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Chapter 13 practice bank

100 questions:
Optimization

Solve on paper first. Reveal only the final answer when you want to check yourself. There are no worked solutions in this bank by design.

34 easy 33 medium 33 advanced

EasyObjectivesoptimization-001

Find the minimiser and minimum of f(x)=(x−1)².

EasyObjective valuesoptimization-002

Evaluate f(x)=(x−2)² at x=4.

EasyGradient descentoptimization-003

For f=(x−4)², take one gradient step from x₀=2 with η=1/4.

EasyDescent directionsoptimization-004

If ∇f=[4,-4], give the unscaled steepest-descent direction.

EasyStationary pointsoptimization-005

Find the stationary point of f(x)=5x².

EasyConvexityoptimization-006

Is f(x,y)=6x²+6y² convex?

EasyFeasibilityoptimization-007

Is x=7 feasible for the constraint x≤8?

EasyProjectionoptimization-008

Project x=5 onto the interval [0,2].

EasyMini-batchesoptimization-009

A two-example batch has scalar gradients 4 and 5. What is the batch gradient?

EasyMomentumoptimization-010

With β=1/2, old velocity 6, and gradient 4, compute v_new=βv+g.