MATLAB: Projection of a point onto a closed convex

projection

I have a point P(50, 5000) and a curve (x.^2 – 2*x – 1) with x = [0:100]. How do I find the closest point on that curve to point P?

Thank you.

Best Answer

Use this FEX

Q=[50;5000];
% Projection candidates on x.^2-2*x-y-1 = 0
P=ConicPrj(Q,[1 0; 0 0],[-2;-1],-1);
% Find the closest
[~,loc]=min(sum((P-Q).^2,1));
P=P(:,loc);
x=P(1)
y=P(2)

The projection is

x = 71.723733062992125
y = 4.999846418365362e+03

Related Solutions

MATLAB: 4 unknown 3 nonlinear equation solution with minimum norm

I change your last equation rhs to 600 (otherwise there is NO solution).

Here is two methods to find the minimum norm solution.

The first method needs this file on FEX, no toolbox is required.
The second method needs FMINCON from the optimization toolbox.

% 1st & 2nd equations: Aeq*x = beq
Aeq = [3  5 11 14; ... 
       7  2  8  5];
beq = [100; 7];
% 3rd equation: x'*Hx*x + gx'*x + cx = 0
Hx = diag([8 7 8 5]);
gx = [9; 5; 9; 6];
cx = -600;
% Method 1, using ConicPrj
% on FEX https://www.mathworks.com/matlabcentral/fileexchange/27711-euclidian-projection-on-ellipsoid-and-conic
x0 = pinv(Aeq)*beq;
Q = null(Aeq);
Hy = Q'*Hx*Q;
gy = Q'*(gx + 2*Hx*x0);
cy = cx + x0'*(Hx*x0 + gx);
% File exchange to be downloaded
y = ConicPrj([0; 0], Hy, gy, cy);
[~,imin] = min(sum(y.^2,1));
xellprj = x0+Q*y(:,imin)
% Method 2: FMINCON
[xfmincon,a,b,c] = fmincon(@(x) sum(x.^2), zeros(4,1), ...
    [], [], ...
    Aeq, beq, ...
    [], [], @(x) deal([], x'*Hx*x + gx'*x + cx));
xfmincon

Both gives the same solution

>> test
xellprj =
   -6.3652
    2.9411
    2.0611
    5.8370
Local minimum found that satisfies the constraints.
Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.
<stopping criteria details>
xfmincon =
   -6.3652
    2.9411
    2.0611
    5.8370
>>

MATLAB: In an assignment A(I) = B, the number of elements in B and I must be the same.

I cannot follow what you are doing, and I cannot run your code.

However, the most likely problem is that you need to vectorize your code, since otherwise MATLAB will interpret multiply, divide, and exponentiation operations as matrix rather than element-wise array operations. That means using the ‘dot operator’, substituting ‘.*’ for ‘*’, ‘./’ for ‘/’, and ‘./’ for ‘^’.

Use the vectorize (link) fiunction, and see the documentation on Array vs. Matrix Operations (link) for details.

Best Answer

Related Solutions

MATLAB: 4 unknown 3 nonlinear equation solution with minimum norm

MATLAB: In an assignment A(I) = B, the number of elements in B and I must be the same.

Related Question