In this paper we consider three problems in continuous multi-criteria optimization: An application of the Berge Maximum Theorem, properties of Pareto-retract mappings, and the structure of Pareto sets. The key goal of...In this paper we consider three problems in continuous multi-criteria optimization: An application of the Berge Maximum Theorem, properties of Pareto-retract mappings, and the structure of Pareto sets. The key goal of this work is to present the relationship between the three problems mentioned above. First, applying the Maximum Theorem we construct the Pareto-retract mappings from the feasible domain onto the Pareto-optimal solutions set if the feasible domain is compact. Next, using these mappings we analyze the structure of the Pareto sets. Some basic topological properties of the Pareto solutions sets in the general case and in the convex case are also discussed.展开更多
用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归...用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。展开更多
文摘In this paper we consider three problems in continuous multi-criteria optimization: An application of the Berge Maximum Theorem, properties of Pareto-retract mappings, and the structure of Pareto sets. The key goal of this work is to present the relationship between the three problems mentioned above. First, applying the Maximum Theorem we construct the Pareto-retract mappings from the feasible domain onto the Pareto-optimal solutions set if the feasible domain is compact. Next, using these mappings we analyze the structure of the Pareto sets. Some basic topological properties of the Pareto solutions sets in the general case and in the convex case are also discussed.
文摘用5个定理给出最小一乘线性回归的相关性质,为其工程应用奠定了基础。文中首先证明了“由“最小一乘”准则确定的直线y=b1x1+ b2x2经过其两个样本点”以及“由最小一乘准则确定的直线y=b1x1+ b2x2+a经过其三个样本点”。然后应用数学归纳法得到如下定理:设有n(n>P)个样本点(x1i, x2i, ? xP i, yi,),则由最小一乘准则确定的线性非奇次模型y=b1x1+b2x2+?bPxP+a经过其P+1个样本点,而相应的奇次模型必经过其P个样本点。通过大量工程实例证实了最小一乘具有较强的稳健性,同时也证实了定理的正确性。
