min S(x,y,z), st. V(x,y,z) = V0, h0 <= y + z <= h1
max V(x,y,z), st. S(x,y,z) = S0, h0 <= y + z <= h1
using the augmented Lagrange multiplier method. You may use matlab
function fmincon.m with bound constraints only.
Also use the augmented Lagrange multiplier method to solve the LP:
max b'*y: -1 <= A'*y <= 1.
min S(x,y,z), st. V(x,y,z) = V0, y + z = h0
max V(x,y,z), st. S(x,y,z) = S0, y + z = h0
where S is the surface area and V is the volume
of the tent which has a square base of 2x by 2x,
a height of y for the main body, and a height of
z for the pyramid top. The tent does not have a bottom.
[S,x,y,z,u] = Newton_tent1(V0,h0); [V,x,y,z,u] = Newton_tent2(S0,h0);where u is the Langrange multiplier vector. You may use finite difference approximation for the Jacobian of the KKT system.