Optimisation

Maximise $2x + y$ subject to $x + y \le 4$, $x \le 3$, $x, y \ge 0$. What is the optimal value?

Which of these functions $\mathbb{R} \to \mathbb{R}$ are convex? (Choose all that apply.)

In the workshop problem ($\max 3x + 5y$ subject to $x \le 4$, $2y \le 12$, $3x + 2y \le 18$, $x, y \ge 0$) the optimal dual solution is $y = (0, \tfrac32, 1)$. By how much does the maximum profit increase if workshop C gets one extra hour (so $3x + 2y \le 19$)?

In the proof of weak duality, $c^{\mathsf T}x \le (A^{\mathsf T}y)^{\mathsf T}x = y^{\mathsf T}(Ax) \le y^{\mathsf T}b$. Which hypotheses justify the last inequality $y^{\mathsf T}(Ax) \le y^{\mathsf T}b$?

Minimise $(x - 3)^2 + (y - 1)^2$ subject to $x + y \le 2$. Which point is the minimiser?

The Lean theorem local_min_convex_global assumes hf : ConvexOn ℝ Set.univ f and hx : IsLocalMin f x and concludes f x ≤ f y. Which function and point show that the convexity hypothesis hf cannot be dropped?

Gradient descent with step $1/L$ is run on a convex $L$-smooth function with $L = 4$, starting at distance $\|x_0 - x^*\| = 3$ from a minimiser. Using Theorem 9, what is the smallest number of steps $k$ that guarantees $f(x_k) - f(x^*) \le 0.01$?

Write gradient_descent(grad, x0, step, iters) that applies $x \mapsto x - \text{step} \cdot \text{grad}(x)$ to a number x0 iters times and returns the final value. Then use it on $f(x) = 2(x - 3)^2$, whose gradient is $4(x - 3)$ (so $L = 4$), and observe that steps below $2/L = 0.5$ converge while larger steps diverge.