Plotting and maximizing a function subject to constraints (using an anonymous function)?

I have a function f(x) = -(x-alpha)^2 and I want to maximize it subject to the constraints x >= 0 and x<= 4. Let's say for starters the value of alpha = 5.

I want to graph the function f(x) and vertical lines marking the lower and upper boundaries of the constraints (so basically a line at x = 0 and x =4) and then a dot at the point where the function is maximized, subject to those constraints.

I am trying to define an anonymous function f = @(x) - 4 (x - alpha)^.2 after setting the value of alpha, so that I can then use if statements to calculate the constrained maximum and then plot everything... I need help! Thanks!

Answer 1

[Star Strider]

The easy way:

f = @(x) - 4 .* (x(1) - x(2)).^2;
x = fmincon(@(x)-f(x), [2; 5], [], [], [], [], [0; -Inf], [4; Inf])

x =
     2.0075
     2.0075

Here, x(1) is ‘x’, and x(2) is ‘alpha’. The solutions are not unique, as you can easily demonstrate with different initial parameter estimates. (I suspect this is a ‘proxy’ Question for a different problem.)

[Rohini Ghosh]

The optimization problem isn't actually my issue, finding the maximum is fine but it's how to plot it. I want to plot the function, two vertical lines representing the constraints and a dot on the point of the calculated maximum. Thanks!

[Star Strider]

My plesure!

The plot is easier than the fmincon call:

f = @(x) - 4 .* (x(1) - x(2)).^2;
x = fmincon(@(x)-f(x), [1; 5], [], [], [], [], [0; -Inf], [4; Inf])

xlims = [0, 4];

figure(1)
plot(x(1), x(2), 'bp')
hold on
plot(xlims(1)*[1, 1], ylim, 'r-',   xlims(2)*[1, 1], ylim,'r-', 'LineWidth',2)
hold off
grid
axis([-1  5    ylim])

That should do what you want.