During the last decades, research in multi-objective optimisation has seen considerable growth. However, this activity has been focused on linear, non-linear, and combinatorial optimisation with multiple objectives. Multi-objective mixed integer (linear or non-linear) programming has received considerably less attention. In this paper we propose an algorithm to compute a finite set of non-dominated points/efficient solutions of a bi-objective mixed binary optimisation problems for which the sub-problems obtained when fixing the binary variables are convex, and there is a finite set of feasible binary variable vectors. Our method uses bound sets and exploits the convexity property of the sub-problems to find a set of efficient solutions for the main problem. Our algorithm creates and iteratively updates bounds for each vector in the set of feasible binary variable vectors, and uses these bounds to guarantee that a set of exact non-dominated points is generated. For instances where the set of feasible binary variable vectors is too large to generate such provably optimal solutions within a reasonable time, our approach can be used as a matheuristic by heuristically selecting a promising subset of binary variable vectors to explore. This investigation is motivated by the problem of beam angle optimisation arising in radiation therapy planning, which we solve heuristically to provide numerical results.

在过去的几十年中，多目标优化的研究有了长足的发展。但是，此活动集中于具有多个目标的线性，非线性和组合优化。多目标混合整数（线性或非线性）编程受到的关注很少。在本文中，我们提出了一种算法，用于计算双目标混合二进制优化问题的非控制点/有效解的有限集，该问题的固定二进制变量时得到的子问题是凸的，并且存在一个有限集可行的二进制变量向量的集合。我们的方法使用边界集并利用子问题的凸性来找到针对主要问题的一组有效解。我们的算法为可行二元变量向量集中的每个向量创建并迭代更新边界，并使用这些边界来确保生成一组精确的非支配点。对于可行的二进制变量向量的集合太大而无法在合理的时间内生成此类可证明的最优解的情况，我们的方法可以通过启发式地选择有希望的二进制变量向量子集进行探索，从而用作数学方法。这项研究的动机是放射治疗计划中出现的束角优化问题，我们通过试探性地求解以提供数值结果。对于可行的二进制变量向量的集合太大而无法在合理的时间内生成此类可证明的最优解的情况，我们的方法可以通过启发式地选择有希望的二进制变量向量子集进行探索，从而用作数学方法。这项研究的动机是放射治疗计划中出现的束角优化问题，我们通过试探性地求解以提供数值结果。对于可行的二进制变量向量的集合太大而无法在合理的时间内生成此类可证明的最优解的情况，我们的方法可以通过启发式地选择有希望的二进制变量向量子集进行探索，从而用作数学方法。这项研究的动机是放射治疗计划中出现的束角优化问题，我们通过试探性地求解以提供数值结果。