A Lagrange–Newton algorithm for sparse nonlinear programming

The sparse nonlinear programming (SNP) problem has wide applications in signal and image processing, machine learning finance, etc. However, the computational challenge posed by SNP not yet been well resolved due to nonconvex discontinuous $$\ell _0$$ -norm involved. In this paper, we resolve numerical developing a fast Newton-type algorithm. As theoretical cornerstone, establish first-order optimality condition for based on concept of strong $$\beta $$ -Lagrangian stationarity via Lagrangian function, reformulate it as system equations called equations. nonsingularity corresponding Jacobian is discussed, which Lagrange–Newton algorithm (LNA) then proposed. Under mild conditions, locally quadratic convergence its iterative complexity estimation. To further demonstrate efficiency superiority our proposed algorithm, apply LNA two specific problems arising from compressed sensing high-order portfolio selection, significant benefits accrue restricted Newton step.