A Unified Framework for Lossless Discrete Optimization: Mixed-Integer Quadratic Programming in Signal Processing, Resource Allocation, and Packing

This research presents a generalized mathematical framework for lossless discrete optimization utilizing Mixed-Integer Quadratic Programming (MIQP).We address the fundamental conflict between computational efficiency and data integrity by modeling high-dimensional systems through a symmetric block-diagonal matrix architecture.The framework is demonstrated across four distinct domains: lossless video encoding under 1MB constraints, compressive sensing for medical MRI, energy-efficient sensor network routing, and the resolution of the industrial Packing Open Problem.By enforcing strict integer constraints and leveraging L 1 -norm sparsity, the proposed method achieves bit-perfect reconstruction with a Mean Squared Error (MSE) of exactly 0.0.Our results confirm that complex discrete problems involving up to 500 items can be solved within stable numerical tolerances using specialized second-order cone solvers like Clarabel.1