**ABSTRACT**

This work presents the design and synthesis of a processing unit for numbers encoded according to the recently introduced BAN format.

Such an encoding allows one to represent numbers which are not only finite (as the reals) but also infinitely large or infinitely small, i.e., non-Archimedean.

The motivation behind this study is the significant burst the non-Archimedean numerical computations have received in the last 20 years and the applications that have been found.

With a hardware support, this operations would significantly increase in speed, enlarging the spectrum of possible applications to industrial and real-time ones.

**KEYWORDS**

Non-Archimedean fields, Alpha Theory, Bounded Algorithmic Number (BAN), Arithmetic Unit, FPGA.

]]>**ABSTRACT**

This work aims at reviewing the state of the art of the field of lexicographic multi/many-objective optimization.

The discussion starts with a review of the literature, emphasizing the numerous application in the real life and the recent burst received by the advent of new computational frameworks which work well in such contexts, e.g., Grossone Methodology.

Then the focus shifts on a new class of problems proposed and studied for the first time only recently: the Priority-Levels Mixed-Pareto-Lexicographic Multi-Objective-Problems (PL-MPL-MOPs).

This class of programs preserves the original preference ordering of pure many-objective lexicographic optimization, but instantiates it over multi-objective problems rather than scalar ones.

Interestingly, PL-MPL-MOPs seem to be very well qualified for modeling real world tasks, such as the design of either secure or fast vehicles.

The work also describes the implementation of an evolutionary algorithm able to solve PL-MPL-MOPs, and reports its performance when compared against other popular optimizers.

**KEYWORDS**

Many-Objective Optimization, Lexicographic Optimization, Evolutionary Computation, Paretian Optimization, Grossone Methodology

**LINK**

Pure and mixed lexicographic-paretian many-objective optimization: state of the art | SpringerLink

]]>**ABSTRACT**

As is well known, players involved in a prisoners' dilemma mayAs is well known, players involved in a prisoners' dilemma maysolve it through its repetition over an infinite horizon, provided theirdiscount factors meet the requirement arising from the version of thefolk theorem being adopted. We prove that if the dilemma is impure,players may indeed neglect this route to achieve the Pareto-optimaloutcome of the constituent stage game through a correlation devicewhich does not identify a correlated equilibrium strategy in the stagegame but does so in the repeated game and, in presence of a Nashbargaining solution, may indeed make time preferences immaterial.Then, we show that the same conclusion holds in continuous strategygames as well, by means of an example based upon an oligopoly modelwith rational expectations and network externalities, with and withoutproduct di¤erentiation, and either quantity or price competition. Thisexample additionally shows that firms may activate a form of collusionwhich is not only independent of their time preferences but also verydifficult to interpret from the standpoint of an antitrust authority.

**KEYWORDS**

Impure prisoners' dilemma; Correlation; RepeatedImpure prisoners dilemma; Correlation; Repeatedgames; Network externalities; Expectations.

]]>**ABSTRACT**

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.

**KEYWORDS**

game theory; path planning; level sets; autonomous vehicles; underwater driving; marine surface driving; coverage games; search games; patrolling; coordination control; pursuer–evader

**LINK**

**ABSTRACT**

Alpha-Theory has been introduced in 1995 to provide a simplified version of Robinson’s Non-Standard Analysis which overcomes the technicalities of symbolic logic. The theory has been improved during the years, and recently it has been used also to solve practical problems in a pure numerical way, thanks to the introduction of the algorithmic numbers. In this paper, we introduce Alpha-Theory using a novel axiomatic approach oriented towards real-world applications, to avoid the need to master mathematical logic and model theory. To corroborate the strong link of this Alpha-Theory axiomatization and scientific computations, we report numerical illustrative applications never carried out by means of non-standard numbers within a computer, i.e., the computation of the eigenvalues of a non-Archimedean matrix, some computations related to non-Archimedean Markov Chains, and the Cholesky factorization of a non-Archimedean matrix. We also highlight the differences between our numerical routines and pure symbolic approaches: as expected, the former scales better when the dimension of the problem increases.

**KEYWORDS**

Alpha-Theory, Non-Standard Analysis, Non-Archimedean Analysis, Algorithmic Numbers, Non-Archimedean Scientific Computing

**LINK**

To appear

]]>**ABSTRACT**

As is well known, zero-sum games are appropriate instruments for the analysis of several issues across areas including economics, international relations and engineering, among others. In particular, the Nash equilibria of any two-player finite zero-sum game in mixed-strategies can be found solving a proper linear programming problem. This chapter investigates and solves non-Archimedean zero-sum games, i.e., games satisfying the zero-sum property allowing the payoffs to be infinite, finite and infinitesimal. Since any zero-sum game is coupled with a linear programming problem, the search for Nash equilibria of non-Archimedean games requires the optimization of a non-Archimedean linear programming problem whose peculiarity is to have the constraints matrix populated by both infinite and infinitesimal numbers. This fact leads to the implementation of a novel non-Archimedean version of the Simplex algorithm called Gross-Matrix-Simplex. Four numerical experiments served as test cases to verify the effectiveness and correctness of the new algorithm. Moreover, these studies helped in stressing the difference between numerical and symbolic calculations: indeed, the solution output by the Gross-Matrix Simplex is just an approximation of the true Nash equilibrium, but it still satisfies some properties which resemble the idea of a non-Archimedean ε-Nash equilibrium. On the contrary, symbolic tools seem to be able to compute the “exact” solution, a fact which happens only on very simple benchmarks and at the price of its intelligibility. In the general case, nevertheless, they stuck as soon as the problem becomes a little more challenging, ending up to be of little help in practice, such as in real time computations. Some possible applications related to such non-Archimedean zero-sum games are also discussed.

**KEYWORDS**

None

**LINK**

To appear

]]>**ABSTRACT**

This chapter introduces a new class of optimization problems, called Mixed-Pareto-Lexicographic Multi-objective Optimization Problems (MPL-MOPs), to provide a suitable model for scenarios where some objectives have priority over some others. Specifically, this work focuses on two relevant subclasses of MPL-MOPs, namely optimization problems having the objective functions organized as priority chains or priority levels. A priority chain (PC) is a sequence of objectives ordered lexicographically by importance; conversely, a priority level (PL) is a group of objectives having the same importance in terms of optimization, but a lexicographic ordering exists between the PLs. After describing these problems and discussing why the standard algorithms are inadequate, an innovative approach to deal with them is introduced: it leverages the Grossone Methodology, a recent theory that allows handling priorities by means of infinite and infinitesimal numbers. Most interestingly, this technique can be easily embedded in most of the existing evolutionary algorithms, without altering their core logic. Three algorithms for MPL-MOPs are shown: the first two, called PC-NSGA-II and PC-MOEA/D, are the generalization of NSGA-II and MOEA/D, respectively, in the presence of PCs; the third, named PL-NSGA-II, generalizes instead NSGA-II when PLs are present. Several benchmark problems, including some from the real world, are used to evaluate the effectiveness of the proposed approach. The generalized algorithms are compared to other famous evolutionary ones, either priority-based or not, through a statistical analysis of their performances. The experiments show that the generalized algorithms are consistently able to produce more solutions and of higher quality.

**KEYWORDS**

None

**LINK**

To appear

]]>**ABSTRACT**

This paper concerns the study of Mixed Pareto-Lexicographic Multi-objective Optimization Problems where the objectives must be partitioned in multiple priority levels.

A priority level (PL) is a group of objectives having the same importance in terms of optimization and subsequent decision-making, while between PLs a lexicographic ordering exists. A naive approach would be to define a multi-level dominance relationship and apply a standard EMO/EMaO algorithm, but the concept does not conform to a stable optimization process as the resulting dominance relationship violates the transitive property needed to achieve consistent comparisons.

To overcome this, we present a novel approach which merges a custom non-dominance relation with the Grossone methodology, a mathematical framework to handle infinite and infinitesimal quantities.

The proposed method is implemented on a popular multi-objective optimization algorithm (NSGA-II), deriving a generalization of it called by us PL-NSGA-II.

We also demonstrate the usability of our strategy by quantitatively comparing the results obtained by PL-NSGA-II against other priority and non-priority-based approaches.

Among the test cases, we include two real-world applications: one 10-objective aircraft design problem and one 3-objective crash safety vehicle design task.

The obtained results show that PL-NSGA-II is more suited to solve lexicographical many-objective problems than the general purpose EMaO algorithms.

**KEYWORDS**

Multi-Objective Optimization, Lexicographic Optimization, Evolutionary Computation, Genetic Algorithms, Numerical Infinitesimals, Grossone Methodology

**LINK**

**ABSTRACT**

Zero-sum games are a well known class of game theoretic models, which are widely used in several economics and engineering applications.

It is known that any two-player finite zero-sum game in mixed-strategies can be solved, i.e., one of its Nash equilibria can be found solving a linear programming problem associated to it.

The idea of this work is to propose and solve zero-sum games which involve infinite and infinitesimal payoffs too, that is non-Archimedean payoffs.

Since to find a Nash equilibrium a non-Archimedean linear programming problem needs to be solved, we implement and extend a more powerful version of an already existing non-Archimedean Simplex algorithm, namely the Gross-Simplex one.

In particular, the new algorithm, called Gross-Matrix-Simplex, is able to handle the constraint matrix A when it is made of non-Archimedean quantities.

To test the correctness and the efficiency of the Gross-Matrix-Simplex algorithm, we provide four numerical experiments, which have been run on an Infinity Computer simulator.

Furthermore, we stressed the difference between numerical and symbolic calculations, characterizing the solutions that an algorithm is able to output running over a finite-precision machine.

In particular, we showed that the numerical solutions are particular approximations of the true Nash equilibrium which satisfy some properties which make them interestingly close to the concept of an non-Archimedean ε-Nash equilibrium.

Finally, we also discuss several examples based on well known models related to economics, politics and engineering, where a non-Archimedean zero-sum model appears to be a reasonable, powerful and flexible representation.

**KEYWORDS**

Game Theory, Zero-Sum Matrix Games, Linear Programming, Non-Archimedean Analysis, Grossone Methodology, Infinity Computer

**LINK**

**ABSTRACT**

This paper studies a class of mixed Pareto-Lexicographic multi-objective optimization problems where the preference among the objectives is available in different *priority levels* (PLs) before the start of the optimization process -- akin to many practical problems involving domain experts.

Each priority level (PL) is a group of objectives having an identical importance in terms of optimization, so that they must be optimized in the standard Pareto sense. However, between two PLs, a lexicographic preference structure exists. Clearly, finding the entire set of Pareto optimal solutions first and then choosing the lexicographic solutions using the given PL structure is not computationally efficient.

A new efficient algorithm is presented here using a recent mathematical breakthrough in handling infinite and infinitesimal quantities: the *Grossone* methodology.

The proposal has been implemented within a popular multi-objective optimization algorithm (NSGA-II), thereby obtaining its generalized version named PL-NSGA-II, although other EMO or EMaO algorithms could have also been used instead.

A quantitative comparison of PL-NSGA-II performance against existing algorithms is made. Results clearly show the advantage of the proposed Grossone-based methodology in solving such priority-level many-objective problems.

**KEYWORDS**

Multi-Objective Optimization, Lexicographic Optimization, Evolutionary Computation, Grossone Methodology

**LINK**