约束DC优化的双束法及对偶问题

针对带有凸不等式约束的非光滑DC优化问题,提出了一种基于罚函数的凸约束DC优化问题双束法,同时也刻画了双束法子问题的对偶问题;首先,利用L_1精确罚技巧把凸约束DC优化问题转化成无约束DC优化问题,便于直接对目标函数进行DC分解,然后分别建立了增广目标函数DC分量的凸分段线性近似模型,最后利用Lagrange函数得到了原问题和对偶问题最优解之间的等价关系,说明了利用对偶问题求解搜索方向的可行性和有效性。

Double Bundle Method and Dual Problem for Constrained DC Optimization

Aiming at the nonsmooth DC optimization problem with convex inequality constraints,a double bundle method for convex constrained DC optimization problems based on penalty function is proposed. The dual problem of the double bundle method is also described. This method can effectively utilize the explicit convex DC decomposition structure of the nonconvex objective function. Firstly,the L1exact penalty technique is used to transform the convex constrained DC optimization problem into an unconstrained DC optimization problem,which is convenient for DC decomposition of the objective function directly. Then,the convex piecewise linear approximation model of the DC component of the augmented objective function is established,and finally the Lagrange function is used to obtain the equivalence relation between the optimal solution of the original problem and the optimal solution of the dual problem,which illustrates the feasibility and effectiveness of solving the search direction by using the dual problem.