AUGMENTED LAGRANGE MULTIPLIER FUNCTIONS AND DUALITY IN NONCONVEX PROGRAMMING
AUGMENTED LAGRANGE MULTIPLIER FUNCTIONS AND DUALITY IN NONCONVEX PROGRAMMING
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DOI：
10.1137/0312021
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发表时间：
1974-01-01
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通讯作者：
ROCKAFELLAR, RT
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文献类型：
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作者：
ROCKAFELLAR, RT
如果用普通拉格朗日函数来分析非线性规划问题，通常会存在对偶间隙，除非目标函数和约束函数是凸函数。这里表明，可以通过传递到涉及二次罚项的增广拉格朗日来消除间隙。然后，修改后的对偶问题包括最大化拉格朗日乘子的凹函数和附加变量（惩罚参数）。与经典情况相反，与原数中的不等式约束相对应的乘数在对偶中并不被先验地限制为非负数。如果达到对偶问题的最大值（并且给出了暗示这一点的条件），则原始问题的最优解可以用增广拉格朗日的全局鞍点来表示。这表明计算解决方案现有惩罚方法的可能改进。
If a nonlinear programming problem is analyzed in terms of its ordinary Lagrangian function, there is usually a duality gap, unless the objective and constraint functions are convex. It is shown here that the gap can be removed by passing to an augmented Lagrangian which involves quadratic penalty-like terms. The modified dual problem then consists of maximizing a concave function of the Lagrange multipliers and an additional variable, which is a penalty parameter. In contrast to the classical case, the multipliers corresponding to inequality constraints in the primal are not constrained a priori to be nonnegative in the dual. If the maximum in the dual problem is attained (and conditions implying this are given), optimal solutions to the primal can be represented in terms of global saddle points of the augmented Lagrangian. This suggests possible improvements of existing penalty methods for computing solutions.