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We consider the sparse optimization problem with nonlinear constraints and anobjective function, which is given by the sum of a general smooth mapping andan additional term defined by the $ \ell_0 $-quasi-norm. This term is used toobtain sparse solutions, but difficult to handle due to its nonconvexity andnonsmoothness (the sparsity-improving term is even discontinuous). The aim ofthis paper is to present two reformulations of this program as a smoothnonlinear program with complementarity-type constraints. We show that theseprograms are equivalent in terms of local and global minima and introduce aproblem-tailored stationarity concept, which turns out to coincide with thestandard KKT conditions of the two reformulated problems. In addition, asuitable constraint qualification as well as second-order conditions for thesparse optimization problem are investigated. These are then used to show thatthree Lagrange-Newton-type methods are locally fast convergent. Numericalresults on different classes of test problems indicate that these methods canbe used to drastically improve sparse solutions obtained by some other(globally convergent) methods for sparse optimization problems.

The Sparse(st) Optimization Problem: Reformulations, Optimality, Stationarity, and Numerical Results