MAGNEO: A Unified Framework for Combinatorial Optimization via Continuous Energy Minimization

MAGNEO (Method of Approximation by Gradient for NEO-energetic Optimization) presents a unified theoretical framework for solving NP-hard combinatorial optimization problems by transforming them into differentiable continuous optimization problems. Main Contribution This revolutionary method relaxes discrete problems to the continuous space [0,1]ⁿ via a parameterized energy function: E(x;t) = O(x) + Σₖ λₖ(t)Cₖ(x) where penalty coefficients λₖ(t) evolve progressively to maintain feasibility while preserving differentiability. Major Theoretical Results Guaranteed convergence to critical points under the Kurdyka-Łojasiewicz property Adiabatic behavior avoiding abrupt bifurcations during slow parameter variations Quality bounds controlling degradation after discrete projection Complete proofs with detailed analysis of discrete and continuous convergence Three-Phase Algorithm Continuous energy descent: Gradient minimization via ẋ(t) = -∇ₓE(x(t);Λ(t)) Projection/rounding: Transition from [0,1]ⁿ to {0,1}ⁿ Local repair: Correction of violated constraints Potential Applications Graph optimization: Maximum cut, coloring, partitioning Machine Learning: Feature selection, binary clustering Logistics: Assignment and routing problems Networks: Optimal routing, resource placement Distinctive Advantages Unified framework applicable to diverse combinatorial problem classes Differentiability enabling use of modern optimizers (Adam, BFGS) Flexibility in penalty function design Theoretical guarantees with rigorous convergence analysis Innovative adiabatic principle for combinatorial optimization Document Content Complete mathematical formalization of the framework Detailed proofs of main theorems Adapted Kurdyka-Łojasiewicz property analysis Technical appendix with extended demonstrations Discussion of limitations and future directions Impact and Originality MAGNEO elegantly reconciles continuous and discrete optimization, opening new perspectives for treating NP-hard problems. This theoretically grounded approach could become a reference for developing generic solvers and teaching combinatorial optimization.