مسابقة دكتوراه 2026 — المدرسة الوطنية العليا للرياضيات

I- Consider the problem of minimizing a general quadratic function subject to a linear constraint:

$$\text{minimize} \quad \frac{1}{2}x^\top Qx - c^\top x + d$$ $$\text{subject to} \quad Ax=b$$

where $Q=Q^\top>0\ (PD)$, $A\in\mathbb{R}^{m\times n}$, $m<n$, $\operatorname{rank}(A)=m$, $b\in\mathbb{R}^m$, $c\in\mathbb{R}^n$, and $d$ is a real constant.

i) Is this optimization problem convex?

ii) Prove that this problem admits a unique global minimizer.

iii) Show that any feasible point is regular.

iv) Derive a closed-form solution to this problem.

II- Consider the following optimization problem:

$$\text{Minimize} \quad x_1^2+2x_1x_2+3x_2^2+2x_3^2+4x_1+5x_2+6x_3$$ $$\text{subject to} \quad x_1+2x_2=3$$ $$4x_1+5x_3=6$$

a) Show that this problem has the same form as the problem in Part I. Verify whether all conditions in Part I are satisfied.

b) Solve this optimization problem using the results from Part I.

Hint: If

$$Q= \begin{bmatrix} 2&2&0\\ 2&6&0\\ 0&0&4 \end{bmatrix}, \qquad A= \begin{bmatrix} 1&2&0\\ 4&0&5 \end{bmatrix},$$

then

$$Q^{-1} = \frac14 \begin{bmatrix} 3&-1&0\\ -1&1&0\\ 0&0&1 \end{bmatrix}, \qquad B=AQ^{-1}A^\top = \frac14 \begin{bmatrix} 3&4\\ 4&73 \end{bmatrix}.$$

Exercice 2.

For each of the following statements, indicate whether it is true or false, and provide a justification for your answer.

1. If $G_1$ and $G_2$ are finite cyclic groups, then the product group $G_1 \times G_2$ is cyclic.

2. The fraction field of the Gaussian ring $\mathbb{Z}[i]$ is $\mathbb{C}$.

3. All subrings of $\mathbb{Z} \times \mathbb{Z}$ are ideals of $\mathbb{Z} \times \mathbb{Z}$.

4. Let $H_1$ and $H_2$ be subgroups of a group $G$. If $G = H_1 \cup H_2$, then $G = H_1$ or $G = H_2$.

5. Let $E$ be a $\mathbb{K}$-vector space and $f$ is an endomorphism of $E$. Then $\lambda$ is an eigenvalue of $f$ if and only if the endomorphism $(f - \lambda\, \mathrm{id}_E)$ is not injective.

Exercice 3.

Let $\mathbb{K}$ be a field and $E$ a $\mathbb{K}$-vector space of finite dimension $n$ $(n \geq 2)$. Let $f$ be an endomorphism of $E$ having $n$ pairwise distinct eigenvalues.

1. Show that there exists $x \in E$ for which $(x, f(x), f^2(x), \ldots, f^{n-1}(x))$ is a basis of $E$.

2. Determine the matrix associated with $f$ with respect to this basis.