On a Reformulation of Mathematical Programs with Cardinality Constraints

• Mathematical programs with cardinality constraints are optimization problems with an additional constraint which requires the solution to be sparse in the sense that the number of nonzero elements, i.e. the cardinality, is bounded by a given constant. Such programs can be reformulated as a mixed-integer ones in which the sparsity is modeled with the use of complementarity-type constraints. It is shown that the standard relaxation of the integrality leads to a nonlinear optimization program of the striking property that its solutions (global minimizers) are the same as the solutions of the original program with cardinality constraints. Since the number of local minimizers of the relaxed program is typically larger than the number of local minimizers of the cardinality-constrained problem, the relationship between the local minimizers is also discussed in detail. Furthermore, we show under which assumptions the standard KKT conditions are necessary optimality conditions for the relaxed program. The main result obtained for such conditions is significantly different from the existing optimality conditions that are known for the somewhat related class of mathematical programs with complementarity constraints.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Cardinality constraints

global minima

local minima

stationary points

M-stationarity

relaxation

regularization method

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