nlp作业

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IE521:Nonlinear Programming Homework1
Professor Terry L.Friesz
15Jan2014
Instructions:The due date of this assignment is Tuesday,Jan.27nd,2015. Please submit your written work to the teaching assistant or in one pdf…le to yiw5120@.
1.Consider the following nonlinear program:
min f(x)= x1
such that
x21+x22 1
(x1 1)3 x2 0
Solve this problem graphically.
2.Review the de…nition of gradient and the Hessian matrix for a function,
determine the gradient and the Hessian for each function:(a)f(x)= 3x1x2+4e x1x2;(b)f(x)=x x21+ln x1x2:
3.Review the simplex method in tablau form and solve the following problem
with it:
max2x1+3x2
such that
x1+2x2 6
2x1+x2 9
x1;x2 0
4.Review the revised simplex method and solve the following problem with
it:
max2x1+4x2+3x3+x4
such that
3x1+x2+x3+4x4 12
x1 3x2+2x3+3x4 7
2x1+x2+3x3 x4 10
x i 0
1
5.Consider the following linear program and its dual:
min cx s.t.Ax b;x 0
(LP)max wb s.t.wA c;w 0(D-LP)
suppose the optimal feasible solution for (LP)is x ;the optimal solution for the dual is w ;and they are non-degenerate,also suppose z is the opti-mal value of (LP):(a)Show that @z @b i =w i (b)What is your interpretation of the above result?
6.Suppose you are stranded on a desert island.In order to return to civiliza-tion,you must invert a large (n-by-n)matrix.However,the only software available is a linear programming solver written in a programming lan-guage developed by space aliens and which you cannot comprehend.The matrix is M =0B B B B @m 11m 12 m 1n m 21...m 2n .........m n 1m n 2 m nn
1C C C C A Find M 1by providing only input data to the LP solver and interpreting only output results from the LP solver.
2。