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 \(\epsilon \)-solution to partial optimum of biconvex programming, and their convergence is also proved. Finally, numerical experiments show that an \(\epsilon \)-feasible solution is obtained by the proposed algorithm.

双凸规划是描述许多实际问题的非凸优化。现有的研究表明，求解双凸规划的难度使得寻找新的理论和求解方法成为一个非常有价值的课题。本文首先通过目标惩罚函数得到了关于双凸规划部分优化的两个重要理论结果。一个结果认为部分 Karush-Kuhn-Tucker (KKT) 条件等价于双凸规划目标惩罚函数的部分精确性。另一个结果认为，部分稳定条件等价于双凸规划目标惩罚函数的部分精确性。这些结果为求解双凸规划局部最优的算法收敛性提供了保证。然后，\(\epsilon \) -双凸规划部分最优解的解，并证明了它们的收敛性。最后，数值实验表明，所提出的算法获得了一个\(\epsilon \) -可行解。