有限维互补性问题的独特解
Distinct solutions of finite-dimensional complementarity problems
摘要：
互补性问题通常有不同的解，这在最优化、对策论和其他领域中具有重要意义。在本文中，我们发展了一个数值技术计算多个孤立的解决方案的互补性问题，从相同的初始猜测。这种技术，称为通货紧缩，适用于与现有的算法，重新制定的互补性问题的寻根问题的半光滑残留。找到一个解后，收缩的思想是将算子应用于相应半光滑重构的参数，以确保求解器不会再次收敛到同一个解。这确保了如果半光滑求解器从相同的初始猜测重新开始并且收敛，则它将收敛到不同的解。我们证明了理论结果的有效性的方法，并将其应用到几个困难的有限维互补性问题的文献。虽然通货紧缩并不能保证找到所有的解决方案，对于每一个问题，考虑有限数量的解决方案，我们确定初始猜测，所有已知的解决方案与通货紧缩计算。
英文摘要：
Complementarity problems often permit distinct solutions, a fact of major significance in optimization, game theory and other fields. In this paper, we develop a numerical technique for computing multiple isolated solutions of complementarity problems, starting from the same initial guess. This technique, called deflation, is applied in conjunction with existing algorithms that reformulate the complementarity problem as the rootfinding problem of a semismooth residual. After one solution is found, the idea of deflation is to apply operators to the arguments of the corresponding semismooth reformulation to ensure that solvers will not converge to that same solution again. This ensures that if the semismooth solver is restarted from the same initial guess and it converges, it will converge to a different solution. We prove theoretical results on the effectiveness of the method, and apply it to several difficult finite-dimensional complementarity problems from the literature. While deflation is not guaranteed to find all solutions, for every problem considered with a finite number of solutions, we identify initial guesses from which all known solutions are computed with deflation.
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