Special Session 91: Advances on Explainable Artificial Intelligence and related Mathematical Modeling

Quasivariational Inequalities for Dynamic Competitive Economic Equilibrium Problems in Discrete Case

David Barilla Messina Italy Co-Author(s):    Shapour Heidarkhani, Giuseppe Caristi

Abstract: Equilibrium is a central concept in numerous disciplines including economics, management science, operations research, and engineering. We are concerned with an evolutionary quasivariational inequality which is connected discrete dynamic competitive economic equilibrium problem in terms of maximization of utility functions and of excess demand functions. We study the discrete equilibrium problem by means of a discrete time-dependent quasivariational inequality in the discrete space $\ell^2([0,T]_{\mathbb{Z}},\mathbb{R})$. We ensure an existence result of discrete time-dependent equilibrium solutions. Finally, We show the stability of equilibrium in a completely decentralized Walrasian general equilibrium economy in which prices are fully controlled by economic agents, with production and trade occurring out of equilibrium.

Environmental Policy: The Coevolution of Pollution and Compliance

Marta Biancardi Univerity of Bari Italy Co-Author(s):    Marta Biancardi,Yannis Petrohilos-Andrianos, Giovanni Villani , Anastasios Xepapadeas

Abstract: The paper studies the evolution of compliance and non-compliace firms in a setup of taxable emissions. Firms can either choose to comply with the emissions standard imposed by a public authority or violate them. Two different scenarios are analyzed, in fact, violation is considered either as a single option or is let to vary between low and high emissions. The latter induce a different level of fine if firms get caught. In a context of replicator dynamics, firms can switch between strategies according to a proportional rule of strategy evolution and the conditions for stability are investigated accounting for two distinct types of probability of inspection.

On Duality for Nonsmooth Mathematical Problems with Vanishing Constraints

Giuseppe Caristi University of Messina Italy Co-Author(s):    Nader Kanzi, Hamed Soroush, Giuseppe Caristi and David Barilla

Abstract: In this paper, we provide a duality theory for nonsmooth optimization problems with vanishing constraints (MPVC) defined by locally Lipschitz functions. In order to do this, we first formulate a new mixed-type dual problem for an MPVC, which is a generalization of Wolf and Mond-Weir dual problems. Since this dual problem depends on the feasible points of the primal problem, we introduce another mixed-type dual problem that does not have this dependence. Then, we present the weak, the strong, the converse, the restricted converse, and the strict converse duality results for these parametric dual problems. Finally, we compare the results of the article written by Mishra (2016) with our results and state the correct version of some of its incorrect theorems.

Explainable Artificial Intelligence and Mathematical Modeling: New Challenges of Research on

Massimiliano Ferrara University Mediterranea of Reggio Calabria Italy Co-Author(s):

Abstract: \begin{abstract} The increasing complexity of artificial intelligence (AI) models has led to a rising demand for explainability in AI (XAI). Explainable Artificial Intelligence aims to make AI`s decision-making processes transparent and understandable to humans. This talk examines the integral connection between XAI and mathematics, highlighting how mathematical principles can enhance the interpretability, transparency, and trustworthiness of AI models. We explore the mathematical foundations that underpin XAI techniques, examine case studies where mathematics has improved explainability, and propose future directions for integrating mathematics into XAI frameworks. \end{abstract}

EvoFolio: a portfolio optimization method based on multi-objective evolutionary algorithms

Luca Grilli University of Foggia Italy Co-Author(s):    Alfonso Guarino, Domenico Santoro, Luca Grilli, Rocco Zaccagnino, Mario Balbi

Abstract: \textit{Optimal portfolio selection} -- composing a set of stocks/assets that provide high yields/returns with a reasonable risk -- has attracted investors and researchers for a long time. As a consequence, a variety of methods and techniques have been developed, spanning from purely mathematics ones to computational intelligence ones. In this paper, we introduce a method for optimal portfolio selection based on multi-objective evolutionary algorithms, specifically \textit{Nondominated Sorting Genetic Algorithm}-II (NSGA-II), which tries to \textit{maximize} the yield and \textit{minimize} the risk, simultaneously. The system, named \textit{EvoFolio}, has been experimented on stock datasets in a three-years time-frame and varying the configurations/specifics of NSGA-II operators. \textit{EvoFolio} is an \textit{interactive} genetic algorithm, i.e., users can provide their own insights and suggestions to the algorithm such that it takes into account users` preferences for some stocks. We have performed tests with optimizations occurring quarterly and monthly. The results show how \textit{EvoFolio} can significantly reduce the risk of portfolios consisting only of stocks and obtain very high performance (in terms of return). Furthermore, considering the investor`s preferences has proved to be very effective in the portfolio`s composition and made it more attractive for end-users. We argue that \textit{EvoFolio} can be effectively used by investors as a support tool for portfolio formation.

Variational approaches to a dynamic Sturm-Liouville boundary value problem related to time scales and machine learning algorithms

Shapour Heidarkhani Razi University Iran Co-Author(s):    Giuseppe Caristi - Massimiliano Ferrara - Sharin Moradi - David Barilla

Abstract: In this talk, using a consequence of the local minimum theorem due Bonanno, we establish the existence of one solution under algebraic conditions on the nonlinear term. Additionally, we obtain two solutions for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Moreover, by utilizing two critical point theorems, one due Averna and Bonanno, and another one due Bonanno, we ensure the existence of two and three solutions for dynamic Sturm--Liouville boundary value problems on time scales which turns out as an optimization problem in a special case. We provide an example to elucidate the results we have obtained. Our findings are relevant to economics applications. Time scale calculus may be used in any subject where dynamic processes are characterized using discrete or continuous models. Therefore, the conclusions of time scale calculus are immediately applicable to economics since many economic models are dynamic models. At the end, we introduce a Matlab package to calculate as many eigenvalues of the problem as needed and create the required data for training and validating the machine learning algorithms.

Cellular Automata on Probability Measures: Induced Dynamics on Random Graphs and Applications in Explainable AI

Davide La Torre SKEMA Business School, Cote d`Azur University France Co-Author(s):    Enrico Formenti

Abstract: In this talk, we present an extension of the traditional concept of cellular automata (CAs) by broadening the range of possible states. This new framework is inspired by the need to establish dynamics on random graphs. We propose that the state space consists of all probability measures defined over a fundamental alphabet A. This definition simplifies to the classical notion of cellular automata when we use Dirac probability measures for individual elements of the alphabet. We will discuss several results related to the convergence of these generalized cellular automata, particularly under specific conditions on the transition map. Additionally, we will provide various examples to illustrate how this extended approach can be applied in different scenarios. Furthermore, we will explore how cellular automata can be utilized to enhance the explainability of machine learning training processes. This talk aims to enhance our understanding of cellular automata while exploring their potential applications in complex systems.

Coideals as remainders of groups distinguishing between combinatorial covering properties

Giovanni Molica Bisci University of Urbino Carlo Bo Italy Co-Author(s):

Abstract: In this talk we construct consistent examples of subgroups of $2^\omega$ with Menger remainders which fail to have other stronger combinatorial covering properties. This answers several open questions asked by Bella, Tokgoz and Zdomskyy (Arch. Math. Logic 55 (2016), 767-784). The results are presented and proved in Molica Bisci, Repov\v{s}, and Zdomskyy (Topology Appl. 340 (2023), art. 108725, 15 pp.).

Dynamics of a New Keynesian model with heterogeneous expectations: the role of monetary policy

Nicolo Pecora Catholic University Italy Co-Author(s):    Anna Agliari, Alessandro Spelta

Abstract: Modern monetary policy has emphasized that maintaining a stable monetary environment depends crucially on the ability of the policy regime to control inflation (and output) expectations. In fact, the activity of modern Central Banks is a form of management of expectations. The present work considers a standard New Keynesian model, described by a two-dimensional nonlinear map, to analyze the bifurcation structure when agents own heterogeneous expectations on inflation and output gap, and update their beliefs based on past performance. Agents are then allowed to switch among predictors over time. Depending on the degree of reactivity of the monetary policy to inflation and output deviations from the target equilibrium, different kind of dynamics may occur. Multiple equilibria and complicated dynamics, associated to codimension-2 bifurcations, may arise even if the monetary policy adheres to the Taylor principle. We show that if the monetary policy accommodates for a sufficient degree of output stabilization, complicated dynamics can be avoided and the number of coexisting equilibria reduces. In the second part of the analysis, an arbitrarily large number of agents` beliefs is considered by applying the concept of Large Type Limit. In this respect, the intensity of choice or the spread of beliefs is crucial for the extent of the Central Bank to stabilize the economy. When the predictors are largely dispersed around the target, the Taylor principle is a requisite for stability; when the set of beliefs is somehow anchored to the target, stability can be achieved with a weaker monetary policy.

More or Less. A comparison between Machine and Deep Learning Models on high stationarity data

Domenico Santoro University of Foggia Italy Co-Author(s):    Domenico Santoro, Tiziana Ciano, Massimiliano Ferrara

Abstract: Advances in sensor, computing, and communication technologies are enabling big data analytics by providing time series data. However, conventional models struggle to identify sequence features and forecast accuracy. This paper investigates time series features and shows that some Machine Learning algorithms can outperform Deep Learning models. In particular, the problem analyzed concerned the prediction of the number of vehicles passing through an Italian tollbooth in 2021. The dataset, composed of 8766 rows and 6 columns relating to additional tollbooths, proved to have high stationarity and was treated through Machine Learning methods such as Support Vector Machine, Random Forest, and eXtreme Gradient Boosting (XGBoost), as well as Deep Learning through Recurrent Neural Networks with Long Short-Term Memory (RNN-LSTM) cells. From the comparison of these models, the prediction through the XGBoost algorithm outperforms competing algorithms, particularly in terms of MAE and MSE. The result highlights how a shallower algorithm than a Neural Network is, in this case, able to obtain a better adaptation to the time series instead of a much deeper model that tends to develop a smoother prediction.

Ant colony optimization for Chinese postman problem

Giacinto Angelo GA Sgarro University of studies of Foggia Italy Co-Author(s):    Giacinto Angelo Sgarro, Luca Grilli, and Domenico Santoro

Abstract: This paper aims to solve the Chinese Postman Problem (CPP) using an Ant Colony Optimization (ACO) algorithm. In graph theory, the CPP looks for the shortest closed path that visits every edge of a connected undirected graph. This problem has many applications, including route optimization, interactive system analysis, and flow design. Although numerous algorithms aimed at solving CPP are present in the literature, very few meta-heuristic algorithms are proposed, and no ACO applications have been proposed to solve them. This paper tries to fill this gap by presenting an ACO algorithm that solves CPP (ACO-CPP). To prove its consistency and effectiveness, ACO-CPP is compared with a Genetic Algorithm (GA) and a recursive algorithm throughout three experiments: (1) recursive-ACO-GA comparisons over randomly generated graphs for the attainment of the global optimum; (2) ACO-GA statistical comparisons over specifically generated graphs; (3) recursive-ACO-GA comparisons by changing ACO hyperparameters over randomly generated graphs for the attainment of the global optimum. The experiments prove that the ACO-CPP algorithm is efficient and exhibits a consistency similar to GA when the number of possible solutions to explore is relatively low. However, when that number greatly exceeds those explored, ACO outperforms GA. This suggests that ACO is more suitable for solving problems with a CPP structure.

Developing Neural Network Approaches for Analyzing Piecewise Functions in Tuberculosis Treatment Outcomes

Ramsha Shafqat Thammasat University, Rangsit Campus, Thailand Thailand Co-Author(s):    Ramsha Shafqat, Ateq Alsaadi

Abstract: The paper presents a mathematical analysis of tuberculosis, caused by Mycobacterium TB, which primarily affects the lungs. The bacteria spreads through coughing, sneezing, or speaking, and is classified into five categories: susceptible, infected with DS, infected with MDR, isolated, and recovered. The study uses a novel piecewise derivative approach that considers both singular and non-singular kernels to improve our understanding of rabies spread dynamics. The uniqueness of the solution is established using fixed point and piecewise derivative frameworks. A piecewise numerical iteration scheme based on Newton interpolation polynomials is used to obtain the approximate solution. The study contributes to understanding crossover dynamics and the concept of piecewise derivatives. Additionally, an Artificial Neural Network approach is employed for training, testing, and validating data to investigate the disease problem, ensuring high accuracy in training, testing, and validation.