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A code-constrained inverse filter criterion based approach is presented for blind detection of asynchronous short-code DS-CDMA (direct sequence code division multiple access) signals in multipath channels. Only the spreading code of the desired user is assumed to be known; its transmission delay may be unknown. We focus on minimization of the negative normalized fourth cumulant of inverse filtered (equalized) data with respect to the equalizer coefficients subject to the equalizer lying in a subspace associated with the desired user's code sequence. Previously we presented a projection method for constrained optimization. In this paper we investigate a penalty function-based approach to constrained optimization. Global minima and some of the local minima of the cost function are investigated. A simulation example is presented.