes possible to extend the approach to non-smooth functions and sets.
Overall, the key idea is to find suitable approximations or representations that allow the non-convex and non-differentiable elements to be transformed into convex formulations, enabling the application of convexification techniques in a broader range of scenarios.

The computational advantages of solving the convexified instances compared to directly tackling the original non-convex GNEP lie in the ability to leverage existing optimization algorithms and tools designed for convex problems.
Efficiency: Convex optimization problems have well-developed theory and efficient algorithms for finding optimal solutions. By convexifying the GNEP, it can be transformed into a form that is more amenable to these optimization techniques, leading to potentially faster and more reliable solutions.
Global Optimality: Convex optimization often guarantees global optimality or provides bounds on the optimality gap, which can be advantageous in ensuring the quality of the solutions obtained.
Simplicity: Convex problems have unique properties that make them easier to solve, such as the absence of local minima, which can simplify the optimization process.
However, there are also limitations to consider:
Approximation Errors: Convexification involves approximating non-convex functions and sets, which can introduce errors in the solution. The quality of the approximation can impact the accuracy of the final equilibrium solution.
Computational Complexity: The process of convexification itself may introduce additional computational complexity, especially for highly non-convex or non-differentiable functions.
Feasibility: While convexification can make the problem more tractable, there is no guarantee that the convexified instance will always have feasible solutions for the original non-convex GNEP.

The concepts of k-restrictive-closed and restrictive-closed GNEPs can be generalized to identify broader classes of GNEPs that admit computationally tractable convexified instances by considering different types of restrictions and constraints on the players' strategy spaces.
Generalized Restrictions: By expanding the definition of restrictions beyond joint constraints, such as considering individual player constraints or more complex interdependencies among players, a wider range of GNEPs can be classified into k-restrictive-closed and restrictive-closed categories.
Structural Properties: Identifying specific structural properties of GNEPs that lead to the existence of convexified instances can help in generalizing the concepts to a broader class of games. This could involve exploring the relationships between the convex hull operator, feasible strategy subsets, and convexified cost functions in different types of games.
Algorithmic Development: Developing algorithms and computational methods that can efficiently handle the convexification process for a wider range of GNEPs is crucial. This may involve exploring different convexification techniques, optimization approaches, and approximation methods tailored to the specific characteristics of the game.
By extending the concepts of k-restrictive-closed and restrictive-closed GNEPs to encompass a more diverse set of game structures and constraints, it becomes possible to identify and solve a broader class of GNEPs using convexification techniques.

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