The escalating complexity of modern machine learning necessitates solving challenging non-convex optimization problems, particularly in high-dimensional regimes and scenarios contaminated by gross outliers. Traditional approaches, relying on convex relaxations or specialized local search heuristics, frequently succumb to suboptimal local minima and fail to recover the true underlying discrete structures. In this paper, we propose treating these non-convex challenges as a global search problem and introduce a unified framework based on Quantum-Inspired Evolutionary Optimization (QIEO). By leveraging a probabilistic representation inspired by quantum superposition, QIEO maintains a global view of the search space, enabling it to tunnel through local optima that trap conventional gradient-based and greedy solvers. We comprehensively evaluate QIEO across diverse non-convex applications, including sparse signal recovery (gene expression analysis and compressed sensing) and robust linear regression. Extensive benchmarking against state-of-the-art continuous solvers (ADAM, Differential Evolution), classical metaheuristics (Genetic Algorithms), and specialized non-convex algorithms (Iterative Hard Thresholding) demonstrates that QIEO consistently achieves superior structural fidelity, lower mean squared error, and enhanced robustness without support inflation. Our findings suggest that embracing a quantum-inspired global search provides a resilient, unified paradigm for overcoming the inherent intractability of discrete nonconvex machine learning landscapes.
We introduce a physics-inspired continuous relaxation framework that yields substantially improved solutions for NP-hard combinatorial optimization problems, including Quadratic Unconstrained Binary Optimization (QUBO), binary sparse coding, and planted-solution Ising models. By parameterizing discrete binary variables as continuous wave-like states on the complex unit circle, we inherently smooth highly non-convex energy landscapes. We show that representing binary variables as complex phases reveals an implicit regularization mechanism that promotes convergence toward discrete states. Extracting this mechanism yields significant improvements even within standard real-valued optimization frameworks, using this regularizer explicitly. Empirically, this regularization yields vastly higher ground-state convergence rates than standard real-valued alternatives. Our models achieved zero error in large-scale 160x160 QUBO tasks under severe noise (sigma=0.25), and outperformed traditional algorithms (OMP and LASSO) in underdefined sparse coding with perfect recovery at sigma=0.15. The solver's robustness was further validated by recovering exact ground-state configurations in 8 out of 11 rigorously engineered planted-solution benchmarks.
This paper introduces a structural taxonomy for constrained non-convex optimization based on the signature of Lagrange multipliers at KKT stationary points. Leveraging a unified game-theoretic interpretation of eight classical algorithm families--including block coordinate descent, ADMM, generalized Benders decomposition, successive convex approximation, interior-point methods, mirror descent, Frank-Wolfe, and Riemannian gradient descent--we show that the normalized multiplier vector carries an algorithm-independent structural fingerprint. Four scale-free shape features of this vector partition the dual space into five operational regimes: Unconstrained, Resource-Limited, Saturation, Strongly-Coupled, and Hybrid. We establish four structural theorems characterizing the partition: invariance under natural KKT symmetries, local stability under data perturbation with explicit Lipschitz margins from Robinson's strong regularity, codimension-one regime transitions, and the topological identification of the Hybrid regime as the Lebesgue-null boundary of the core regimes. A linear-time classifier is proposed with provable guarantees on correctness, iteration stabilization, sample complexity, and online tracking under data drift. Numerical experiments on 104 mixed-integer nonlinear programs and a downlink beamforming instance validate the theoretical predictions. The framework provides a foundational tool for regime-aware algorithm design and robustness analysis in non-convex optimization.
Seyed Mohsen Kazemi, Ali Movaghar, Shaahin hessabi
Nonconvex methods have emerged as a dominant approach for low-rank matrix estimation, a problem that arises widely in machine learning and AI for learning and representing high-dimensional data. Existing analyses for these methods often require additional regularization to mitigate nonconvexity, even though such regularization is often unnecessary in practice. Moreover, most analyses rely on problem-specific arguments that are difficult to generalize to more complex settings. In this paper, we develop a theoretical framework for studying nonconvex procedures across a broad class of low-rank matrix estimation problems. Rather than focusing on a specific model, we reveal a fundamental mechanism that explains why nonconvex procedures can behave well in low-rank estimation. Our key device is a {\it benign regularizer} that does not alter the original update rule, but yields an equivalent locally strongly convex formulation of the algorithm. This perspective uncovers a disguised convexity inherent in the nonconvex procedure and provides a new route to theoretical guarantees for nonconvex low-rank matrix estimation.