Several Problems Of Continuous And Discrete Optimization

Continuous and discrete optimization problems is very active and rich research content. They involve a very wide range of applications.We study three aspects about the continuous and discrete optimization problems in this paper. The specific content is divided into three parts, as follows.First, the notion of generalized convex functions is an essential condition of establishing multi-objective optimization conditions and duality theory and setting theory foundation about effective algorithm and stable analyze for (VP1). We explore a kind of generalized convex function (generalized essentially pseudo convex function and generalized essentially quasi convex function).We set up new sufficient condition and its saddle point theorem about generalized convex function. We study Wolf duality and Mond-Weir duality of the nonsmooth generalized convex multi-objective programming and proof dualily theorem.Combinatorial optimization is quickly select the optimal approach of effective decision-making program to in many discrete decision plan.The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems.The second part is the famous conjecture about extremal problems of the combinatorial optimization. Frankl and Furedi conjectured that the r-graph with m edges formed by taking the first m sets in the col ex ordering of N(r) has the largest Lagrangian of all r-graphs with m edges in1989. In chapter4, we proved that Frankl and Furedi’s conjecture holds for nearly colex ordering3-graphs. In most applications, we need an upper bound for the Lagrangian of a hypergraph. In chapter5, we proved that the r-uniform graph with m edges formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-uniform graphs on t vertices with m edges when the r-dimensional hypergraph meet the given conditions. Calculation about Lagrangian of the general hypergraphs is difficult, but our approach provide theoretical guide for a class of optimization problems to industry.The third part is critical conjecture about the minimum edge chromatic number.Very old famous minimum edge chromatic number was given by the smallest integerprogramming portrayed of2-graphs, but the specific solving is NP-hard. It has a seriesof very interesting conjecture. In1977, Professor Hilton proposed a few criticalityconjectures with respect to the minimum number edge-coloring. This attracted higherattention. Specifically, Hilton proposed the following conjecture: a simple graphwithout containing spanning subgraph is VI-critical if and only if it is VC–critical. Inchapter6, we prove some results. Support2-graph is a star multigraph of meeting thenumber of vertices, then2-graph is VI-critical if and only if it is VC-critical. Weprove that professor Hilton’s conjecture holds for the given conditions.Finally, we give the conclusion and deal with the major conclusions of thepresent study.