C.I.A.O.: Constrained Intelligent Aggregative Optimization

🚧 Work In Progress!

This repository accompanies my thesis work on distributed coordination algorithms for constrained aggregative optimization problems.

🔎 Motivation

We study a cloud offloading scenario, where multiple users (e.g., robots, IoT devices, satellites) decide how much computation to execute locally vs. offload to a cloud server.

• Local execution → higher energy consumption, reduced battery lifetime.
• Cloud offloading → possible congestion and capacity limits.

The challenge is to balance local and cloud execution to minimize costs while respecting global cloud capacity. The goal is to add a constraint that involves the aggregation term e.g. the mean of the agents' state.

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🎯 Goal

• Formalize constrained aggregative optimization problems, where costs depend both on individual decisions and on an aggregate variable.
• Design and implement distributed primal-dual algorithms that allow each agent to optimize locally with limited communication.
• Compare with centralized solutions for benchmarking.

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✏️ Problem Formulations

We consider several variants:

1. Box constraints

   • Decision variables constrained to ([0,1]).
   • Centralized: Projected Gradient Descent.
   • Distributed: Aggregative Tracking (consensus-based).

2. Affine constraints

   • Agents subject to linear coupling constraints.
   • Centralized: Saddle-Point Dynamics / Augmented Lagrangian Primal-Dual Gradient Dynamics (Aug-PDGD) [Qu & Li, 2019].
   • Distributed: Distributed Aggregative Primal-Dual Algorithm [Du & Meng, 2025].

3. Aggregate constraints

   • Explicit constraint on the aggregate variable (\sigma(z)).
   • Centralized: Primal-Dual Gradient Method.
   • Distributed: Primal-Dual with Consensus.

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🚀 Algorithms Implemented

• Saddle-Point Dynamics (PDGD)

  • Based on Qu & Li (2019).
  • Implemented in continuous time and discretized via Forward Euler.

• Discrete-Time Primal-Dual (Arrow-Hurwicz-Uzawa)

  • Based on Notarnicola (2024).
  • Direct discrete-time formulation with simplified Lagrangian.

• Distributed Primal-Dual Algorithms

  • Local primal-dual updates.
  • Consensus for tracking global aggregates.

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📚 References

• G. Qu, N. Li, On the Exponential Stability of Primal-Dual Gradient Dynamics, IEEE Control Systems Letters, 2019.
• B. Notarnicola, Semiglobal Exponential Stability of Discrete-Time Primal-Dual Algorithms for Constrained Optimization, Automatica, 2024.
• K. Du, M. Meng, Distributed Aggregative Optimization with Affine Coupling Constraints, Neural Networks, 2025.

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👥 Authors

• Luca Fantini (Master student)
• Gianluca Bianchin (Supervisor)