Summary: The successive quadratic programming (SQP) method is used with the finite element method (FEM) to solve frictionless geometrically nonlinear contact problems involving large deformations of the elastica in the presence of flat rigid walls. To formulate the SQP problems, the potential energy (PE) is expanded in a Taylor series of second order in displacement increments about a configuration near a contact solution. The SQP problems consist of minimizing the Taylor expansion of the PE subject to the inequality constraints which represent contact. The quadratic programming (QP) method is made part of a Newton-Raphson search (NR) in which the QP corrections are made when a NR step does not satisfy the constraints. A revised simplex method developed by {\it M. H. Rusin} [SIAM J. Appl. Math. 20, 143-160 (1971; Zbl 0221.90037)] is used to solve the QP problems. The elastica is modelled with a total Lagrangian FEM developed by {\it I. Fried} [Comput. Methods Appl. Mech. Eng. 38, 29-44 (1983; Zbl 0512.73071)]. Solutions are obtained for the end loaded buckled elastica in point contact with a rigid wall and for a uniformly loaded elastica in regional contact with a rigid wall. The problems are also solved using a penalty method. The results obtained for the point contact problem are compared to an analytical solution. Calculations were made to obtain numerical information on maximum load step size and the number of inverse operations required for each load step. Cases in which the elastica stiffened substantially as a result of the initiation of contact are also discussed.