Quadratic optimization

If the highest power is 1, then we call it a linear model. On the other hand, if the highest power is 2, we call it a quadratic function. The R optim() function can be used to find a solution for a minimization problem. For example, we have the following objective function:

Since there is only one variable, we can solve it manually. Take the first-order derivative and set it to:

x<-seq(-10,10,0.1) 
a<--2 
b<-10 
c<-5 
y<-a*x^2+b*x+c 
plot(x,y,type='l') 

The related graph is shown here:

From the graph, we know that we could get a maximum y value when x is zero:

y<-20-3.5*x^2 
a<--2 
b<-10 
c<-5 
f<-function(x)-(a*x^2+b*x+c) 

In the preceding formula, we use a negative function since the R optim() function would get a minimum value instead of a maximum one:

> optim(0.3,f) 
$par 
[1] 2.500078 
$value 
[1] -17.5 ...