MMiDS 3.4: Self-Assessment Quiz

Which of the following is NOT an operation that preserves the convexity of sets?

• a) Scaling a convex set by a real number.
• b) Translating a convex set by a vector.
• c) Taking the union of two convex sets.
• d) Taking the intersection of two convex sets.

Let \( f : \mathbb{R}^d \to \mathbb{R} \) be twice continuously differentiable. Which of the following conditions is sufficient for \( f \) to be convex?

• a) \( \nabla^2 f(x) \prec 0 \), for all \( x \in \mathbb{R}^d \)
• b) \( \nabla^2 f(x) \preceq 0 \), for all \( x \in \mathbb{R}^d \)
• c) \( \nabla^2 f(x) \succeq 0 \), for all \( x \in \mathbb{R}^d \)
• d) \( \nabla^2 f(x) \succ 0 \), for all \( x \in \mathbb{R}^d \)

Which of the following statements is true about convex functions?

• a) A convex function can have multiple global minima.
• b) A strictly convex function can have at most one global minimum.
• c) A convex function must be differentiable.
• d) A convex function must be twice differentiable.

Consider a convex function \( f : D \to \mathbb{R} \), where \( D \subseteq \mathbb{R}^d \) is convex. If \( x_0 \) is a local minimizer of \( f \) over \( D \), then:

• a) \( x_0 \) may not be a global minimizer of \( f \) over \( D \)
• b) \( x_0 \) is always a global minimizer of \( f \) over \( D \)
• c) \( x_0 \) is a global minimizer of \( f \) over \( D \) if and only if \( \nabla f(x_0) = 0 \)
• d) \( x_0 \) is a global minimizer of \( f \) over \( D \) if and only if \( \nabla^2 f(x_0) \succeq 0 \)

Let \( f : \mathbb{R}^d \to \mathbb{R} \) be a continuously differentiable, convex function. Which of the following is a necessary and sufficient condition for \( x_0 \) to be a global minimizer of \( f \)?

• a) \( \nabla f(x_0) \neq 0 \)
• b) \( \nabla f(x_0) = 0 \)
• c) \( \nabla^2 f(x_0) \succeq 0 \)
• d) \( \nabla^2 f(x_0) \succ 0 \)

A function \( f : \mathbb{R}^d \to \mathbb{R} \) is \( m \)-strongly convex if:

• a) \( \nabla^2 f(x) \succeq mI_{d \times d} \), for all \( x \in \mathbb{R}^d \) and some \( m > 0 \)
• b) \( \nabla^2 f(x) \preceq mI_{d \times d} \), for all \( x \in \mathbb{R}^d \) and some \( m > 0 \)
• c) \( \nabla^2 f(x) \succeq -mI_{d \times d} \), for all \( x \in \mathbb{R}^d \) and some \( m > 0 \)
• d) \( \nabla^2 f(x) \preceq -mI_{d \times d} \), for all \( x \in \mathbb{R}^d \) and some \( m > 0 \)

Let \( f : \mathbb{R}^d \to \mathbb{R} \) be twice continuously differentiable. Which of the following is equivalent to \( f \) being \( m \)-strongly convex?

• a) \( f(y) \leq f(x) + \nabla f(x)^T(y - x) + \frac{m}{2}\|y - x\|^2 \), for all \( x, y \in \mathbb{R}^d \)
• b) \( f(y) \geq f(x) + \nabla f(x)^T(y - x) + \frac{m}{2}\|y - x\|^2 \), for all \( x, y \in \mathbb{R}^d \)
• c) \( f(y) \leq f(x) + \nabla f(x)^T(y - x) - \frac{m}{2}\|y - x\|^2 \), for all \( x, y \in \mathbb{R}^d \)
• d) \( f(y) \geq f(x) + \nabla f(x)^T(y - x) - \frac{m}{2}\|y - x\|^2 \), for all \( x, y \in \mathbb{R}^d \)

Let \( D \subseteq \mathbb{R}^d \) be a nonempty, closed, convex set. For \( x \in \mathbb{R}^d \), the projection of \( x \) onto \( D \), denoted by \( \mathrm{proj}_D(x) \), satisfies:

• a) \( (\mathrm{proj}_D(x) - x)^T(y - \mathrm{proj}_D(x)) \leq 0 \), for all \( y \in D \)
• b) \( (\mathrm{proj}_D(x) - x)^T(y - \mathrm{proj}_D(x)) = 0 \), for all \( y \in D \)
• c) \( (\mathrm{proj}_D(x) - x)^T(y - \mathrm{proj}_D(x)) \geq 0 \), for all \( y \in D \)
• d) None of the above

Which of the following statements is true about the least-squares objective function \( f(x) = \|Ax - b\|_2^2 \), where \( A \in \mathbb{R}^{n \times d} \) has full column rank and \( b \in \mathbb{R}^n \)?

• a) \( f(x) \) is convex but not necessarily strongly convex.
• b) \( f(x) \) is strongly convex.
• c) \( f(x) \) is convex if and only if \( b = 0 \).
• d) \( f(x) \) is strongly convex if and only if \( b = 0 \).