Special Session 46: Advances in Optimization and Equilibrium Problems: methods and applications

Monotonicity of equilibria in nonatomic congestion games

Valerio Dose Sapienza University of Rome Italy Co-Author(s):    Roberto Cominetti, Valerio Dose, Marco Scarsini

Abstract: Equilibria in nonatomic congestion games model how traffic distributes across resources when independent agents seek to minimize their own delays. While one might expect increased demand to uniformly increase resource usage, paradoxical non-monotone behavior is a well-documented phenomenon. In this talk, we investigate the monotonicity of equilibrium loads and costs as functions of total demand. In the classic case of single-commodity routing, it is known that network topology alone dictates these paradoxes. However, with multiple commodities, the combinatorial structure of strategy sets also plays an important role. We first demonstrate that singleton congestion games maintain monotone equilibrium loads with respect to any demand. We then extend this result to a broader class of multi-commodity games, which we define as Constrained Series-Parallel (CSP) games. Finally, we show that CSP games can be represented as variant of multi-commodity routing games on series-parallel networks, bridging the gap between abstract strategy sets and physical network topology.

Ghost penalties and stochastic programming

Francisco Facchinei University of Rome La Sapienza Italy Co-Author(s):

Abstract: After recalling some recent results on ghost penalties, we discuss how this approach to constrained optimization can be used to develop new convergence results in stochastic programming. We cover the fully general case where both the objective function and the inequality constraints are non convex expected value functions.

Advances on Strong Duality Theory: Necessary and Sufficient Conditions and Applications

Sofia Giuffre` University of Reggio Calabria Italy Co-Author(s):    Antonino Maugeri, Attilio Marciano`

Abstract: Aim of the talk is to discuss and refine strong duality theory in infinite-dimensional settings. In particular, we deal with three constraint qualification conditions, Assumption S, Assumption S`, and Condition NES, which are necessary and sufficient conditions for strong duality, highlighting their relationship with the saddle points of the Lagrange functional and the equivalence with a global optimization problem. Moreover, these assumptions prove to be very useful in many applications. Indeed, many equilibrium problems may be expressed in terms of variational or quasi-variational inequalities, in which these assumptions work. In the talk, in particular, we apply strong duality theory to the existence of Lagrange multipliers in the difficult partial differential equations setting of non-constant gradient constrained problem.

Toward an Integrated Reduced Order Modeling Framework for Inverse Problems in Linear Elasticity: A Preliminary Study

Basca Jadamba Rochester Institute of Technology USA Co-Author(s):    Basca Jadamba and Yidan Yang

Abstract: The solution of coefficient inverse problems in partial differential equations (PDEs) remains a significant computational challenge, particularly in the context of linear elasticity where the vector-valued state variable and the high dimensionality of the parameter space result in substantial computational costs. This talk presents an exploratory framework for accelerating the estimation of the shear modulus ($\mu$) in nearly incompressible media by leveraging Reduced Order Modeling (ROM) techniques. While ROM is traditionally applied to accelerate forward simulations, its integration into the optimization loop for inverse problems requires careful alignment between the reduced basis and the trajectory of the optimizer. We discuss an initial approach using Proper Orthogonal Decomposition (POD) to construct a reduced basis for the forward and adjoint operators within a gradient-based optimization routine. Preliminary results comparing the efficiency and reconstruction accuracy of this reduced-order approach for scalar elliptic problems are presented. An ongoing extension of this framework to the nearly incompressible elasticity equations will be discussed. This work represents an initial attempt to move toward a more goal-oriented ROM strategy where the reduced basis is informed by the requirements of the inversion process.

An Augmented Lagrangian Framework for Stochastic Optimization: Applications to Optimal Control and Parameter Estimation

Akhtar A Khan Rochester Institute of Technology USA Co-Author(s):

Abstract: Inversion and control of systems under uncertainty pose significant computational challenges, particularly in high-dimensional stochastic environments. In this talk, I will present a rigorous, end-to-end mathematical and computational framework for solving constrained stochastic optimization problems governed by linear operator equations with uncertain coefficients. We will begin by establishing the problem within the setting of Bochner spaces. By adopting a finite-dimensional noise (FDN) assumption, I will demonstrate how to transform the primary optimization problem into a regularized saddle-point framework via a continuous augmented Lagrangian. I will also share our convergence analysis for the discretized system using a finite-dimensional Galerkin approximation, proving that our discrete solutions reliably converge to the continuous minimizer. To address practical implementation, the second half of the talk will focus on a newly developed Uzawa-type iterative algorithm with guaranteed strong convergence. Finally, I will introduce a fully discrete computational framework utilizing tensor products of spatial and stochastic basis functions. By detailing the explicit assembly of parameter-dependent global block matrices and the derivation of closed-form block gradients and Hessians, this talk will highlight how we can successfully bridge the gap between high-level functional analysis and efficient, robust numerical implementation.

Trust and reputation systems in cloud data centers: a variational approach

Attilio Marciano` University Mediterranea of Reggio Calabria Italy Co-Author(s):

Abstract: The aim of this talk is to present a novel equilibrium model based on trust and reputation mechanisms for the analysis of cloud data centers, namely physical or virtual infrastructures providing computing resources and services. In this framework, servers mutually evaluate each other with the objective of maximizing a performance measure defined as the difference between benefits and costs, expressed in terms of reputation. The reputation of each server is modeled as a weighted aggregation of the trust values it receives, where the weights depend on the distances among servers. A variational approach is adopted to characterize the equilibrium configuration of the system from the user perspective, and theoretical results on existence and uniqueness are discussed. Finally, numerical simulations are presented to illustrate and validate the model, together with a sensitivity analysis highlighting the impact of initial reputation levels and distance parameters on the resulting equilibrium

Competitive Dynamics and Trust Mechanisms in Crowdsourced Delivery Networks: An Uncertain Variational Equilibrium Approach

Laura Rosa Maria Scrimali University of Catania Italy Co-Author(s):

Abstract: The exponential expansion of on-demand delivery services (Uber Eats, Deliveroo, Glovo) has introduced intricate market mechanisms where courier trustworthiness emerges as a pivotal yet ambiguous element. Newly onboarded couriers typically lack substantial track records; platform assessments depend heavily on perception rather than rigorous statistical validation; and reputation metrics remain vulnerable to malicious behavior. This research introduces an innovative competitive equilibrium model grounded in uncertain variational inequalities, in which courier reliability metrics are represented as uncertain variables within Liu's uncertainty framework. Couriers engage in spatial competition for zone assignments while satisfying probabilistic constraints that require minimum trustworthiness thresholds at predetermined confidence levels. Our analytical approach transforms these uncertain restrictions into their deterministic counterparts using inverse uncertainty distributions, subsequently expressing the Nash equilibrium solution as a variational inequality problem. Computational simulations demonstrate how entry barriers tied to reputation scores influence both equilibrium resource distribution and platform efficiency. The findings highlight that carefully tuned reliability thresholds combined with adaptive confidence-level adjustments serve as effective policy instruments for equitable market participation, yielding actionable insights for digital platform governance and economic regulation.