Solving Constrained Optimization for 4 variables (or 3?)

I am trying to solve the question from the picture below.

I dont know how to modify the original code from solving 2 variables into solving 4 variables. My friend said it was supposedly only 3 variables (x4 supposedly was x3), maybe because the typo.

There are my Matlab code

Answer 1

[Torsten]

It's a simple linear optimization problem. If you are allowed to, use "linprog" to solve.

f = [1; 1; 1; 0];
Aeq = [3 1 0 -1;1 -2 1 0];
beq = [-5;1];
lb = [-Inf; 0 ;0 ;0];
ub = [Inf ;Inf; Inf ;Inf];
sol = linprog(f,[],[],Aeq,beq,lb,ub)

[Tristofani Agasta]

I tried with another code from forum, but still not giving the same solution like yours

clc
clear
close all
syms l b h k lambda
A = l + b + h + 0; 
V = 3*l + b - k == 0;
G = l - 2*b + h == 0;
L = A + lambda * lhs(V) + lambda * lhs(G);

dL_dl = diff(L,l) == 0; 
dL_db = diff(L,b) == 0; 
dL_dh = diff(L,h) == 0; 
dL_dk = diff(L,k) == 0;
dL_dlambda = diff(L,lambda) == 0;

system = [dL_dl; dL_db; dL_dh; dL_dk; dL_dlambda];

[l_val, b_val, h_val, k_val, lambda_val] = solve(system, [l b h k lambda], 'Real', true) 
results_numeric = double([l_val, b_val, h_val,k_val, lambda_val]) 

V_res = l_val + b_val + h_val + 0
V = 3*l_val + b_val - k_val;
G = l_val - 2*b_val + h_val;

[Torsten]

This code is absolutely inadequate for the problem. Don't stick to it.

And I also don't see how to implement the non-negative constraints for x2 - x4 in the Lagrange approach.

[Tristofani Agasta]

Thank you for your response, perhaps I must find a book which tell me how to do with negative constraints?

[Torsten]

Is there any reason not to use linprog ?

[Tristofani Agasta]

Yes, because my lecturer demanding to use basic programming with steps and can be explained mathematically. Just like Gauss Elimination with some iterations without using library or toolbox

[Torsten]

Then look up "simplex algorithm".

Programming it, everything is hand-made.

[Tristofani Agasta]

Thank you for your reply