How to use quadprog for this multi-dimension problem?

I am trying to use quadprog.m to solve the following 2 dimensional optimization problem.

where $(x_i,y_i)$ is the data point.

How may I implement this 2 dimensional problem in matlab by quadprog? I can do for 1 dimensional problem, i.e., only with $\alpha_i$. But now I have $\alpha_i$ and $\alpha_i'$, i.e., two sets of unknown.

Answer 1

[Alan Weiss]

All Optimization Toolbox™ solvers require that you put your control variables into a single vector, typically called x. For your case, I suggest that you set the column vector x to be

 x = [alpha;alphaprime];

In other words, just concatenate your two existing column vectors of unknowns into one longer column vector. Given this transformation, it is not too difficult to write matrices H and f to represent your problem in the requisite form

 1/2*x'*H*x + f*x

Remember, quadprog minimizes, so if you are looking for a maximum, take the negative of your objective function.

If you cannot figure out how to do it with quadprog, you can always use fmincon.

Alan Weiss

MATLAB mathematical toolbox documentation

[dragoon]

I can write down f very quick. But could you help me a bit on how to write H?

[Matt J]

You say you can write the Hessian H0 for the single unknown vector case. For this case, they will be related by

H = [I,-I].' * H0 * [I,-I]

where I=eye(n);

[dragoon]

Yes, I see! Thank you!