Proximity search is an iterative method to solve complex mathematical programming problems. At each iteration, the objective function of the problem at hand is replaced by the Hamming distance function to a given solution, and a cutoff constraint is added to impose that any new obtained solution improves the objective function value. A mixed integer programming solver is used to find a feasible solution to this modified problem, yielding an improved solution to the original problem. This paper introduces the concept of weighted Hamming distance that allows to design a new method called weighted proximity search. In this new distance function, low weights are associated with the variables whose value in the current solution is promising to change in order to find an improved solution, while high weights are assigned to variables that are expected to remain unchanged. The weights help to distinguish between alternative solutions in the neighborhood of the current solution, and provide guidance to the solver when trying to locate an improved solution. Several strategies to determine weights are presented, including both static and dynamic strategies. The proposed weighted proximity search is compared with the classic proximity search on instances from three optimization problems: the p-median problem, the set covering problem, and the stochastic lot-sizing problem. The obtained results show that a suitable choice of weights allows the weighted proximity search to obtain better solutions, for 75\(\%\) of the cases, than the ones obtained by using proximity search and for 96\(\%\) of the cases the solutions are better than the ones obtained by running a commercial solver with a time limit.

邻近搜索是解决复杂数学编程问题的一种迭代方法。在每次迭代中，将手头问题的目标函数替换为给定解的汉明距离函数，并添加了截止约束以强加任何新获得的解都会改善目标函数值。混合整数规划求解器用于找到此修改后问题的可行解，从而得到对原始问题的改进解。本文介绍了加权汉明距离的概念，该概念允许设计一种称为加权邻近搜索的新方法。在这个新的距离函数中，低权重与变量相关联，这些变量的当前解决方案中的值有望改变以找到改进的解决方案，而较高的权重将分配给预期保持不变的变量。权重有助于区分当前解决方案附近的替代解决方案，并在尝试查找改进的解决方案时为求解器提供指导。提出了几种确定权重的策略，包括静态和动态策略。在三个优化问题上，将拟议的加权邻近搜索与经典邻近搜索进行了比较：p-中值问题，集合覆盖问题和随机批量确定问题。将所得到的结果表明，权重的适当选择允许加权邻近搜索，以获得更好的解决方案，为75 \（\％\）的情况下，不是通过使用邻近搜索和96中获得的那些\（\％\）的在这种情况下，解决方案要比通过运行有时间限制的商用求解器获得的解决方案更好。