Phase 10 Exam

Convex Optimization & Numerical Methods - Phase Exam

15 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches

Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.

Q1 Easy

A set \(C\) is convex iff:

It is bounded

For any \(x,y\in C\), the segment \([x,y]\subseteq C\)

It contains the origin

It is closed

Q2 Easy

For a convex optimization problem, a local minimum is:

Possibly not global

Always also a global minimum

Never unique

A saddle point

Q3 Medium

Weak duality states that for \(\lambda\ge0\):

\(g(\lambda,\nu)\ge p^\star\)

\(g(\lambda,\nu)\le p^\star\)

\(g=p^\star\) always

\(d^\star\gt p^\star\)

Q4 Medium

For a strongly convex \(f\), gradient descent with a suitable fixed step converges:

Sublinearly

Linearly, at rate depending on \(\kappa=L/\mu\)

Quadratically

Not at all

Q5 Medium

The minimum-variance portfolio weights (fully invested) are:

\(\Sigma^{-1}\mu\)

\(\Sigma^{-1}\mathbf1/(\mathbf1^\top\Sigma^{-1}\mathbf1)\)

\(\mathbf1/n\)

\(\Sigma\mathbf1\)

Q6 Medium

The second-order condition for convexity of twice-differentiable \(f\) is:

\(\nabla f(x)=0\)

\(\nabla^2 f(x)\succeq0\) everywhere

\(\nabla^2 f(x)\preceq0\)

\(f(x)\ge0\)

Q7 Medium

The first-order optimality condition over a convex feasible set \(C\) is:

\(\nabla f_0(x^\star)=0\) always

\(\nabla f_0(x^\star)^\top(y-x^\star)\ge0\ \forall y\in C\)

\(f_0(x^\star)=0\)

\(\nabla^2 f_0\succ0\)

Q8 Medium

Which condition guarantees strong duality for a convex problem?

The objective is bounded

Slater's condition (a strictly feasible point exists)

All constraints are equalities

The Hessian is diagonal

Q9 Medium

Newton's method on a strictly convex QUADRATIC converges in:

Infinitely many steps

One step

Two steps

\(\kappa\) steps

Q10 Medium

Ridge regression differs from OLS by:

Using an \(\ell_1\) penalty

Adding \(\lambda\|\beta\|_2^2\), giving \(\hat\beta=(X^\top X+\lambda I)^{-1}X^\top y\)

Being non-convex

Removing the intercept

Q11 Medium

Which operation can DESTROY convexity?

Adding two convex functions

Taking the pointwise maximum

Multiplying two convex functions

Affine precomposition \(f(Ax+b)\)

Q12 Medium

For \(\min\tfrac12\|x\|_2^2\) s.t. \(a^\top x=1\), the optimum is:

\(a\)

\(a/\|a\|_2^2\)

\(a/\|a\|_2\)

\(0\)

Q13 Medium

Complementary slackness says:

\(\lambda_i^\star=f_i(x^\star)\)

\(\lambda_i^\star f_i(x^\star)=0\)

\(\lambda_i^\star\gt 0\) always

\(f_i(x^\star)=0\) always

Q14 Easy

A large condition number \(\kappa\) primarily makes gradient descent:

Diverge

Zig-zag and converge slowly

Converge quadratically

Ignore the gradient

Q15 Easy

For two assets with equal variances and \(\rho\lt 1\), the minimum-variance weight on asset 1 is:

\(1\)

\(1/2\)

\(0\)

Undefined

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