An Intelligent Sparse Search Framework via Lagrangian Dual Optimization

High-dimensional learning often requires explicit structural control to prevent overfitting and meet deployment constraints. Conventional sparse methods rely on penalty-based formulations where sparsity is indirectly governed by manually tuned coefficients, providing limited control over support size. We propose a dual-driven sparse search framework that formulates feature selection as empirical risk minimization under an explicit cardinality constraint. The problem is solved via a primal-dual saddle-point method with continuous feature gates and adaptive Lagrange multiplier updates that enforce structural feasibility through closed-loop dynamics. A final TopK projection ensures exact budget compliance at deployment. We provide theoretical analysis establishing asymptotic constraint satisfaction under standard smoothness assumptions. Experiments on six high-dimensional benchmarks demonstrate improved predictive accuracy, stronger support stability, precise budget adherence, and comparable computational cost relative to regularization-based and greedy baselines. The results support adaptive Lagrangian dual optimization as an effective approach for explicit budget-controlled sparse learning.