Modeling using developable surfaces plays an important role in computer graphics and computer aided design. In this paper, we investigate a new problem called variational developable surface interpolation (VDSI). For a polyline boundary P, different from previous work on interpolation or approximation of discrete developable surfaces from P, the VDSI interpolates a subset of the vertices of P and approximates the rest. Exactly speaking, the VDSI allows to modify a subset of vertices within a prescribed bound such that a better discrete developable surface interpolates the modified polyline boundary. Therefore, VDSI could be viewed as a hybrid of interpolation and approximation. Generally, obtaining discrete developable surfaces for given polyline boundaries are always a time-consuming task. In this paper, we introduce a dynamic programming method to quickly construct a developable surface for any boundary curves. Based on the complexity of VDSI, we also propose an efficient optimization scheme to solve the variational problem inherent in VDSI. Finally, we present an adding point condition, and construct a G1 continuous quasi-Coons surface to approximate a quadrilateral strip which is converted from a triangle strip of maximum developability. Diverse examples given in this paper demonstrate the efficiency and practicability of the proposed methods.