{"title":"Finding Multiple Optimal Solutions to an Integer Linear Program by Random Perturbations of Its Objective Function.","authors":"Noah Schulhof, Pattara Sukprasert, Eytan Ruppin, Samir Khuller, Alejandro A Schäffer","doi":"10.3390/a18030140","DOIUrl":null,"url":null,"abstract":"<p><p>Integer linear programs (ILPs) and mixed integer programs (MIPs) often have multiple distinct optimal solutions, yet the widely used Gurobi optimization solver returns certain solutions at disproportionately high frequencies. This behavior is disadvantageous, as, in fields such as biomedicine, the identification and analysis of distinct optima yields valuable domain-specific insights that inform future research directions. In the present work, we introduce MORSE (Multiple Optima via Random Sampling and careful choice of the parameter Epsilon), a randomized, parallelizable algorithm to efficiently generate multiple optima for ILPs. MORSE maps multiplicative perturbations to the coefficients in an instance's objective function, generating a modified instance that retains an optimum of the original problem. We formalize and prove the above claim in some practical conditions. Furthermore, we prove that for 0/1 selection problems, MORSE finds each distinct optimum with equal probability. We evaluate MORSE using two measures; the number of distinct optima found in <math><mi>r</mi></math> independent runs, and the diversity of the list (with repetitions) of solutions by average pairwise Hamming distance and Shannon entropy. Using these metrics, we provide empirical results demonstrating that MORSE outperforms the Gurobi method and unweighted variations of the MORSE method on a set of 20 Mixed Integer Programming Library (MIPLIB) instances and on a combinatorial optimization problem in cancer genomics.</p>","PeriodicalId":7636,"journal":{"name":"Algorithms","volume":"18 3","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11970949/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/a18030140","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/3/4 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 0
Abstract
Integer linear programs (ILPs) and mixed integer programs (MIPs) often have multiple distinct optimal solutions, yet the widely used Gurobi optimization solver returns certain solutions at disproportionately high frequencies. This behavior is disadvantageous, as, in fields such as biomedicine, the identification and analysis of distinct optima yields valuable domain-specific insights that inform future research directions. In the present work, we introduce MORSE (Multiple Optima via Random Sampling and careful choice of the parameter Epsilon), a randomized, parallelizable algorithm to efficiently generate multiple optima for ILPs. MORSE maps multiplicative perturbations to the coefficients in an instance's objective function, generating a modified instance that retains an optimum of the original problem. We formalize and prove the above claim in some practical conditions. Furthermore, we prove that for 0/1 selection problems, MORSE finds each distinct optimum with equal probability. We evaluate MORSE using two measures; the number of distinct optima found in independent runs, and the diversity of the list (with repetitions) of solutions by average pairwise Hamming distance and Shannon entropy. Using these metrics, we provide empirical results demonstrating that MORSE outperforms the Gurobi method and unweighted variations of the MORSE method on a set of 20 Mixed Integer Programming Library (MIPLIB) instances and on a combinatorial optimization problem in cancer genomics.
整数线性规划(ILPs)和混合整数规划(MIPs)通常有多个不同的最优解,但广泛使用的Gurobi优化求解器以不成比例的高频率返回某些解。这种行为是不利的,因为在生物医学等领域,识别和分析不同的最优值会产生有价值的领域特定见解,为未来的研究方向提供信息。在本工作中,我们引入了MORSE (Multiple Optima via Random Sampling and careful choice of parameter Epsilon)算法,这是一种随机的、可并行的算法,可以有效地为ILPs生成多个最优解。MORSE将乘法扰动映射到实例目标函数的系数上,生成一个保留原始问题最优值的修改实例。我们在一些实际条件下形式化并证明了上述论断。进一步,我们证明了对于0/1选择问题,MORSE以等概率找到每个不同的最优。我们用两种方法来评估MORSE;在r个独立运行中发现的不同最优值的数量,以及通过平均成对汉明距离和香农熵计算的解决方案列表的多样性。利用这些指标,我们提供了实证结果,证明MORSE在一组20个混合整数规划库(MIPLIB)实例和癌症基因组学的组合优化问题上优于Gurobi方法和MORSE方法的未加权变体。