Combinatorial and global optimization with applications to data mining

建议的研究重点是方法的发展（即定理，精确算法和启发式）和数学规划的应用，其目的是解决多变量，约束优化问题。我在几个分支领域工作，有很多合作者。第一个子域是全局优化，其目的是确定非线性、非凸函数在线性或非线性、凸或非凸约束下的全局最优解。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。这将导致算法的一般化、简化和统一，如凹规划、多准则线性规划有效集的优化等。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、内点法和全局优化的结合。第三个子领域是元启发式。我们将以可变邻域搜索（VNS）元启发式的成功为基础，获得各种问题的近最优解，同时也增强精确算法（例如，通过稳定或聚焦列生成），并为启发式提供性能保证。第四个子领域是实验数学。我们将追求一种新型应用的发展，即利用基于vns的autographhix系统在计算机上寻找图论中的猜想。自1997年开发以来，它已经发现了1600多个猜想，其中800多个已经被自动证明，400多个是在40篇论文中手工证明的。这些猜想正越来越多地被一些国家的数学家，特别是代数图论的数学家所解决。

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.