APM 462 Nonlinear Optimization Summer 2025 Final Project Due date: August 23, 2025; 11.59 pm All problems are to be handed in on Quercus. HW 1 (25 pts

APM 462 Nonlinear Optimization Summer 2025 Final Project Due date: August 23, 2025; 11.59 pm All problems are to be handed in on Quercus. HW 1 (25 pts) Consider the problem min f(x, y) s.t. (x, y) ∈ Ω where f(x, y) = x2 + 9y2 and Ω = { (x, y) ∈ R2 : 2x+ y ≥ 1, x+ 3y ≥ 1, x, y ≥ 0}. (a) (2 pts) Draw Ω. (b) (3 pts) Why is there a unique global minimizer for this problem? (c) (2 pts) Write this problem as a constrained optimization problem. (d) (8 pts) Find candidates for minimizers for the constrained optimization prob- lem using the KKT conditions. (e) (5 pts) Find all minimizers for the constrained optimization problem using the second order necessary and sufficient conditions. (f) (5 pts) Show that at the unique global minimizer for this problem, the first order necessary condition for unconstrained problems (using feasible directions) holds. 1 HW 2 (15 pts) Consider the problem of minimizing f(x) = 1 2 xTQx− bTx with variable x ∈ Rn, and Q ∈ Rn×n symmetric positive definite and b ∈ Rn fixed. Let {e0, . . . , en−1} be a basis of Rn (ei is not necessarily a unit vector) and define B0 := {0}, Bi = span{e0, . . . , ei−1} for i = 1, . . . , n. Fix x0 ∈ Rn and let xi be the minimizer of f over x0 +Bi for i = 1, . . . , n. (a) (5 pts) Find the unique α0 ∈ R such that x1 = x0 + α0e0. (b) (5 pts) Find the unique α′0, α ′ 1 ∈ R such that x1 = x0 + α′0e0 + α′1e1. (c) (5 pts) Under what conditions on e0 and e1 is α0 = α ′ 0? 2 HW 3 (20 pts) Consider the problem min f(x) (1) s.t. gi(x) ≥ 0 i = 1, . . . ,m where x ∈ Rn, f is convex and gi is concave for i = 1, . . . ,m. Assume that the strong duality theorem holds and that the maximum in the dual problem is attained at µ¯ = (µ¯1, . . . , µ¯m). We fix t > 0 and define a second (unconstrained problem) min rt(x) (2) s.t. x ∈ Rn where rt(x) := f(x) + t max i=1,…,m {(gi(x))−} and where for z ∈ R the function h(z) = z− := −min(z, 0) denotes the negative part of z. We can express (2) as min f(x) + ty (3) s.t. gi(x) ≥ −y i = 1, . . . ,m (x, y) ∈ Ω = Rn × R≥0. (a) (5 pts) Show that rt is a convex function. (b) (5 pts) Find the dual function for (3) and express it in terms of the dual function for (1). (c) (10 pts) Use the result from (b) to show the following: If t > µ¯1+µ¯2+· · ·+µ¯m then any minimizer of (2) is also a minimizer of (1). Note: Part of this exercise is to discuss the relationships between minimizers of (2) and (3). 3 HW 4 (20 pts) Let Ω ⊂ R2 be a ”nice” domain (connected, open, with smooth boundary) and consider the problem max J(u) s.t. u ∈ S = {u ∈ C2(Ω¯) : u(x, y) = u0(x, y) (x, y) ∈ ∂Ω} where u0 : ∂Ω→ R is a given smooth function and J(u) = ∫ Ω u(x, y)f(x, y)dxdy − ∫ Ω 1 2 (|∇u(x, y)| − 1)2 + dxdy where f : Ω→ R is a given smooth function. Here z+ := max(z, 0) denotes the positive part of z. (a) (10 pts) Find and simplify the Euler-Lagrange equation for J . Hint: You can use without proving that for z ∈ R the function g(z) = z2+ is differentiable with derivative g ′(z) = 2z+. You can also use that for v ∈ R2 the function h(v) = |v| is differentiable if v ̸= 0 with ∇h(v) = v|v| . The final equation should be in the form div(. . . ) = . . . . Note: The integration by parts in the derivation of the Euler-Lagrange equation is purely formal, as neither u nor the integrand of J are smooth enough to justify it. (b) (5 pts) Is the functional J convex or concave? Justify your answer. (c) (5 pts) Is the functional J strictly convex or strictly concave? Justify your answer. 4 HW 5 (20 pts) Consider the problem min J(y) (1) s.t. y ∈ S = {y ∈ X : y(a) = ya, y(b) = yb} where X = C2([a, b]), a, b, ya, yb ∈ R with a < b and J(y) = ∫ b a f(x, y(x), y′(x)) dx with f ∈ C2([a, b]×R×R). In this exercise we will investigate the relation- ship between the optimality conditions in the calculus of variations and the optimality conditions for finite-dimensional optimization problems. In order to approximate a solution y to (1) we discretize the interval [a, b] with a fixed step size h > 0. We define N := b− a h (assume that N ∈ N) and then we get points xi := a+ ih for i = 0, . . . , N . We define yi := y(xi) and we approximate the first derivative through the forward finite-difference y′(xi) ≈ yi+1 − yi h for i = 0, . . . , N − 1. Then we get J(y) = ∫ b a f(x, y(x), y′(x)) dx ≈ N−1∑ i=0 f ( xi, yi, yi+1 − yi h ) · h. We therefore consider the discretized minimization problem min N−1∑ i=0 f ( xi, yi, yi+1 − yi h ) · h (2) s.t. y0 = ya yN = yb with variable y = (y0, y1, . . . , yN ) ∈ RN+1. (a) (5 pts) Discretize the Euler-Lagrange equation similar to the discretization of J . Hint: You should get a system of equations. (b) (10 pts) Write down the KKT conditions for (2). How are the KKT conditions related to the discretized Euler-Lagrange equation from (a)? (c) (5 pts) Solve the discretized problem (2) for J(y) = ∫ b a (y′)2 dx. 5 HW 6 (25 pts) Consider the problem min J(y) s.t. y ∈ S = { y ∈ X : y(0) = 0, y(1) = c } where c is a given constant, X = C2([0, 1]) and J(y) = ∫ 1 0 (y′)2(1 + y′)2 dx. (a) (5 pts) Is J convex? (b) (5 pts) Show that y¯(x) = cx is an admissible extremal. (c) (5 pts) Show that y¯ is a weak local minimum if c = − 16 . (d) (5 pts) Show that y¯ is a weak local