Algorithms for generalized potential games with mixed-integer variables

Sagratella, Simone

doi:10.1007/s10589-017-9927-4

Cited by 46 publications

(49 citation statements)

References 35 publications

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“…This is accomplished in two steps. First (see subsection 5.1), by referring to [16, Theorem 3.2 and Remark 3.2], we replace the pure hierarchical problem (5) by a Generalized Nash Equilibrium Problem (GNEP for short, see e.g., [1,10,11,12,14,22,23,24,25]) with two players, which is parametric in x. Then (see subsection 5.2), we deal with the resulting problem by replacing the lower level game with its Karush-Kuhn-Tucker (KKT) conditions, thus obtaining a corresponding mathematical program with complementarity constraints.…”

Section: How To Address (Spbp ε )mentioning

confidence: 99%

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The Standard Pessimistic Bilevel Problem

Lampariello¹,

Sagratella²,

Stein³

2019

SIAM J. Optim.

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Pessimistic bilevel optimization problems, as optimistic ones, possess a structure involving three interrelated optimization problems. Moreover, their finite infima are only attained under strong conditions. We address these difficulties within a framework of moderate assumptions and a perturbation approach which allow us to approximate such finite infima arbitrarily well by minimal values of a sequence of solvable single-level problems. To this end, as already done for optimistic problems, for the first time in the literature we introduce the standard version of the pessimistic bilevel problem. For its algorithmic treatment, we reformulate it as a standard optimistic bilevel program with a two follower Nash game in the lower level. The latter lower level game, in turn, is replaced by its Karush-Kuhn-Tucker conditions, resulting in a single-level mathematical program with complementarity constraints. We prove that the perturbed pessimistic bilevel problem, its standard version, the two follower game as well as the mathematical program with complementarity constraints are equivalent with respect to their global minimal points. We study the more intricate connections between their local minimal points in detail. As an illustration, we numerically solve a regulator problem from economics for different values of the perturbation parameters.

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Section: How To Address (Spbp ε )mentioning

confidence: 99%

“…In particular, this GNEP is a generalized potential game, see e.g. [13,23]. Any solution of the following convex optimization problem minimize…”

Section: Problems Relations At a Glancementioning

confidence: 99%

The Standard Pessimistic Bilevel Problem

Lampariello¹,

Sagratella²,

Stein³

2019

SIAM J. Optim.

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“…Actually the fixed point x of {x k } is only an equilibrium of the (potential) generalized Nash equilibrium problem (see e.g. [14,17,36]) whose generic player ν ∈ {1, . .…”

Section: Conclusion and Directions For Future Researchmentioning

confidence: 99%

Distributed algorithms for convex problems with linear coupling constraints

Colombo

Sagratella

2019

J Glob Optim

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Distributed and parallel algorithms have been frequently investigated in the recent years, in particular in applications like machine learning. Nonetheless, only a small subclass of the optimization algorithms in the literature can be easily distributed, for the presence, e.g., of coupling constraints that make all the variables dependent from each other with respect to the feasible set. Augmented Lagrangian methods are among the most used techniques to get rid of the coupling constraints issue, namely by moving such constraints to the objective function in a structured, well-studied manner. Unfortunately, standard augmented Lagrangian methods need the solution of a nested problem by needing to (at least inexactly) solve a subproblem at each iteration, therefore leading to potential inefficiency of the algorithm. To fill this gap, we propose an augmented Lagrangian method to solve convex problems with linear coupling constraints that can be distributed and requires a single gradient projection step at every iteration. We give a formal convergence proof to at least ε-approximate solutions of the problem and a detailed analysis of how the parameters of the algorithm influence the value of the approximating parameter ε. Furthermore, we introduce a distributed version of the algorithm allowing to partition the data and perform the distribution of the computation in a parallel fashion.

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“…. Any ε-global minimizer of the potential function P , i.e., anyx ∈ X such that P (x) ≤ P (x) + ε for all x ∈ X , is an ε-MINE of the generalized mixed-integer potential game [18,Th. 2].…”

Section: Automated Driving As a Generalizedmentioning

confidence: 99%

A Mixed-Logical-Dynamical model for Automated Driving on highways

Fabiani

Grammatico

2018

2018 IEEE Conference on Decision and Control (CDC)

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We propose a hybrid decision-making framework for safe and efficient autonomous driving of selfish vehicles on highways. Specifically, we model the dynamics of each vehicle as a Mixed-Logical-Dynamical system and propose simple driving rules to prevent potential sources of conflict among neighboring vehicles. We formalize the coordination problem as a generalized mixed-integer potential game, where an equilibrium solution generates a sequence of mixed-integer decisions for the vehicles that trade off individual optimality and overall safety.

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Algorithms for generalized potential games with mixed-integer variables

Cited by 46 publications

References 35 publications

The Standard Pessimistic Bilevel Problem

The Standard Pessimistic Bilevel Problem

Distributed algorithms for convex problems with linear coupling constraints

A Mixed-Logical-Dynamical model for Automated Driving on highways

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