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) \( \mathbf{H}_f(\mathbf{x}) \prec 0 \), for all \( \mathbf{x} \in \mathbb{R}^d \)
• b) \( \mathbf{H}_f(\mathbf{x}) \preceq 0 \), for all \( \mathbf{x} \in \mathbb{R}^d \)
• c) \( \mathbf{H}_f(\mathbf{x}) \succeq 0 \), for all \( \mathbf{x} \in \mathbb{R}^d \)
• d) \( \mathbf{H}_f(\mathbf{x}) \succ 0 \), for all \( \mathbf{x} \in \mathbb{R}^d \)

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 \( \mathbf{x}_0 \) is a local minimizer of \( f \) over \( D \), then:

• a) \( \mathbf{x}_0 \) may not be a global minimizer of \( f \) over \( D \)
• b) \( \mathbf{x}_0 \) is always a global minimizer of \( f \) over \( D \)
• c) \( \mathbf{x}_0 \) is a global minimizer of \( f \) over \( D \) if and only if \( \nabla f(\mathbf{x}_0) = \mathbf{0} \)
• d) \( \mathbf{x}_0 \) is a global minimizer of \( f \) over \( D \) if and only if \( \mathbf{H}_f(\mathbf{x}_0) \succeq 0 \)

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 \( \mathbf{x}_0 \) to be a global minimizer of \( f \)?

• a) \( \nabla f(\mathbf{x}_0) \neq \mathbf{0} \)
• b) \( \nabla f(\mathbf{x}_0) = \mathbf{0} \)
• c) \( \mathbf{H}_f(\mathbf{x}_0) \succeq 0 \)
• d) \( \mathbf{H}_f(\mathbf{x}_0) \succ 0 \)

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

• a) \( \mathbf{H}_f(\mathbf{x}) \succeq m I_{d \times d} \), for all \( \mathbf{x} \in \mathbb{R}^d \) and some \( m > 0 \)
• b) \( \mathbf{H}_f(\mathbf{x}) \preceq m I_{d \times d} \), for all \( \mathbf{x} \in \mathbb{R}^d \) and some \( m > 0 \)
• c) \( \mathbf{H}_f(\mathbf{x}) \succeq -m I_{d \times d} \), for all \( \mathbf{x} \in \mathbb{R}^d \) and some \( m > 0 \)
• d) \( \mathbf{H}_f(\mathbf{x}) \preceq -m I_{d \times d} \), for all \( \mathbf{x} \in \mathbb{R}^d \) 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) \( f(\mathbf{y}) \leq f(\mathbf{x}) + \nabla f(\mathbf{x})^T(\mathbf{y} - \mathbf{x}) + \frac{m}{2}\|\mathbf{y} - \mathbf{x}\|^2 \), for all \( \mathbf{x}, \mathbf{y} \in \mathbb{R}^d \)
• b) \( f(\mathbf{y}) \geq f(\mathbf{x}) + \nabla f(\mathbf{x})^T(\mathbf{y} - \mathbf{x}) + \frac{m}{2}\|\mathbf{y} - \mathbf{x}\|^2 \), for all \( \mathbf{x}, \mathbf{y} \in \mathbb{R}^d \)
• c) \( f(\mathbf{y}) \leq f(\mathbf{x}) + \nabla f(\mathbf{x})^T(\mathbf{y} - \mathbf{x}) - \frac{m}{2}\|\mathbf{y} - \mathbf{x}\|^2 \), for all \( \mathbf{x}, \mathbf{y} \in \mathbb{R}^d \)
• d) \( f(\mathbf{y}) \geq f(\mathbf{x}) + \nabla f(\mathbf{x})^T(\mathbf{y} - \mathbf{x}) - \frac{m}{2}\|\mathbf{y} - \mathbf{x}\|^2 \), for all \( \mathbf{x}, \mathbf{y} \in \mathbb{R}^d \)

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

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

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

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