EQUIVALENT DIFFERENTIABLE OPTIMIZATION PROBLEMS AND DESCENT METHODS FOR ASYMMETRIC VARIATIONAL INEQUALITY PROBLEMS
EQUIVALENT DIFFERENTIABLE OPTIMIZATION PROBLEMS AND DESCENT METHODS FOR ASYMMETRIC VARIATIONAL INEQUALITY PROBLEMS
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DOI：
10.1007/bf01585696
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发表时间：
1992-01-06
影响因子：
2.7
通讯作者：
FUKUSHIMA, M
中科院分区：
文献类型：
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作者：
FUKUSHIMA, M
一般的非对称变分不等式问题能否转化为可微优化问题一直是一个悬而未决的问题。本文对这一问题作了肯定的回答。我们提供了一个新的优化问题的变分不等式问题的制定，并表明，其目标函数是连续可微的映射在后一个问题是连续可微的。我们还表明，在适当的假设下，后者的映射，任何稳定点的优化问题是一个全局最优解，从而解决变分不等式问题。我们讨论下降法解决等价的优化问题和评论系统的非线性方程和非线性互补问题。
Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.