Combinatorial and global optimization with applications to data mining

拟议的研究重点是方法（即定理，精确算法和算法）的发展和数学规划的应用，其目的是解决多变量，约束优化问题。我在几个子领域工作，与许多合作者。第一个子领域是全局优化，其目的是确定线性或非线性，凸或非凸约束下的非线性，非凸函数的全局最优解。我们将继续研究具有特定结构的问题，并使用算法嵌入的概念将问题转化为另一个问题。这将导致几个问题的算法的推广，简化和统一，例如凹规划，多目标线性规划的有效集的优化等。第二个子领域是大规模组合优化问题的分解方法，以及它们与整数规划、内点方法和全局优化的结合。- 第三个子领域是元分析学。我们将建立在可变邻域搜索（VNS）元启发式的成功，以获得各种问题的接近最优的解决方案，但也提高了精确的算法（例如，通过稳定或集中列生成），并提供性能可靠性的分析。- 第四个分支是实验数学。我们将致力于开发一种新型的应用程序，即使用基于VNS的AutoGraphiX系统通过计算机查找图论中的图形。自1997年开发以来，它已经发现了1600多个证明，其中800多个已经被自动证明，400多个在40篇论文中被手工证明。这些问题越来越多地被解决的数学家在几个国家，特别是那些代数图论。

The proposed research focusses on development of methods (i.e. theorems, exact algorithms and heuristics)and applications of mathematical programming, the aim of which is to solve multivariate, constrainedoptimization problems. I work in several subfields, with many collaborators.- The first subfield is global optimization whose purpose if to determine global optima of nonlinear, nonconvexfunctions subject to linear or nonlinear, convex or nonconvex constraints. We will continue our study of problems with particular structure and transformation of problems one into the other, using the concept of embedding of algorithms. This should lead to generalization, simplification and unification of algorithms for several problems as e.g. concave programming, optimization of the efficient set of multicriteria linear programs, etc.- The second subfield is decomposition methods for large-scale combinatorial optimization problems, and their combination with integer programming, interior point methods and global optimization. - A third subfield is metaheuristics. We will build upon the success of the Variable Neighborhood Search (VNS) metaheuristic to obtain near optimal solutions to various problems but also to enhance exact algorithms (e.g. by stabilizing or focusing column generation) and provide performance guaranties for heuristics. - A fourth subfield is experimental mathematics. We will pursue the development of a new type of application, i.e. finding conjectures in graph theory by computer with the VNS-based AutoGraphiX system. Developed since 1997 it has already led to finding over 1600 conjectures, more than 800 of which have been proved automatically and over 400 by hand in forty papers. These conjectures are increasingly being solved by mathematicians in several countries, particularly those on algebraic graph theory.