In this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as CPLEX and GUROBI, are unable to solve to optimality even test instances with \(n=50\) variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

在本文中，我们解决了网络安全背景下出现的博弈论问题。在传统的博弈论问题中，给定一个防御者和一个攻击者，人们寻求使线性收益函数最小化的混合策略。在本文解决的问题中，最小化问题增加了一个二次项。此术语表示转换成本，即防御者在Nash游戏的连续回合中从给定策略转换为另一策略的成本。所产生的问题是具有线性约束的非凸QP问题，结果是非常具有挑战性的。我们将展示使多凸面上的非凸QP功能最小化的最新方法，包括CPLEX和GUROBI等商业求解器，甚至无法测试具有\（n = 50 \）的实例变量。因此，我们建议与他们一起扩展当前针对QP问题的基准测试实例集。我们还提出了解决这些问题的空间分支定界方法，其中基于最优性的域约简起着主要作用，在分支定界树的每个节点上都有LP问题的多个解决方案。当然，域缩减是空间分支定界方法中的标准工具。但是，我们的贡献在于观察到，从计算的角度来看，对这些工具进行相当激进的应用似乎是解决所建议实例的最佳方法。确实，根据我们的实验，尽管它们使每个节点的计算成本很高，但很大程度上是由分支定界树中的节点数量的相当缓慢的增长所补偿的，