I have a simple optimization problem. The objective function consists of $4$ variables, say $a,b,c$ and $d$. So the objective function $y=f(a,b,c,d)$ is a linear function of $a,b,c,d$.
The constraints for these variables are only their range, say: $$3 \leq a \leq 10,$$ $$25 \leq b \leq 35,$$ $$80 \leq c \leq 100$$ $$30 \leq d \leq 45.$$ My question is which optimization technique/method can I use to get maximum value of $y$.
Say $$ f(a,b,c,d)=2a-3b+5c-d. $$ Since the domain you prescribed is compact, and since the function is continuous, there is indeed a maximum. You want to maximize $2a$ and $5c$, and minimize $3b$ and $d$. For instance, $2a$ is maximum for $a=10$ , while $d$ is minimum for $d=30$. Hence $$ \max f=2\cdot 10-3\cdot 25+5\cdot 100-30. $$ More generally, to get the maximum of you linear functional, you need to maximize each variable corresponding to a positive coefficient, and you minimize each variable corresponding to a negative coefficient.
It's a linear program. There are several ways to solve it.
If you want a really simple solution, evaluate your objective function on each corner of your hyperbox, i.e. find the max of $f(3,25,80,30), f(10,25,80,30), \ldots $. Since it's a 4D box, you have 16 corners and thus 16 possible solutions.
Since you're solving a linear optimization problem, you're guaranteed an optimal solution on one of the corners.
That is a very simple linear programming problem. Just take the maximum value of the variables with positive coefficients in the objective function, and the minimal value of those with negative coefficients.
