Constrained Optimization Involving Nonconvex ℓp Norms: Optimality Conditions, Algorithm and Convergence

This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an ℓp 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 ℓp norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the ℓp norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the ℓp norm and the normal cones of the ℓp 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.