24 KiB
24 KiB
title: Midterm -- Optimization Methods author: Claudio Maggioni header-includes:
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Acknowledgements on group work
- Gianmarco De Vita suggested me the use of MATLAB's equation solver for parts
of
dogleg.m's implementation. - I have discussed my solutions for exercise 1.2 and exercise 3 with several
people, namely:
- Gianmarco De Vita
- Tommaso Rodolfo Masera
- Andrea Brites Marto
- This song for the divine inspiration that made me write the proof for Exercise 1.2.
Exercise 1
Point 1
Question (a)
As already covered in the course, the gradient of a standard quadratic form at a
point x_0 is equal to:
\nabla f(x_0) = A x_0 - b Plugging in the definition of x_0 and knowing that $\nabla f(x_m) = A x_m - b
= 0$ (according to the first necessary condition for a minimizer), we obtain:
\nabla f(x_0) = A (x_m + v) - b = A x_m + A v - b = b + \lambda v - b =
\lambda v $$
### Question (b)
The steepest descent method takes exactly one iteration to reach the exact
minimizer $x_m$ starting from the point $x_0$. This can be proven by first
noticing that $x_m$ is a point standing in the line that first descent direction
would trace, which is equal to:
$$g(\alpha) = - \alpha \cdot \nabla f(x_0) = - \alpha \lambda v$$
For $\alpha = \frac{1}{\lambda}$, and plugging in the definition of $x_0 = x_m +
v$, we would reach a new iterate $x_1$ equal to:
$$x_1 = x_0 - \alpha \lambda v = x_0 - v = x_m + v - v = x_m $$
The only question that we need to answer now is why the SD algorithm would
indeed choose $\alpha = \frac{1}{\lambda}$. To answer this, we recall that the
SD algorithm chooses $\alpha$ by solving a linear minimization option along the
step direction. Since we know $x_m$ is indeed the minimizer, $f(x_m)$ would be
obviously strictly less that any other $f(x_1 = x_0 - \alpha \lambda v)$ with
$\alpha \neq \frac{1}{\lambda}$.
Therefore, since $x_1 = x_m$, we have proven SD
converges to the minimizer in one iteration.
## Point 2
The right answer is choice (a), since the energy norm of the error indeed always
decreases monotonically.
The proof of this that I will provide is independent from the provided objective
or the provided number of iterations, and it works for all choices of $A$ where
$A$ is symmetric and positive definite.
Therefore, first of all I will prove $A$ is indeed SPD by computing its
eigenvalues.
$$CP(A) = det\left(\begin{bmatrix}2&-1&0\\-1&2&-1\\0&-1&2\end{bmatrix} - \lambda I \right) =
det\left(\begin{bmatrix}2 - \lambda&-1&0\\-1&2 -
\lambda&-1\\0&-1&2-\lambda\end{bmatrix} \right) = -\lambda^3 + 6
\lambda^2 - 10\lambda + 4$$
$$CP(A) = 0 \Leftrightarrow \lambda = 2 \lor \lambda = 2 \pm \sqrt{2}$$
Therefore we have 3 eigenvalues and they are all positive, so A is positive
definite and it is clearly symmetric as well.
Now we switch to the general proof for the monotonicity.
To prove that this is true, we first consider a way to express any iterate $x_k$
in function of the minimizer $x_s$ and of the missing iterations:
$$x_k = x_s + \sum_{i=k}^{N} \alpha_i A^i p_0$$
This formula makes use of the fact that step directions in CG are all
A-orthogonal with each other, so the k-th search direction $p_k$ is equal to
$A^k p_0$, where $p_0 = -r_0$ and $r_0$ is the first residual.
Given that definition of iterates, we're able to express the error after
iteration $k$ $e_k$ in a similar fashion:
$$e_k = x_k - x_s = \sum_{i=k}^{N} \alpha_i A^i p_0$$
We then recall the definition of energy norm $\|e_k\|_A$:
$$\|e_k\|_A = \sqrt{\langle Ae_k, e_k \rangle}$$
We then want to show that $\|e_k\|_A = \|x_k - x_s\|_A > \|e_{k+1}\|_A$, which
in turn is equivalent to claim that:
$$\langle Ae_k, e_k \rangle > \langle Ae_{k+1}, e_{k+1} \rangle$$
Knowing that the dot product is linear w.r.t. either of its arguments, we pull
out the sum term related to the k-th step (i.e. the first term in the sum that
makes up $e_k$) from both sides of $\langle Ae_k, e_k \rangle$,
obtaining the following:
$$\langle Ae_{k+1}, e_{k+1} \rangle + \langle \alpha_k A^{k+1} p_0, e_k \rangle
+ \langle Ae_{k+1},\alpha_k A^k p_0 \rangle > \langle Ae_{k+1}, e_{k+1}
\rangle$$
which in turn is equivalent to claim that:
$$\langle \alpha_k A^{k+1} p_0, e_k \rangle
+ \langle Ae_{k+1},\alpha_k A^k p_0 \rangle > 0$$
From this expression we can collect term $\alpha_k$ thanks to linearity of the
dot-product:
$$\alpha_k (\langle A^{k+1} p_0, e_k \rangle
+ \langle Ae_{k+1}, A^k p_0 \rangle) > 0$$
and we can further "ignore" the $\alpha_k$ term since we know that all
$\alpha_i$s are positive by definition:
$$\langle A^{k+1} p_0, e_k \rangle
+ \langle Ae_{k+1}, A^k p_0 \rangle > 0$$
Then, we convert the dot-products in their equivalent vector to vector product
form, and we plug in the definitions of $e_k$ and $e_{k+1}$:
$$p_0^T (A^{k+1})^T (\sum_{i=k}^{N} \alpha_i A^i p_0) +
p_0^T (A^{k})^T (\sum_{i=k+1}^{N} \alpha_i A^i p_0) > 0$$
We then pull out the sum to cover all terms thanks to associativity of vector
products:
$$\sum_{i=k}^N (p_0^T (A^{k+1})^T A^i p_0) \alpha_i+ \sum_{i=k+1}^N
(p_0^T (A^{k})^T A^i p_0) \alpha_i > 0$$
We then, as before, can "ignore" all $\alpha_i$ terms since we know by
definition that
they are all strictly positive. We then recalled that we assumed that A is
symmetric, so $A^T = A$. In the end we have to show that these two
inequalities are true:
$$p_0^T A^{k+1+i} p_0 > 0 \; \forall i \in [k,N]$$
$$p_0^T A^{k+i} p_0 > 0 \; \forall i \in [k+1,N]$$
To show these inequalities are indeed true, we recall that A is symmetric and
positive definite. We then consider that if a matrix A is SPD, then $A^i$ for
any positive $i$ is also SPD[^1]. Therefore, both inequalities are trivially
true due to the definition of positive definite matrices.
[^1]: source: [Wikipedia - Definite Matrix $\to$ Properties $\to$
Multiplication](
https://en.wikipedia.org/wiki/Definite_matrix#Multiplication)
Thanks to this we have indeed proven that the delta $\|e_k\|_A - \|e_{k+1}\|_A$
is indeed positive and thus as $i$ increases the energy norm of the error
monotonically decreases.
# Question 2
## Point 1
### (a) For which kind of minimization problems can the trust region method be used? What are the assumptions on the objective function?
The trust region method is an algorithm that can be used for unconstrained
minimization. The trust region method uses parts of the gradient descent and
Newton methods, and thus it accepts basically the same domain of objectives
that these two methods accept.
These constraints namely require the objective function $f(x)$ to be twice
differentiable in order to make building a quadratic model around an arbitrary
point possible. In addition, our assumptions w.r.t. the scope of this course
require that $f(x)$ should be continuous up to the second derivatives.
This is needed to allow the Hessian to be symmetric (as by the Schwartz theorem)
which is an assumption that simplifies significantly proofs related to the
method (like namely Exercise 3 in this assignment).
Finally, as all the other unconstrained minimization methods we covered in this
course, the trust region method is only able to find a local minimizer close to
he chosen starting points, and by no means the computed minimizer is guaranteed
to be a global minimizer.
### (b) Write down the quadratic model around a current iterate xk and explain the meaning of each term.
$$m(p) = f + g^T p + \frac12 p^T B p \;\; \text{ s.t. } \|p\| \leq \Delta$$
Here's an explaination of the meaning of each term:
- $\Delta$ is the trust region radius, i.e. an upper bound on the step's norm
(length);
- $f$ is the energy function value at the current iterate, i.e. $f(x_k)$;
- $p$ is the trust region step, the solution of $\arg\min_p m(p)$ with $\|p\| <
\Delta$, i.e. the optimal step to take;
- $g$ is the gradient at the current iterate $x_k$, i.e. $\nabla f(x_k)$;
- $B$ is the hessian at the current iterate $x_k$, i.e. $\nabla^2 f(x_k)$.
### (c) What is the role of the trust region radius?
The role of the trust region radius is to put an upper bound on the step length
in order to avoid "overly ambitious" steps, i.e. steps where the the step length
is considerably long and the quadratic model of the objective is low-quality
(i.e. the performance measure $\rho_k$ in the TR algorithm indicates significant
energy difference between the true objective and the quadratic model).
In layman's terms, the trust region radius makes the method switch more gradient
based or more quadratic based steps w.r.t. the "confidence" (measured in terms
of $\rho_k$) in the computed
quadratic model.
### (d) Explain Cauchy point, sufficient decrease and Dogleg method, and the connection between them.
The Cauchy point and Dogleg method are algorithms to compute iteration steps
that are in the bounds of the trust region. They allow to provide an approximate
solution to the minimization of the quadratic model inside the TR radius.
The Cauchy point is a method providing sufficient decrease (as per the Wolfe
conditions) by essentially performing a gradient descent step with a
particularly chosen step size limited by the TR radius. However, since this
method basically does not exploit the quadratic component of the objective model
(the hessian is only used as a term in the step length calculation), even if it
provides sufficient decrease and consequentially convergence it is rarely used
as a standalone method to compute iteration steps.
The Cauchy point is therefore often integrated in another method called Dogleg,
which uses the former algorithm in conjunction with a purely Newton step to
provided a steps obtained by a blend of linear and quadratic information.
This blend is achieved by choosing the new iterate by searching along a path
made out of two segments, namely the gradient descent step with optimal step
size and a segment pointing from the last point to the pure netwon step. The
peculiar angle between this two segments is the reason the method is nicknamed
"Dogleg", since the final line resembles a dog's leg.
In the Dogleg method, the Cauchy point is used in case the trust region is small
enough not to allow the "turn" on the second segment towards the Netwon step.
Thanks to this property and the use of the performance measure $\rho_k$ to grow
and shrink the TR radius, the Dogleg method performs well even with inaccurate
quadratic models. Therefore, it still satisfies sufficient decrease and the
Wolfe conditions while delivering superlinear convergence, compared to the
purely linear convergence of Cauchy point steps.
### (e) Write down the trust region ratio and explain its meaning.
$$\rho_k = \frac{f(x_k) - f(x_k + p_k)}{m_k(0) - m_k(p_k)}$$
The trust region ratio and performance measure $\rho_k$ measures the quality of
the quadratic model built around
the current iterate $x_k$, by measuring the ratio between the energy difference
between the old and the new iterate according to the real energy function w.r.t. the quadratic model around $x_k$.
The ratio is used to test the adequacy of the current trust region radius. For
an inaccurate quadratic model, the predicted energy decrease would be
considerably higher than the effective one and thus the ratio would be low. When
the ratio is lower than a predetermined threshold ($\frac14$ is the one chosen
by Nocedal) the trust region radius is divided by 4. Instead, a very accurate
quadratic model would result in little difference with the real energy function
and thus the ratio would be close to $1$. If the trust region radius is higher
than a certain predetermined threshold ($\frac34$ is the one chosen by Nocedal),
then the trust region radius is doubled in order to allow for longer steps,
since the model quality is good.
### (f) Does the energy decrease monotonically when Trust Region method is employed? Justify your answer.
When using the trust region method, the energy of the iterates decreases
monotonically. This is true because by construction the algorithm either makes
the next iterate equal to the current one (i.e. when the performance measure
$\rho_k$ is too poor to accept a step) or applies a linear, quadratic, or
blended descending step to the current iterate.
When a step is taken, the step
by definition should be a solution (or a close approximation of such solution)
of the energy minimization problem inside the trust region itself. Therefore,
the step cannot lead to a point that has higher energy than the one from the
current iterate.
Therefore, the energy either stays constant or decreases at every single
iteration, and therefore the energy decreases monotonically.
## Point 2
The trust region algorithm is the following:
\begin{algorithm}[H]
\SetAlgoLined
Given $\hat{\Delta} > 0, \Delta_0 \in (0,\hat{\Delta})$,
and $\eta \in [0, \frac14)$\;
\For{$k = 0, 1, 2, \ldots$}{%
Obtain $p_k$ by using Cauchy or Dogleg method\;
$\rho_k \gets \frac{f(x_k) - f(x_k + p_k)}{m_k(0) - m_k(p_k)}$\;
\uIf{$\rho_k < \frac14$}{%
$\Delta_{k+1} \gets \frac14 \Delta_k$\;
}\Else{%
\uIf{$\rho_k > \frac34$ and $\|\rho_k\| = \Delta_k$}{%
$\Delta_{k+1} \gets \min(2\Delta_k, \hat{\Delta})$\;
}
\Else{%
$\Delta_{k+1} \gets \Delta_k$\;
}}
\uIf{$\rho_k > \eta$}{%
$x_{k+1} \gets x_k + p_k$\;
}
\Else{
$x_{k+1} \gets x_k$\;
}
}
\caption{Trust region method}
\end{algorithm}
The Cauchy point algorithm is the following:
\begin{algorithm}[H]
\SetAlgoLined
Input $B$ (quadratic term), $g$ (linear term), $\Delta_k$\;
\uIf{$g^T B g \geq 0$}{%
$\tau \gets 1$\;
}\Else{%
$\tau \gets \min(\frac{\|g\|^3}{\Delta_k \cdot g^T B g}, 1)$\;
}
$p_k \gets -\tau \cdot \frac{\Delta_k}{\|g\|^2 \cdot g}$\;
\Return{$p_k$}
\caption{Cauchy point}
\end{algorithm}
Finally, the Dogleg method algorithm is the following:
\begin{algorithm}[H]
\SetAlgoLined
Input $B$ (quadratic term), $g$ (linear term), $\Delta_k$\;
$p_N \gets - B^{-1} g$\;
\uIf{$\|p_N\| < \Delta_k$}{%
$p_k \gets p_N$\;
}\Else{%
$p_u = - \frac{g^T g}{g^T B g} g$\;
\uIf{$\|p_u\| > \Delta_k$}{%
compute $p_k$ with Cauchy point algorithm\;
}\Else{%
solve for $\tau$ the equality $\|p_u + \tau * (p_N - p_u)\|^2 =
\Delta_k^2$\;
$p_k \gets p_u + \tau \cdot (p_N - p_u)$\;
}
}
\caption{Dogleg method}
\end{algorithm}
## Point 3
The trust region, dogleg and Cauchy point algorithms were implemented
respectively in the files `trust_region.m`, `dogleg.m`, and `cauchy.m`.
## Point 4
### Taylor expansion
We first compute the gradient and the hessian of the energy function:
$$\nabla f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix}
\frac{d f(x)}{d x_1} \\ \frac{d f(x)}{d x_2} \end{bmatrix} =
\begin{bmatrix}48x_1^3 - 16x_1x_2 + 2x_1 - 2\\2x_2 - 8x_1^2\end{bmatrix}$$
$$ \nabla^2 f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix}
\frac{d^2 f(x)}{d x_1^2} & \frac{d^2 f(x)}{dx_2 x_1} \\ \frac{d^2 f(x)}{dx_1
x_2} & \frac{d^2 f(x)}{d x_2^2}\end{bmatrix} = \begin{bmatrix}144x_1^2 -16x_2 +
2 - 16 & -16 \\ -16 & 2 \end{bmatrix}$$
The Taylor expansion up the second order of the function is the following:
$$f(x_0, w) = f(x_0) + \langle \nabla f(x_0), w\rangle + \frac12 \langle
\nabla^2 f(x_0) w, w\rangle$$
### Minimization
The code used to minimize the function can be found in the MATLAB script
`main.m` under section 2.4. The resulting minimizer (found in 10 iterations) is:
$$x_m = \begin{bmatrix}1\\4\end{bmatrix}$$
### Energy landscape
The following figure shows a `surf` plot of the objective function overlayed
with the iterates used to reach the minimizer:
The code used to generate such plot can be found in the MATLAB script `main.m`
under section 2.4c.
## Point 5
### Minimization
The code used to minimize the function can be found in the MATLAB script
`main.m` under section 2.5. The resulting minimizer (found in 25 iterations) is:
$$x_m = \begin{bmatrix}1\\5\end{bmatrix}$$
### Energy landscape
The following figure shows a `surf` plot of the objective function overlayed
with the iterates used to reach the minimizer:
The code used to generate such plot can be found in the MATLAB script `main.m`
under section 2.5b.
### Gradient norms
The following figure shows the logarithm of the norm of the gradient w.r.t.
iterations:
The code used to generate such plot can be found in the MATLAB script `main.m`
under section 2.5c.
Comparing the behaviour shown above with the figures obtained in the previous
assignment for the Newton method with backtracking and the gradient descent with
backtracking, we notice that the trust-region method really behaves like a
compromise between the two methods. First of all, we notice that TR converges in
25 iterations, almost double of the number of iterations of regular NM +
backtracking. The actual behaviour of the curve is somewhat similar to the
Netwon gradient norms curve w.r.t. to the presence of spikes, which however are
less evident in the Trust region curve (probably due to Trust region method
alternating quadratic steps with linear or almost linear steps while iterating).
Finally, we notice that TR is the only method to have neighbouring iterations
having the exact same norm: this is probably due to some proposed iterations
steps not being validated by the acceptance criteria, which makes the method mot
move for some iterations.
# Exercise 3
We first show that the lemma holds for $\tau \in [0,1]$. Since
$$\|\tilde{p}(\tau)\| = \|\tau p^U\| = \tau \|p^U\| \text{ for } \tau \in [0,1]$$
Then the norm of the step $\tilde{p}$ clearly increases as $\tau$
increases. For the second criterion, we compute the quadratic model for a
generic $\tau \in [0,1]$:
$$m(\tilde{p}(\tau)) = f + g^T * \tilde{p}(\tau) + \frac{1}{2}
\tilde{p}(\tau)^T B \tilde{p}(\tau) = f + g^T \tau p^U + \frac12
\tau (p^U)^T B \tau p^U $$
We then recall the definition of $p^U$:
$$ p^U = -\frac{g^Tg}{g^TBg}g: $$
and plug it into the expression:
$$ = f + g^T \tau \cdot -
\frac{g^Tg}{g^TBg}g + \frac{1}{2} \tau \cdot -\left(\frac{g^Tg}{g^TBg}g\right)^T
\cdot B \tau \cdot - \frac{g^Tg}{g^TBg}g $$$$ = f - \tau \cdot \frac{\| g
\|^4}{g^TBg} + \frac{1}{2} \tau^2 \cdot \left(\frac{\|g\|^2}{g^T B g}\right) g^T \cdot B
\frac{g^T g}{g^TBg}g $$$$ = f - \tau \cdot \frac{\| g \|^4}{g^TBg} +
\frac{1}{2} \tau^2 \cdot \frac{\| g \|^4}{(g^TBg)^2} \cdot g^TBg $$$$ = f -
\tau \cdot \frac{\| g \|^4}{g^TBg} + \frac{1}{2} \tau^2 \cdot \frac{\| g
\|^4}{g^TBg} $$$$ = f + \left(\frac{1}{2} \tau^2 - \tau\right) \cdot \frac{\| g
\|^4}{g^TBg}$$$$= f + \left(\frac12\tau^2 - \tau\right) z \; \text{where } z =
\frac{\| g \|^4}{g^TBg}$$
We then compute the derivative of the model:
$$\frac{dm(\tilde{p}(\tau))}{d\tau} = z\tau - z = z(\tau - 1)$$
We know $z$ is positive since $g^T B g > 0$, since we assume $B$ is positive
definite. Then, since $\tau \in [0,1]$, the derivative is always $\leq 0$ and
therefore we have proven that the quadratic model of the Dogleg step decreases
as $\tau$ increases.
Now we show that the two claims on gradients hold also for $\tau \in [1,2]$. We
define a function $h(\alpha)$ (where $\alpha = \tau - 1$) with same gradient
"sign"
as $\|\tilde{p}(\tau)\|$ and we show that this function increases:
$$h(\alpha) = \frac12 \|\tilde{p}(1 + \alpha)\|^2 = \frac12 \|p^U + \alpha(p^B -
p^U)\|^2 = \frac12 \|p^U\|^2 + \frac12 \alpha^2 \|p^B - p^U\|^2 + \alpha (p^U)^T
(p^B - p^U)$$
We now take the derivative of $h(\alpha)$ and we show it is always positive,
i.e. that $h(\alpha)$ has always positive gradient and thus that it is
increasing w.r.t. $\alpha$:
$$h'(\alpha) = \alpha \|p^B - p^U\|^2 + (p^U)^T (p^B - p^U) \geq (p^U)^T (p^B -
p^U) = \frac{g^Tg}{g^TBg}g^T\left(- \frac{g^Tg}{g^TBg}g + B^{-1}g\right) =$$$$=
\|g\|^2 \frac{g^TB^{-1}g}{g^TBg}\left(1 -
\frac{\|g\|^2}{(g^TBg)(g^TB^{-1}g)}\right) $$
Since we know $B$ is symmetric and positive definite, then $B^{-1}$ is as well.
Therefore, we know that the term outside of the parenthesis is always positive
or 0. Therefore, we now only need to show that:
$$\frac{\|g\|^2}{(g^TBg)(g^TB^{-1}g)} \leq 1 \Leftrightarrow \|g\|^2 \leq
(g^TBg)(g^TB^{-1}g)$$
since both factors in the denominator are positive for what we shown before.
We now define a inner product space $\forall a, b \in R^N, \; {\langle a,
b\rangle}_B = a^T B b$. We now prove that this is indeed a linear product space
by proving all properties of such space:
- **Linearity w.r.t. the first argument:**
$\alpha {\langle x, y \rangle}_B + \beta {\langle z,
y \rangle}_B = \alpha \cdot x^TBy + \beta \cdot z^TBy = (\alpha x + \beta z)^TBy
= {\langle (\alpha x + \beta z), y \rangle}_B$;
- **Symmetry:**
${\langle x, y \rangle}_B = x^T B y = (x^T B y)^T = y^TB^Tx$, and
since $B$ is symmetric, $y^TB^Tx = y^TBx = {\langle y,x \rangle}_B$;
- **Positive definiteness:**
${\langle x, x \rangle_B} = x^T B x > 0$ is true since B is positive definite
for all $x \neq 0$.
Since ${\langle x, y \rangle}_B$ is indeed a linear product space, then:
$${\langle g, B^{-1} g \rangle}_B \leq {\langle g, g \rangle}_B {\langle B^{-1}
g, B^{-1} g \rangle}_B$$
holds according to the Cauchy-Schwartz inequality. Now, if we expand each inner
product we obtain:
$$g^T B B^{-1} g \leq (g^T B g) (g^T (B^{-1})^T B B^{-1} g)$$
Which, since $B$ is symmetric, in turn is equivalent to writing:
$$g^T g \leq (g^TBg) (g^T B^{-1} g)$$
which is what we needed to show to prove that the first gradient constraint
holds for $\tau \in [1,2]$.
For the second constraint, we adopt a similar strategy as for before and we
define a new function $\hat{h}(\alpha) = m(\tilde{p}(1 + \alpha))$, thus
plugging the Dogleg step in the quadratic model:
$$\hat{h}(\alpha) = m(\tilde{p}(1+\alpha)) = f + g^T (p^U + \alpha (p^B - p^U)) +
\frac12 (p^U + \alpha (p^B - p^U))^T B (p^U + \alpha (p^B - p^U)) = $$$$ =
f + g^T p^U + \alpha g^T (p^B - p^U) + \frac12 (p^U)^T B p^U + \frac12 \alpha (p^U)^T B
(p^B - p^U) + \frac12 \alpha (p^B - p^U)^T B p^U + \frac12 \alpha^2
(p^B - p^U)^T B (p^B - p^U)$$
We now derive $\hat{h}(\alpha)$:
$$\hat{h}'(\alpha) = g^T (p^B - p^U) + \frac12 (p^U)^T B (p^B - p^U) + \frac12
(p^B - p^U)^T B p^U + \alpha (p^B - p^U)^T B (p^B - p^U) = $$$$
= (p^B - p^U)^T g + \frac12 ((p^U)^T B (p^B - p^U))^T +
\frac12 (p^B - p^U)^T B p^U + \alpha (p^B - p^U)^T B (p^B - p^U) = $$$$
= (p^B - p^U)^T g + \frac12 (p^B - p^U) B^T (p^U)^T +
\frac12 (p^B - p^U)^T B p^U + \alpha (p^B - p^U)^T B (p^B - p^U) = $$$$
= (p^B - p^U)^T (g + \frac12 \cdot 2 \cdot B p^U) + \alpha (p^B - p^U)^T B
(p^B - p^U) \leq $$$$
\leq (p^B - p^U)^T(g + B p^U + B (p^B - p^U)) = $$$$
=(p^B - p^U)^T(g+Bp^B) = $$$$
=(p^B - p^U)^T(g+B \cdot (-1) \cdot B^{-1} g) = $$$$
=(p^B - p^U)^T(g - g) = 0$$
and we therefore obtain $\hat{h}(\alpha) \leq 0$, thus finding that the
$m(\tilde{p})$ is indeed a decreasing function of $\tau$ or $\alpha = \tau - 1$
also for $\tau \in [1,2]$, thus completing the proof for the lemma.
<!--
$$\hat{h}'(\alpha) = g^T (p^B - p^U) + \frac12 (p^U)^T B (p^B - p^U) + \frac12
(p^B - p^U)^T B p^U + \alpha (p^B - p^U)^T B (p^B - p^U) = $$$$
= (p^B - p^U)^T g + \frac12 ((p^U)^T B (p^B - p^U))^T +
\frac12 (p^B - p^U)^T B p^U + \alpha (p^B - p^U)^T B (p^B - p^U) = $$$$
= (p^B - p^U)^T g + \frac12 (p^B - p^U) B^T (p^U)^T +
\frac12 (p^B - p^U)^T B p^U + \alpha (p^B - p^U)^T B (p^B - p^U) = $$$$
= (p^B - p^U)^T (g + \frac12 \cdot 2 \cdot B p^U) + \alpha (p^B - p^U)^T B
(p^B - p^U) \leq $$$$
\leq (p^B - p^U)^T(g + B p^U + B (p^B - p^U)) = $$$$
=(p^B - p^U)T(g+Bp^B) = 0$$
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