I have attempted the following question multiple times and I am very confused about the proof please help me solve it.

Let $V(x)=a+b^{T}x+\frac{1}{2}x^{T}Cx$ for some $a \in \mathbb{R}, b \in \mathbb{R}^{n}, C \in \mathbb{R}^{nxn}$ that for $V$ to have a strict unique minimum it is imperative that $C>0$.


We know that:

  1. A twice differentiable function of several variables is strictly convex on a convex set if and only if its Hessian matrix is positive definite on the interior of the convex set.

  2. Any local minimum of a convex function is also a global minimum

  3. A strictly convex function will have at most one global minimum.

So, basically, to guarantee that $V$ has a unique minimum we need its Hessian to be positive definite.

We have that $x = \left( {{x_1}, \ldots ,{x_n}} \right) \in {\mathbb{R}^n}$, so $V = V\left( x \right) = V\left( {{x_1}, \ldots ,{x_n}} \right).$

$$V\left( x \right) = a + {b^T}x + \frac{1}{2}{x^T}Cx = a + \sum\limits_{i = 1}^n {{b_i}{x_i}} + \frac{1}{2}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {{c_{ij}}{x_i}{x_j}} } $$

$$\frac{\partial }{{\partial {x_k}}}V\left( x \right) = {b_k} + \frac{1}{2} \cdot 2\sum\limits_{i = 1}^n {{c_{ki}}{x_i}} $$

$$\frac{\partial }{{\partial {x_l}}}\left( {\frac{\partial }{{\partial {x_k}}}V\left( x \right)} \right) = {c_{kl}}$$

Thus, the Hessian of $V$, which by definition has entries $${\left( {{H_V}\left( x \right)} \right)_{i,j}} = \frac{\partial }{{\partial {x_j}}}\left( {\frac{\partial }{{\partial {x_i}}}V\left( x \right)} \right)$$ is $${H_V}\left( x \right) = C.$$

Hence, for $V$ to have a unique global minimum, $C$ has to be positive definite.

  • $\begingroup$ Thank You very much, I was wondering what you ment by $d/d_{k}$ and by $b_{k}$ ? Why do we take a double derivative? Can't we take one derivative and conclude there is a minimum when it equals $0$. So $d/dx_{k} V(x)=b_{k} +1/2 *2 Cx = 0$. Then we can say that this occurs when $x=-C^{-1}b$ and re plug this back into the first equation? I am just a bit confused how to continue from there. $\endgroup$ – TorqueNoFriction Feb 1 '14 at 22:30
  • $\begingroup$ By $\frac{\partial }{{\partial {x_k}}}$ I mean the partial derivative with respect to $x_k$. Bear in mind that $V$ is a function of $x$ and $x =({x_1}, \ldots ,{x_n}) \in {\mathbb{R}^n}$ is a vector, so $V$ is a function of several variables! $\nabla V( x ) =( {\frac{{\partial V}}{{\partial {x_1}}},\frac{{\partial V}}{{\partial {x_n}}}, \ldots ,\frac{{\partial V}}{{\partial {x_n}}}}) =(0,0, \ldots ,0)$ is just a necessary condition, that is: if you know there is a minimum then this necessarily happens, but just the fact it happens is not enough to guarantee that a minimum exists. $\endgroup$ – etothepitimesi Feb 1 '14 at 22:47
  • $\begingroup$ Thank You. I am still confused on what the second derivative is proving? $\endgroup$ – TorqueNoFriction Feb 1 '14 at 23:04
  • $\begingroup$ Do you know what the Hessian of a function of several variables is? Perhaps you should read more about it: en.wikipedia.org/wiki/Hessian_matrix The partial derivatives of second order are used to compute the Hessian. If the Hessian of a function is positive definite in a convex set, then the function is strictly convex; and when know that a function is strictly convex then, if it has one minimum, that minimum is unique. $\endgroup$ – etothepitimesi Feb 1 '14 at 23:11
  • $\begingroup$ Do you need the second result? "Any local minimum of a convex function is also a global minimum" $\endgroup$ – Rodrigo May 18 '19 at 9:23

I'll start with the assumption that $C$ is symmetric, so that it has an orthonormal basis $\{ v_{1},\cdots,v_{n}\}$ of eigenvectors with corresponding eigenvalues $\{\lambda_{1},\cdots,\lambda_{n}\}$. Then, using this basis, $$ V(\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n})=a+\sum_{j=1}^{n}\beta_{j}\alpha_{j}+\frac{1}{2}\sum_{j=1}^{n}\lambda_{j}\alpha_{j}^{2}, \; \mbox{ where } \beta_{j}=b^{T}v_{j}. $$ If one eigenvalue $\lambda_{k}$ is strictly negative, then you cannot have a minimum because the linear terms become neglible in the following limit: $$ \lim_{\alpha_{k}\rightarrow\infty}V(\cdots)=-\infty .$$ So it is necessary that $C \ge 0$ in order to have a minimum. If $\lambda_{k}=0$ for some $k$, then $V$ will not have a maximum or a minimum if $\beta_{k}\ne 0$ because, in such a case, $(V-a)=\beta_{k}\alpha_{k}$ is linear in $\alpha_{k}$ while keeping the other $\alpha_{j}$ fixed at 0. If $\lambda_{k}=0$ and $\beta_{k}=0$, then you cannot have a unique minimum because varying $\alpha_{k}$ won't affect the expression for $V$ at all. So, an absolute minimum requires $C > 0$. And you can show that such an absolute minimum exists in that case because, assuming $\alpha_{j}\ne 0$ for all $j$, allows you write $$ V(\alpha_1 v_1+\cdots+\alpha_n v_n)=\sum_{j=1}^{n}\lambda_j\left(\alpha_j+\frac{\beta_{j}}{2\lambda_{j}}\right)^{2}+K, $$ where $K$ is a constant which does not depend on the $\alpha_{j}$. Clearly the above has an absolute minimum if $\lambda_{j} > 0$ for all $j$.


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