Branch-and-Cut Approaches for Nonconvex Nonlinear and Mixed-Integer Nonlinear Programming

这笔赠款为发展新的理论和计算方法提供资金，以通过分支切割(B&A；C)方法解决一些非凸非线性和混合整数非线性规划问题(NCNLP和MINLP)，这些问题具有重要的理论和实践意义。具体地说，将为它们开发新的公式、前处理、分支方案、原始启发式和切割平面。该项目将集中于以下方向：(1)近似求解NCNLP和MINLP的分段线性优化；(2)求解非凸二次规划的线性互补问题；(3)有界纯整数(相对于混合整数)非线性规划的割平面；(4)特殊结构的问题，包括双矩阵对策和投资组合优化。部分资金将用于为工程、管理、经济、数学和科学专业的学生建造离散非线性优化领域的新本科生和研究生课程和教材。该项目有助于了解求解这种结合了离散变量和非线性函数的优化模型的算法。因此，它将在离散和非线性优化的组合领域开辟新的研究场所。该项目的成果将大大改进实践中使用的优化工具。最终结果将是高效的B&A；C战略，这些战略将提高我们解决金融、生物信息学和数据挖掘等领域最困难的模型的能力，因此这些模型是我们经济和福利的核心。通过该项目开发的成果和工具，包括新的理论发展、软件和新的教学资源，将通过为该项目创建的网站向公众开放。与该项目有关的教育材料的编写和传播将有助于形成业务研究领域的人员，他们在工业界和学术界都是必不可少的。

This grant provides funds for the development of new theory and computational methods to solve a number of nonconvex nonlinear and mixed-integer nonlinear programming problems (NCNLP and MINLP) through the branch-and-cut (B&C) approach, which are of significant theoretical and practical interest. Specifically, new formulation, pre-processing, branching scheme, primal heuristic, and cutting planes will be developed for them. The project will focus on the following directions: (1) piecewise linear optimization to solve NCNLP and MINLP approximately; (2) linear complementarity problem to solve nonconvex quadratic programming; (3) cutting planes for bounded purely integer (as opposed to mixed-integer) nonlinear programming; (4) specially structured problems, including bimatrix games and portfolio optimization. Part of the funds will be applied towards building new undergraduate and graduate curricula and educational materials in the area of discrete nonlinear optimization for students in engineering, management, economics, mathematics, and science. The project contributes to the knowledge of algorithms for solving such optimization models that combine discrete variables and nonlinear functions. As such, it will open new venues of research in the combined fields of discrete and nonlinear optimization. The results of the project will provide significant enhancements to the optimization tools that are used in practice. The final outcome will be efficient B&C strategies that will advance our ability to solve models that stand as some of the most difficult in fields such as finance, bioinformatics, and data mining, and therefore are at the core of our economy and welfare. The results and tools developed through the project, including the new theoretical developments, software, and the new teaching resources, will be publicly available through a website that will be created for the project. The development and dissemination of the educational material related to the project will contribute to the formation of personnel in the field of operations research who are essential both in industry and academia.