Assuming I have an optimization problem of the form f(x,y) subject to I.) g(x,y)-c=0 ; II.) h(x,y)-d=0 (x,y being variables, c and d constants).
The Lagrangian will look like this L(x,y,l,m)=f(x,y)-l(g(x,y)-c)-m(h(x,y)-d) with l and m being the respective Lagrange multipliers.
Setting the Gradient of L(x,y,l,m) = 0 and solving for x,y,l,m will give me the critical point(s).
For the Second-Order-Condition, I would however get a Bordered Hessian of k+n (=4) rows/columns dimensions, k (=2) being the number of constraints and n (=2) the number of variables. Hence, there is no second derivative test possible as the smallest principal minor whose determinant I would have to evaluate is given by 2k+1 (=5) rows/columns, which is greater than the entire Hessian! What can I do in these type of cases?
Any help would be greatly appreciated!!
The Lagrangian will look like this L(x,y,l,m)=f(x,y)-l(g(x,y)-c)-m(h(x,y)-d) with l and m being the respective Lagrange multipliers.
Setting the Gradient of L(x,y,l,m) = 0 and solving for x,y,l,m will give me the critical point(s).
For the Second-Order-Condition, I would however get a Bordered Hessian of k+n (=4) rows/columns dimensions, k (=2) being the number of constraints and n (=2) the number of variables. Hence, there is no second derivative test possible as the smallest principal minor whose determinant I would have to evaluate is given by 2k+1 (=5) rows/columns, which is greater than the entire Hessian! What can I do in these type of cases?
Any help would be greatly appreciated!!