Constrained Optimization Involving Nonconvex $\ell_p$ Norms: Optimality Conditions, Algorithm and Convergence
This work addresses optimization difficulties in sparse solution promotion for various applications, but it is incremental as it builds on existing theory for nonconvex norms.
The paper tackles the problem of constrained optimization with nonconvex ℓp norms (0<p<1), which are challenging due to nonsmoothness and non-Lipschitz properties, by deriving first-order necessary and sequential optimality conditions, and shows that these conditions facilitate global convergence for iteratively reweighted algorithms.
This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an $\ell_p$ norm ($0<p<1$) of the variables, which may appear in either the objective or the constraint. This kind of problems have strong applicability to a wide range of areas since usually the $\ell_p$ norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the $\ell_p$ norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the $\ell_p$ norm and the normal cones of the $\ell_p$ ball. For both problems, we derive the first-order necessary conditions under various constraint qualifications. We also derive the sequential optimality conditions for both problems and study the conditions under which these conditions imply the first-order necessary conditions. We point out that the sequential optimality conditions can be easily satisfied for iteratively reweighted algorithms and show that the global convergence can be easily derived using sequential optimality conditions.