连续与离散最优化中的几个问题
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摘要
连续与离散最优化问题的研究内容十分活跃和丰富，其所涉及的范围及其应用十分广泛。本文主要研究了连续与离散最优化领域中的三方面问题，具体内容如下。
第一部分是引进了一类非光滑广义凸函数，并且对这类非光滑广义本性凸规划的锥有效解给出了最优性条件与鞍点定理,同时建立了Mond-Weir型对偶和Wolf型对偶；具体讲，在第三章中，我们首先引进了广义本性伪凸函数，广义本性拟凸函数的概念，其次对包含这类广义本性伪凸，广义本性拟凸的多目标Lipschitz规划的弱有效解给出了最优性充分条件，同时给出了多目标最优化问题(VP1)的向量值Lagrangian鞍点定理，最后，对这类非光滑广义凸多目标规划建立了Wolf型对偶和Mond-Weir型对偶，并证明了对偶定理。
第二部分是有关组合极值问题的著名猜想：在所有m条边的r维超图当中，按照图的Colex排序得到的r维超图具有最大的Lagrangian值。此猜想是由Frankl-Füredi于1989年提出。这个猜想对于r维超图的Lagrangian值的估计依赖于对r-次多变量函数的最优化极值的估计。具体讲：（一）我们证明了当所给定的3维超图与由Colex排序决定的3维超图比较接近时，Frankl–Füredi猜想对3维超图成立；（二）我们证明得到了具有t个顶点的r维一致超图的边满足一定条件时，按照图的Colex排序形成的r维超图，具有最大的Lagrangian值。因此，Frankl-Füredi猜想在满足给定的限制条件下对于r维超图成立。虽然对一般超图的Lagrangian值的估算很难，但是我们的方法能为工业界的一类优化问题提供理论指南。
第三部分是有关最小边着色数的临界猜想。非常古老著名的最小边着色数最初是通过2-图的最小整数规划刻画出来的，但是具体的求解是NP-难题。这方面有着一系列非常有意义的猜想，1977年，英国的Hilton教授提出了几个有关最小边着色数的临界猜想，引起了大家的关注。其中Hilton提出了下面猜想：一个不包含生成子图K1,的简单图G是VI-临界的当且仅当它是VC-临界的。在这一部分中，我们证明了部分结果：设2-图G是顶点个数满足一定条件的星多重图，那么2-图G是VI-临界的当且仅当G是VC-临界的。也就是说，1977年Hilton教授提出的猜想，在给定的条件下是成立的。我们改进了关于星多重图的边着色的顶点临界性的一个结果，改进了文[82]中Hilton所给结果的下界，使得它的下界变得越来越小，得到的结果是目前较好的结果。
最后，我们探讨了进一步研究的可能性。
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.
第一部分是引进了一类非光滑广义凸函数，并且对这类非光滑广义本性凸规划的锥有效解给出了最优性条件与鞍点定理,同时建立了Mond-Weir型对偶和Wolf型对偶；具体讲，在第三章中，我们首先引进了广义本性伪凸函数，广义本性拟凸函数的概念，其次对包含这类广义本性伪凸，广义本性拟凸的多目标Lipschitz规划的弱有效解给出了最优性充分条件，同时给出了多目标最优化问题(VP1)的向量值Lagrangian鞍点定理，最后，对这类非光滑广义凸多目标规划建立了Wolf型对偶和Mond-Weir型对偶，并证明了对偶定理。
第二部分是有关组合极值问题的著名猜想：在所有m条边的r维超图当中，按照图的Colex排序得到的r维超图具有最大的Lagrangian值。此猜想是由Frankl-Füredi于1989年提出。这个猜想对于r维超图的Lagrangian值的估计依赖于对r-次多变量函数的最优化极值的估计。具体讲：（一）我们证明了当所给定的3维超图与由Colex排序决定的3维超图比较接近时，Frankl–Füredi猜想对3维超图成立；（二）我们证明得到了具有t个顶点的r维一致超图的边满足一定条件时，按照图的Colex排序形成的r维超图，具有最大的Lagrangian值。因此，Frankl-Füredi猜想在满足给定的限制条件下对于r维超图成立。虽然对一般超图的Lagrangian值的估算很难，但是我们的方法能为工业界的一类优化问题提供理论指南。
第三部分是有关最小边着色数的临界猜想。非常古老著名的最小边着色数最初是通过2-图的最小整数规划刻画出来的，但是具体的求解是NP-难题。这方面有着一系列非常有意义的猜想，1977年，英国的Hilton教授提出了几个有关最小边着色数的临界猜想，引起了大家的关注。其中Hilton提出了下面猜想：一个不包含生成子图K1,的简单图G是VI-临界的当且仅当它是VC-临界的。在这一部分中，我们证明了部分结果：设2-图G是顶点个数满足一定条件的星多重图，那么2-图G是VI-临界的当且仅当G是VC-临界的。也就是说，1977年Hilton教授提出的猜想，在给定的条件下是成立的。我们改进了关于星多重图的边着色的顶点临界性的一个结果，改进了文[82]中Hilton所给结果的下界，使得它的下界变得越来越小，得到的结果是目前较好的结果。
最后，我们探讨了进一步研究的可能性。
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.
引文
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[2] KAUL R N, KAUR S. Optimality criteria in nonlinear programming involvingnonconvex functions. J Journal of Mathematical Analysis and Applications,1985,105:104-112.
[3] BEN-ISRAEL B, Mond B. What is invexity. J Journal of Australian MathematicalSociety.1986,28B:1-9.
[4] CRAVEN B D. Fractional Programming. Sigma Series in Applied Mathematics.Heldermann Verlag Berlin,1988.
[5] CRAVEN B D, GLOVER B M. Invex functions and duality. J. Austral. Math.Soc,1985,24:1-20.
[6] MARTIN D H. The essence of invexity. J Journal Optimization Theory andApplications,1985,47:65-76.
[7] JEYAKUMAR V, MOND B. On generalized convex mathematical programming.J Journal of Australian Mathematical Society Ser B,1992,34:43-53.
[8] CRAVEN B D, GLOVER B M. Invex functions and duality. J. Austral. Math. Soc,1985,24:1-20.
[9] HANSON M A, MOND B. Further generalization of convexity in mathematicalprogramming. J Journal of Information and Optimization Science,1982,3:25-32.
[10] EGUDO R R. Efficiency and generalized convex duality for multiobjectiveprograms. J Journal of Mathematical Analysis and Applications,1989,138:84-94.
[11] MISHRA S K, MUKHERJEE R N. Duality for multiobjective fractionalvariational problems. J Journal of Mathematical Analysis and Applications,1994a,186:711-725.
[12] MISHRA S K, MUKHERJEE R N. On efficiency and duality for multiobjectivevariational problems. J Journal of Mathematical Analysis and Applications,1994b,187:40-54.
[13] MISHRA S K, MUKHERJEE R N. On generalized convex multiobjecive non-smooth programming. J Journal of the Australian Mathematical Society B,1996,38:140–148.
[14] SUNEJA S K, SRIVASTAVA M K. Duality in multiobjective fractionalprogramming involving generalized invexity. J Opsearch,1994,31:127-143.
[15] WEIR T. A note on invex functions and duality in multiple-objective optimization.Opsearch,1988,25:98–104.
[16] WEIR T, MOND B. Pre-invex functions in multiple objective optimi zation.Journal of Mathematical Analysis and Applications,1988,136:29–38.
[17] BECTOR C R, CHONDRA S, Bector M K. Sufficient Optimality Condition andDuality for a Quasiconvex Programming Problem. JOTA,1988,59:209-221.
[18] EGUDO R R.Efficiency and Generalized Convex Duality for Multiobjectiveprograms. J. Math Anal Appl,1989,138:84-94.
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