1. Do Exercise 4.1
Limit attention to the following three methods.
1.b Steepest descent with line search. You can use fminsearch to find λ.
1.d Conjugate gradient based on steepest descent. You can use fminsearch to find λ.
To see which method does well, try different starting values, and try to find cases that lead to correct solution under some method(s) but not other(s). Specifically, try starting from (0,0), (-1,-1) and (-1,2), and report (xk, y k) for the first 5 iterations.
You do not need to plot the function, but if you want to have a look at it, the command are countour() for contour plot, and mesh() for 3D plot.
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1. Do Exercise 4.2
Solve the problem for m =n = 3 using: (A) Penalty method, (B) Matlab’s fmincon.
The preference parameters are:
aii
=
1
aii+1
=
0.2;a1N = 0.2
aij
=
0 in all other cases
vji
=
0.5
The weights are equal for all: λi = 1. Endowments are equally distributed: eji = n1. The constraints include the market clearing:
iN=1xji ≤iN=1 eji, j = 1,...,M
and non-negativity: xji ≥ 0.
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