A DERIVATIVE-FREE ALGORITHM FOR INEQUALITY CONSTRAINED NONLINEAR PROGRAMMING VIA SMOOTHING OF AN ∞ PENALTY FUNCTION∗
A DERIVATIVE-FREE ALGORITHM FOR INEQUALITY CONSTRAINED NONLINEAR PROGRAMMING VIA SMOOTHING OF AN ∞ PENALTY FUNCTION∗
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一种通过 ∞ 惩罚函数平滑的不等式约束非线性规划的无导数算法*
DOI：
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发表时间：
2009
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通讯作者：
O. SIAMJ.
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作者：
O. SIAMJ.
在这篇文章中，我们考虑了不能使用目标函数和约束的一阶导数的不等式约束的非线性优化问题。我们的出发点是将原来的约束问题转化为非光滑精确罚函数的无约束或线性约束最小化问题。这种方法有两个主要困难：一是这类精确罚函数的非光滑性，这可能导致无导数的代码收敛到问题的非平稳点；二是约束问题的驻点与精确罚函数的驻点之间的等价性只能在罚参数小于一个先验未知的阈值时才能得到。在这篇文章中，我们提出了一个免导数的算法，它克服了前面的困难，产生了一个点序列，它允许一个子序列收敛到约束问题的Karush-Kuhn-Tucker点。特别地，所提出的算法基于不可微精确惩罚函数的光滑化，并且包括更新规则，该更新规则在至多有限次更新之后能够确定惩罚参数的“正确值”。此外，我们还给出了关于人体胰岛素-葡萄糖模型中参数估计的真实世界问题的结果。
In this paper we consider inequality constrained nonlinear optimization problems where the first order derivatives of the objective function and the constraints cannot be used. Our starting point is the possibility to transform the original constrained problem into an unconstrained or linearly constrained minimization of a nonsmooth exact penalty function. This approach shows two main difficulties: the first one is the nonsmoothness of this class of exact penalty functions which may cause derivative-free codes to converge to nonstationary points of the problem; the second one is the fact that the equivalence between stationary points of the constrained problem and those of the exact penalty function can only be stated when the penalty parameter is smaller than a threshold value which is not known a priori. In this paper we propose a derivative-free algorithm which overcomes the preceding difficulties and produces a sequence of points that admits a subsequence converging to a Karush–Kuhn–Tucker point of the constrained problem. In particular the proposed algorithm is based on a smoothing of the nondifferentiable exact penalty function and includes an updating rule which, after at most a finite number of updates, is able to determine a “right value” for the penalty parameter. Furthermore we present the results obtained on a real world problem concerning the estimation of parameters in an insulin-glucose model of the human body.