We study a class of single-objective non-linear optimization problems, the so-called Integer Linear Generalized Maximum Multiplicative Programs (IL-GMMP). This class of optimization problems has a significant number of applications in different fields of study including but not limited to game theory, systems reliability, and conservation planning. An IL-GMMP can be reformulated as a mixed integer Second-Order Cone Program (SOCP) and therefore, can be solved effectively by commercial solvers such as IBM ILOG CPLEX, Gurobi, and FICO Xpress. In this study, we show that IL-GMMPs can be viewed as special cases of the problem of optimization over the efficient (or Pareto-optimal) set in multi-objective integer linear programming. Based on this observation, we develop three exact solution approaches with a desirable property: they only solve a finite number of single-objective integer linear programs to compute an optimal solution of an IL-GMMP (which is nonlinear). Through an extensive computational study with 57600 experiments, we compare the performance of all three algorithms using the three main commercial single-objective integer linear programming solvers in the market: CPLEX, Gurobi, and Xpress. We also compare the performance of our algorithms using the mixed integer SOCP solvers of CPLEX, Gurobi, and Xpress. The results show that the choice of a commercial solver impacts the solution time dramatically and that, by the right choice of solver, one of our proposed algorithms is significantly faster than other methods. We also illustrate that although it is possible to linearize IL-GMMPs, commercial solvers struggle to solve such linearized instances.

我们研究一类单目标非线性优化问题，即所谓的整数线性广义最大乘法程序 (IL-GMMP)。这类优化问题在不同的研究领域有大量应用，包括但不限于博弈论、系统可靠性和保护规划。IL-GMMP 可以重新表述为混合整数二阶锥程序 (SOCP)，因此可以由 IBM ILOG CPLEX、Gurobi 和 FICO Xpress 等商业求解器有效求解。在这项研究中，我们表明 IL-GMMP 可以被视为多目标整数线性规划中有效（或帕累托最优）集优化问题的特例。基于这一观察，我们开发了三种具有理想属性的精确解法：他们只求解有限数量的单目标整数线性程序来计算 IL-GMMP（非线性）的最优解。通过包含 57600 次实验的广泛计算研究，我们使用市场上三种主要的商业单目标整数线性规划求解器（CPLEX、Gurobi 和 Xpress）比较了所有三种算法的性能。我们还使用 CPLEX、Gurobi 和 Xpress 的混合整数 SOCP 求解器来比较我们的算法的性能。结果表明，商业求解器的选择会显着影响求解时间，并且通过正确选择求解器，我们提出的算法之一比其他方法要快得多。我们还说明，虽然可以将 IL-GMMP 线性化，但商业求解器很难解决此类线性化实例。