Constrained Optimization Involving Nonconvex ℓ_p Norms: Optimality Conditions, Algorithm and Convergence
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This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an ℓ_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 ℓ_p norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the ℓ_p norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the ℓ_p norm and the normal cones of the ℓ_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.
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