Second Order optimality fmincon

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Sargondjani 2019-5-23

Someone explained the second order condition for optimality of a constrained optimization problem here:

So I should take the Hessian, and ZZ = nullspace of the jacobian of all active constraints, and then it is an optimal point if

Z'*Hessian*Z >= 0.

Here is my code:

[xx,~,~,~,~,~,hess] = fmincon(@(XX)-XX(1)^2-XX(2)^2,[0.3,0.3],[],[],[1,1],1);
ZZ= null([1,1])
ZZ'*hess*ZZ

The active linear constraint: x1 + x2 = Q, so I thought the Jacobian of the only actice constraint is [1,1].

Matlab's nullspace: ZZ = [-sqrt(2)/2;sqrt(2)/2)];

Z'*Hessian*ZZ is 1, but the point is not a local minimum (only stationary point). I want to proof numerically that it is only a staionary point.

What goes wrong?

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