Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems

Abstract

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 and , are unable to solve to optimality even test instances with $$n=50$$ 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.

From Goal Programming for Continuous Multi-Criteria Optimization to the Target Decision Rule for Mixed Uncertain Problems

“…Nevertheless, in many cases mixed strategies may be more effective [ 12 ]. A mixed strategy occurs when the DM chooses and executes a combination of alternatives.…”

Evaluation and Comparison of Integer Programming Solvers for Hard Real-Time Scheduling

“…GUROBI solver's performance was superior to that of the CPLEX solver. In the context of non-convex QP problems, CPLEX and Gurobi are also compared being Gurobi which presents the best results [5]. For solving the problem of mining production planning [6], CPLEX is more efficient than Gurobi solving the proposed model in medium-sized instances.…”

Fix and Bound: An efficient approach for solving large-scale quadratic programming problems with box constraints

“…, is not smaller than the current upper bound, then we can fix all variables in T . In fact, the solution of ( 12) is slightly less informative with respect to the solution of (9). Indeed, in case fixing is not possible, i.e., when T − ε 1 < |T | concerning (9), or α T < U B concerning (12), the solution of (9) allows at least to add the linear cut (10).…”

“…In fact, the solution of ( 12) is slightly less informative with respect to the solution of (9). Indeed, in case fixing is not possible, i.e., when T − ε 1 < |T | concerning (9), or α T < U B concerning (12), the solution of (9) allows at least to add the linear cut (10). On the other hand, recent first-order SDP solvers discussed in Section 3.1 encounter some difficulties when solving (9), while they are able to solve (12) quite efficiently.…”

“…In the bilinear reformulation, each linear function Q i x is set equal to a new variable y i and substituted by it in the objective function (here Q i denotes the i-th row of matrix Q). In spite of its simplicity, the bilinear reformulation turned out to be the best one for a special class of QP problems arising from game theory [9]. In the spectral reformulation (see, e.g., [10]), first a spectral decomposition of matrix Q is performed, then each term u i x, where u i is an eigenvector associated to a negative eigenvalue of Q, is set equal to a variable z i and replaced into the objective by such variable.…”