An objective penalty function method for biconvex programming

Biconvex programming is nonconvex optimization describing many practical problems. The existing research shows that the difficulty in solving biconvex programming makes it a very valuable subject to find new theories and solution methods. This paper first obtains two important theoretical results about partial optimum of biconvex programming by the objective penalty function. One result holds that the partial Karush–Kuhn–Tucker (KKT) condition is equivalent to the partially exactness for the objective penalty function of biconvex programming. Another result holds that the partial stability condition is equivalent to the partially exactness for the objective penalty function of biconvex programming. These results provide a guarantee for the convergence of algorithms for solving a partial optimum of biconvex programming. Then, based on the objective penalty function, three algorithms are presented for finding an approximate ϵ-solution to partial optimum of biconvex programming, and their convergence is also proved. Finally, numerical experiments show that an ϵ-feasible solution is obtained by the proposed algorithm.