Discrete Optimization Algorithms based on Discrete Convex Analysis

In the area of nonlinear programmingthe theory of convex analysis provides a unified framework for well-solved problems. In the area ofdiscrete optimization on the other hand，我们不做一个统一的框架。Matroidal structure, however，“discrete convex”理论是已知的一个良好的结构，在discrete optimizationanalysis is proposed by the head investigator with a view to capturing the continuous optimizationdiscrete optimization from a common viewpoint. discrete convex analysis is a theoreticalframework connecting the theory of convex analysis and the theory of matroids.It is often the casediscrete optimization problems appearing in the real world do not have nice combinatorialstructure such as matroids. Therefore，we need to use -type algorithms such as the branch-and-bound方法or approximationalgorithms such as meta heuristics. In either approach，it is essential to extract tractable part from the discrete structure…More of the problems to be solved. For example，这通常是有效的，类似网络结构的subproblems when we solve generaldiscrete optimization problems.The fundamental idea of our研究is to extract certain结构which can be dealt with discrete convex analysis as subproblems when we solve general discreteoptimization problems. We summarize the main results as follows: the concepts of M-convex andL-convex functions play a central role in the framework of discrete convex analysis. We showedvarious properties of these functions. We proposed efficient algorithms for the minimization ofM-convex functions我们extended the concepts of M-convex and L-convex functions defined over theinteger lattice to functions over the real space.我们generalized the concepts of M-convex andL-convex functions to quasi M-convex/L-convex functions. We obtained some results on the应用程序of discrete convex analysis to economic equilibrium such as the equivalence of gross substitutesproperty and M-convexity. Less

In the area of nonlinear programming, the theory of convex analysis provides a unified framework for well-solved problems. In the area of discrete optimization, on the other hand, we do not have such a unified framework. Matroidal structure, however, is known to be a well-behaved structure in discrete optimization. The theory of "discrete convex analysis" is proposed by the head investigator with a view to capturing the continuous optimization and the discrete optimization from a common viewpoint. Discrete convex analysis is a theoretical framework connecting the theory of convex analysis and the theory of matroids.It is often the case that discrete optimization problems appearing in the real world do not have nice combinatorial structure such as matroids. Therefore, we need to use enumeration-type algorithms such as the branch-and-bound method or approximation algorithms such as meta heuristics. In either approach, it is essential to extract tractable part from the discrete structure … More of the problems to be solved. For example, it is often effective to extract network-like structure as subproblems when we solve general discrete optimization problems.The fundamental idea of our research is to extract certain structure which can be dealt with discrete convex analysis as subproblems when we solve general discrete optimization problems. We summarize the main results as follows :・The concepts of M-convex and L-convex functions play a central role in the framework of discrete convex analysis. We showed various properties of these functions.・We proposed efficient algorithms for the minimization of M-convex functions.・We extended the concepts of M-convex and L-convex functions defined over the integer lattice to functions over the real space.・We generalized the concepts of M-convex and L-convex functions to quasi M-convex/L-convex functions.・We obtained some results on the application of discrete convex analysis to economic equilibrium such as the equivalence of gross substitutes property and M-convexity. Less