What do you mean by a vector is "decreasing"? There is no linear ordering of vectors.
I am trying prove that a function has a unique optimal solution. It is a function with two variables, and it is not necessarily well-behaved, in particular is not concave. Can I argue the following?
The vector of first order derivatives is decreasing in the vector of variables, hence the function is unimodal and would have a unique optimal solution.
In other words, for univariate functions, we can show the unimodality by argueing that the first derivative is decreasing. Can I generalize it to the multivariate case as described above?
Thanks for any help provided.
I am sorry, let me rephrase my question: Using only the first derivatives and the cross partial derivatives, how can I show that a function of two variables is unimodal? More specifically, if the function is z = z(x,y), I am talking about partial derivative with respect to x, z_x, partial derivative with respect to y, z_y, and the cross partial derivative z_xy.