Optimization: certify decisions and audit algorithms
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 210 minutes, as an unvalidated planning estimate. This collection contains 11 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the mixed problems before revealing guidance. State definitions, hypotheses and intermediate arguments, and identify any theorem you use without proving it.
These public tasks require human mathematical review. A saved response is not verified proof quality, a secure examination or evidence of universal course equivalence.
The suggested time is planning guidance; work is untimed. Preserve the initial attempt, consult targeted lessons, then revise your reasoning.
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0 of 11 tasks have recorded responses. No overall mastery score is assigned.
Minimize x²+y² with x+y=2 and3/2−x≤0. Derive the dual, give primal and dual optimizers, and connect their existence to the full-domain Slater proof. Now replace the inequality by(x−3/2)²≤0. The feasible optimum is unchanged: determine whether finite KKT multipliers and dual attainment survive. Prove your answer using the Lagrangian restricted to the equality line, then construct equality multipliers approaching the optimum bound.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Project a=(−2,5) onto C={(x,y):x≥0,y≥0,x+y≤2}. Prove existence and uniqueness and give a first-order certificate, rather than trusting a plot. Contrast this with minimizing exp(−x) on x≥0.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For f(x,y)=x²+3y²−4x−6y, find the minimizer. Derive the error recurrence for gradient descent with step 1/6, and compare exact Newton. State a safe Armijo condition and explain why an arbitrary positive step or a quasi-Newton matrix need not guarantee descent.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Maximize 5x+3y subject to x+y≤6,2x+y≤10,x,y≥0. Starting at the slack basis, perform two improving simplex pivots, then give a dual certificate. Replace 6 by b≥0 and determine the optimal value as a function of b, including where your dual prices change.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For x,y≥0, consider x+y≤1 and x+y≥3. Give a direct infeasibility certificate and formulate a Phase I problem whose positive optimum witnesses it. Explain why a degenerate zero-length simplex pivot or an exhausted ellipsoid iteration budget is not such a certificate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Minimize x²+2y² subject to x+y=3 and x≥0. Find all KKT candidates and identify the minimizer using an appropriate second-order or convex certificate. Then show why minimizing y subject to x²+y²≤0 has no KKT multiplier despite having a minimizer.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Minimize x² subject to x≥3. Derive the Lagrange dual and its optimum. For the logarithmic barrier x²−τ log(x−3), τ>0, derive the central point and primal-dual gap. Explain what happens as τ tends to zero.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Maximize X₁₂ over real symmetric positive-semidefinite 2×2 matrices with X₁₁=4,X₂₂=9. Derive a primal optimum and a dual certificate using the trace inner product. Explain why treating the off-diagonal coefficient as 1 in both entries would change the objective.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For f(x)=1/2‖x−a‖² with a=(−3,2,1/4), compare one projected-gradient step of size 1 onto the nonnegative orthant with the proximal solution of min f(x)+(1/2)‖x‖₁. Derive the proximal coordinates and explain why these two operations solve different constrained or penalized problems.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Maximize 3x+4y over nonnegative integers with 2x+3y≤7. Solve the continuous relaxation, construct an integer incumbent, and certify optimality. Also show a valid branch-and-bound split on the fractional root coordinate, retaining upper and lower bounds explicitly.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For x∈[0,4], minimize the pair (x²,(x−4)²). Determine the Pareto set and the minimizer of weight α on the first objective, 1−α on the second, for 0≤α≤1. Separately derive the robust counterpart of (2+u₁)z₁+(1+u₂)z₂≤5 for all |u₁|≤1/2, |u₂|≤1, allowing signed z.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Minimize x²+y² with x+y=2 and3/2−x≤0. Derive the dual, give primal and dual optimizers, and connect their existence to the full-domain Slater proof. Now replace the inequality by(x−3/2)²≤0. The feasible optimum is unchanged: determine whether finite KKT multipliers and dual attainment survive. Prove your answer using the Lagrangian restricted to the equality line, then construct equality multipliers approaching the optimum bound.
Project a=(−2,5) onto C={(x,y):x≥0,y≥0,x+y≤2}. Prove existence and uniqueness and give a first-order certificate, rather than trusting a plot. Contrast this with minimizing exp(−x) on x≥0.
For f(x,y)=x²+3y²−4x−6y, find the minimizer. Derive the error recurrence for gradient descent with step 1/6, and compare exact Newton. State a safe Armijo condition and explain why an arbitrary positive step or a quasi-Newton matrix need not guarantee descent.
Maximize 5x+3y subject to x+y≤6,2x+y≤10,x,y≥0. Starting at the slack basis, perform two improving simplex pivots, then give a dual certificate. Replace 6 by b≥0 and determine the optimal value as a function of b, including where your dual prices change.
For x,y≥0, consider x+y≤1 and x+y≥3. Give a direct infeasibility certificate and formulate a Phase I problem whose positive optimum witnesses it. Explain why a degenerate zero-length simplex pivot or an exhausted ellipsoid iteration budget is not such a certificate.
Minimize x²+2y² subject to x+y=3 and x≥0. Find all KKT candidates and identify the minimizer using an appropriate second-order or convex certificate. Then show why minimizing y subject to x²+y²≤0 has no KKT multiplier despite having a minimizer.
Minimize x² subject to x≥3. Derive the Lagrange dual and its optimum. For the logarithmic barrier x²−τ log(x−3), τ>0, derive the central point and primal-dual gap. Explain what happens as τ tends to zero.
Maximize X₁₂ over real symmetric positive-semidefinite 2×2 matrices with X₁₁=4,X₂₂=9. Derive a primal optimum and a dual certificate using the trace inner product. Explain why treating the off-diagonal coefficient as 1 in both entries would change the objective.
For f(x)=1/2‖x−a‖² with a=(−3,2,1/4), compare one projected-gradient step of size 1 onto the nonnegative orthant with the proximal solution of min f(x)+(1/2)‖x‖₁. Derive the proximal coordinates and explain why these two operations solve different constrained or penalized problems.
Maximize 3x+4y over nonnegative integers with 2x+3y≤7. Solve the continuous relaxation, construct an integer incumbent, and certify optimality. Also show a valid branch-and-bound split on the fractional root coordinate, retaining upper and lower bounds explicitly.
For x∈[0,4], minimize the pair (x²,(x−4)²). Determine the Pareto set and the minimizer of weight α on the first objective, 1−α on the second, for 0≤α≤1. Separately derive the robust counterpart of (2+u₁)z₁+(1+u₂)z₂≤5 for all |u₁|≤1/2, |u₂|≤1, allowing signed z.
