Abstract:

The study investigates the optimality theory for group-sparse optimization problems with equality and inequality constraints (NL-GSCO) based on group-sparse support projections. It establishes the concept of strong $\gamma$-Lagrange stationary points, characterizes their first-order optimality conditions, and explores the equivalence and progressive relationships between such points and $\alpha$-stationary points as well as F-KKT points. However, the non-differentiability of the group-sparse projection poses challenges for solving the problem. Additionally, considering the computational burden induced by the complementary slackness conditions of inequality constraints, the study focuses on the differentiable Lagrangian equation form of the equality-constrained group-sparse optimization problem (EC-GSCO). Within a local neighborhood of a stationary point, the nonsingularity of the corresponding Jacobian matrix is proven. Finally, the existing gradient projection Newton pursuit (GPNP) algorithm designed for single sparsity is extended to a group-sparse GPNP algorithm, which is shown to converge to a strong $\gamma$-Lagrange stationary point under certain conditions. Numerical experiments conducted on group-sparse nonlinear optimization problems demonstrate its excellent computational performance.

Key words: group-sparse constrained optimization, optimality conditions, stationary points, lagrangian equations, group-sparse GPNP Algorithm