Abstract: Mathematical programming problems with semi-continuous variables and cardinality constraint have many applications, including production planning, portfolio selection, compressed sensing and subset selection in regression. This class of problems can be modeled as mixed-integer programs with special structures and are in general NP-hard. In the past few years, based on new reformulations, approximation and relaxation techniques, promising exact and approximate methods have been developed. We survey in this paper these recent developments for this challenging class of mathematical programming problems.

Key words: Semi-continuous variables , Cardinality and sparsity constraint , Mixed-integer 0–1 quadratic programming , Perspective reformulation , Lagrangian decomposition , Approximate methods