The variational two-electron reduced-density-matrix (2-RDM) method can compute many-electron energies and properties by constraining the 2-RDM with two- or three-particle positivity conditions. While milliHartree accuracy of the energy is obtained with 3-positivity conditions at both equilibrium and highly stretched geometries, the addition of these conditions to the 2-positivity constraints significantly increases the computational work. In this Letter, we relax a subset of three-particle positivity conditions (called T-2) to a recursively generated set of linear inequality constraints that produces most of T-2's accuracy with lower computational cost. Illustrative applications are reported. (C) 2007 Elsevier B.V. All rights reserved.