Non-Archimedean zero-sum games

Cococcioni, Marco; Fiaschi, Lorenzo; Lambertini, Luca

doi:10.1016/j.cam.2021.113483

Cited by 16 publications

(20 citation statements)

References 29 publications

(54 reference statements)

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“…The reasons are two: i) the algorithm should manage and manipulate an infinite amount of data; ii) the machine is finite and cannot store all that information. Notice that, at this stage, the focus is not on variable-length representations of Euclidean numbers, as they would slow the computations down [14]. Fixed-length representations, as the ones discussed in [11] are therefore preferred, also because they are easier to implement in hardware (i.e., they are more "hardware friendly").…”

Section: Non-archimedean Interior Point Methodsmentioning

confidence: 99%

“…Later, Lustig et al [44] showed that directions coincide with those of an infeasible IPM, without solving the unboundedness issue actually. When considering a set of numbers larger than R as E however, an approach in the middle between (14) and the one by Lustig is possible. It consists in the use infinitely big penalizing weights, i.e., in a non-Archimedean embedding.…”

Section: Infeasibility and Unboundednessmentioning

confidence: 99%

“…Recent literature has started to reconsider lexicographic problems, trying to solve the issues coming from both scalarization and preemptive approaches leveraging recent developments in numerical non-Archimedean computations [11,12,13]. Some results involve game theory [14,15,16], evolutionary optimization [17,18,19], linear programming tackled by Simplex-like methods [20,4,21,22]. Actually, the key idea is to scalarize the problem adopting infinite and infinitesimal weights.…”

Section: Introductionmentioning

confidence: 99%

“…Lexicographic interpretation apart, NA-IPM is also able to tackle a huge variety of linear and quadratic non-Archimedean optimization problems as well, as opposed to the non-Archimedean extension of the Simplex algorithm in [4]. Examples of such problems are the non-Archimedean zero-sum games presented in [14].…”

Section: Introductionmentioning

confidence: 99%

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A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

Fiaschi¹,

Cococcioni²

2021

Preprint

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This work presents a generalized implementation of the infeasible primal-dual Interior Point Method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here non-Archimedean IPM (NA-IPM), is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. In addition, infeasibility and unboundedness are no more corner cases to care of: by means of a mild embedding (addition of two variables and one constraint) NA-IPM implicitly and transparently manages their possible presence. To support the theoretical properties of NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool. Finally, the results of this study prove that non-Archimedean scientific computing is an extension of the standard one, suggesting it might have a huge number of practical applications in the near future in coping with both already existing and new real-world problems.

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Section: Non-archimedean Interior Point Methodsmentioning

confidence: 99%

Section: Infeasibility and Unboundednessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

Fiaschi¹,

Cococcioni²

2021

Preprint

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“…Recent advances for multi-objective prioritized optimization, whether lexicographic [110][111][112] or Pareto-lexicographic [113][114][115], have developed programming tools and algorithms that have given new life to the study of lexicographic game theory [116,117]. As a consequence, the numerical study and solution of the practical problems mentioned above seem possible, possibly paving the way for a new approach to autonomous marine path planning.…”

Section: Opportunities and Way Aheadmentioning

confidence: 99%

Game Theory for Unmanned Vehicle Path Planning in the Marine Domain: State of the Art and New Possibilities

Cococcioni

Fiaschi

Lermusiaux

2021

JMSE

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Thanks to the advent of new technologies and higher real-time computational capabilities, the use of unmanned vehicles in the marine domain has received a significant boost in the last decade. Ocean and seabed sampling, missions in dangerous areas, and civilian security are only a few of the large number of applications which currently benefit from unmanned vehicles. One of the most actively studied topic is their full autonomy; i.e., the design of marine vehicles capable of pursuing a task while reacting to the changes of the environment without the intervention of humans, not even remotely. Environmental dynamicity may consist of variations of currents, the presence of unknown obstacles, and attacks from adversaries (e.g., pirates). To achieve autonomy in such highly dynamic uncertain conditions, many types of autonomous path planning problems need to be solved. There has thus been a commensurate number of approaches and methods to optimize this kind of path planning. This work focuses on game-theoretic approaches and provides a wide overview of the current state of the art, along with future directions.

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Computing Optimal Decision Strategies Using the Infinity Computer: The Case of Non-Archimedean Zero-Sum Games

Cococcioni

Fiaschi

Lambertini

2022

Emergence, Complexity and Computation

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Non-Archimedean zero-sum games

Cited by 16 publications

References 29 publications

A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

Game Theory for Unmanned Vehicle Path Planning in the Marine Domain: State of the Art and New Possibilities

Computing Optimal Decision Strategies Using the Infinity Computer: The Case of Non-Archimedean Zero-Sum Games

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