问题 单项选择题

每个线性规划问题需要在有限个线性约束条件下,求解线性目标函数F何处能达到极值。有限个线性约束条件所形成的区域(可行解区域),由于其边界比较简单(逐片平直),人们常称其为单纯形区域。单纯形区域D可能有界,也可能无界,但必是凸集(该区域中任取两点,则连接这两点的线段全在该区域内),必有有限个顶点。

以下关于线性规划问题的叙述中,不正确的是()。

A.若D有界,则F必能在D的某个顶点下达到极值

B.若F在D中A、B点上都达到极值,则在AB线段上也都能达到极值

C.若D有界,则该线性规划问题一定有一个或无穷多个最优解

D.若D无界,则该线性规划问题没有最优解

答案

参考答案:D

解析:

[分析]: 本题旨在从宏观上理解线性规划方法的原理与机制,特别是从二维、三维的直观理解推广到高维的理解。这种宏观的、直观的理解对于深刻认识数学概念、方法是非常重要的,对于创新也会有重要的、奇特的启发作用。

很明显,有界区域内线性函数的值域肯定是有界的。从直观上可以理解,由于线性函数的平坦性,其极值一定会在边界上达到(许多教材上给出了严格证明)。直观的理解有助于形象地感悟某些理论研究的结论。由于单纯形区域的边界是逐片平直的,它对应的线性目标函数值域也会是逐片平直的,人们可以想象,线性函数F会在D区域的顶点处达到极值。所以选项A是正确的。

由于单纯形区域是凸集,只要A、B两点在区域内,则线段AB全在该区域内。由于F(A)与F(B)在线性目标函数值域上,不难看出,线段AB中的任一点C对应的 F(C)就会落在F(A)与F(B)的连线上。所以选项B也是正确的。

选项C可以从选项A与B导出。线性规划问题要么无解,要么只有唯一的最优解,要么会有无穷多个最优解。因为如果有两个最优解,则这两个解的连线段上所有的解都是最优解。所以选项C也是正确的。

选项D不正确。若区域D无界,则线性规划问题可能无解,也可能有解(唯一解或无穷多个解)。

例如,线性规划问题:

MAn Z=X+Y s.t. X≥0,Y≥0

的可行解区域是无界的,但在X=0,Y=0时有唯一的最优解(极小值)Z=0。

又例如,线性规划问题:

Max Z=2X

s.t. X≤4,X≥0,Y≥0

的可行解区域是无界的,但在X=4,Y≥0处有无穷多个最优解(极大值)Z=8。

线性规划问题:

Max Z=X+Y

s.t. X≤4,X≥0,Y≥0

的可行解区域是无界的,不存在最优解。

在坚实的理论基础上,直观、形象、宏观地看问题不仅能深刻理解问题的实质,有时还能启发新的思路,创立新的问题求解方法。

例如,用单纯形方法求解线性规划问题的过程,实际上就是在单纯形区域D的边界上先选一个初始顶点,再通过迭代计算,沿着D的边界逐个顶点行进,直到达到最优解的那个顶点为止。

在企业实际应用中,一般会有大量的变量,区域D的顶点也很多,这种方法的计算量足很大的。

从直观上看,从区域D的一个顶点出发,沿D的边界前进直到最优解顶点,一般都是绕弯的。人们会想到,应该有更捷径的路,而这条路可能是从区域D内穿越过去的。从区域D的一个点出发,沿什么方向走会使线性函数值F增长最快(或下降最快)呢显然,应该沿函数F的梯度方向(或负梯度方向)前进,直到区域D的边界,会有更好的效果。据此,我们认为,可以获得比单纯形法更快的迭代求解方法。当然,单有直观思维是不够的,还需要在这种思维的指导F,去寻求实际可行的求解方法。现在,也确实有人按这种思路获得了新的解法。

单项选择题

Questions 1 to 5 are based on the following conversation.

M: You know, in designing this new town, we’ve tried to look backward at small-town America and take the best of those planning elements: houses close together; sidewalks, front porches, tree-lined streets, easy, non-automobile-dependent access to the town center and to your neighbors and to the school and the other institutions that are vital. And we’ve tried to take some of those ideas and update them and come up with a livable, workable place where people can go and rekindle the sense of community that seems to be missing from suburbs all across the country.

F: So give us a sense of how this new town was designed.

M: Houses are all very close together. We were just 10 feet apart from our neighbors on either side of us, and that’s pretty much the standard for the town. So, you have houses that are close together, houses that surround open areas. They have a lot of big parks, a lot of common areas. The theory is that you’re willing to sacrifice your private yard space--you don’t need a quarter of an acre or half an acre--if you have a public area where you can go and enjoy the facilities there and, most importantly, you can interact with your neighbors. That helps to create this sense of community that’s so important to many of these "new town" developments.

F: Now, I see this whole sense of community: It’s going to be a new town, but we’re going to do it with a sense of nostalgia for the past. Like, a lot of the houses had porches.

M: Yes, it would create a "front porch culture"--that people would be out on their porches talking to their neighbors next door and to people walking down the street or people riding their bikes, and there would be this culture that existed 40, 50, or 60 years ago. But that really has been one of the failures that we observed during our two years in this new town, and the people don’t spend very much time at all on their front porches. There are a couple of things going on. One is it’s central Florida, and it’s hotter than hell a good part of the year, and sitting on your front porch, even if you have a fan going, can be a very uncomfortable thing. People prefer to be inside in the air- conditioning.

F: What were some of the rules you had to live by in the new project And did any of those rules bother you

M: Well, the developer and I have different feelings about rules. His feeling is--now, if I can summarize his feelings for him--that you move in there knowing the rules, and if you don’t like them, you shouldn’t move in. I have some problems with rules. I just sometimes like to break them. And they just bother me because they’re in existence. But the rules sometimes were silly and sometimes weren’t. They dictated what color your curtains could be facing the street and actually asked a woman with red curtains to remove them.

F: Really Is that true

M: And they dictated where you could park your car and for how long. They dictated any sort of thing you could attach to your house. You couldn’t attach a satellite dish to your house. They dictated forever the color of your house. And they dictate how often you have to repaint your house. They’ve tried to go a step further and remove plastic flowers and plastic furniture from those all-important front porches. Some rules seem to go a little too far.

Question No. 1 What are the two speakers talking about

3().

A. It has nothing to do with a sense of nostalgia for the past.

B. It has failed in the new town mentioned in the conversation.

C. People prefer to stay in an air-conditioned front porch.

D. People spend very much time on front porches in hot climates.