Optimal　investment　strategies　for　enhanced　indexation　problems　have　attracted　considerable　attentions　over　the　last　decades　in　the　field　of　fund　management.　In　this　paper,　a　featured　difference　from　the　existing　literature　is　that　our　main　concern　of　the　investigation　is　the　development　of　a　sparse-enhanced　indexation　model　to　describe　the　process　of　assets　selection　by　introducing　a　sparse　(Formula　presented.)　regularization　instead　of　binary　variables,　which　is　expected　to　avoid　the　over-fitting　and　promote　a　better　out-of-sample　performance　for　the　resulting　tracking　portfolio　to　some　extent.　An　Alternating　Quadratic　Penalty　(AQP)　method　is　proposed　to　solve　the　corresponding　nonconvex　optimisation　problem,　into　which　the　Block　Coordinate　Descent　(BCD)　algorithm　is　integrated　to　solve　a　sequence　of　penalty　subproblems.　Under　some　suitable　assumptions,　we　establish　that　any　accumulation　point　of　the　sequence　generated　by　the　AQP　method　is　a　KKT　point　of　the　proposed　model.　Computational　results　on　five　typical　data-sets　are　reported　to　verify　the　efficiency　of　the　proposed　AQP　method,　including　the　superiority　of　the　sparse　(Formula　presented.)　model　with　the　AQP　method　over　one　cardinality　constrained　quadratic　programming　model　with　one　of　its　solution　methods　in　terms　of　computational　costs,　out-of-sample　performances,　and　the　consistency　between　in-sample　and　out-of-sample　performances　of　the　resulting　tracking　portfolios.　©　Operational　Research　Society　2018