Optimization investigation: build, break and defend a certificate
Produce one coherent decision-and-certificate dossier for a synthetic two-product resource model. Derive candidates independently, submit your own exact primal/dual vectors, preserve failed checks, analyze a degenerate budget, diagnose an infeasible demand change, and defend an integer transfer. The supplied checker verifies entered certificates and never generates an optimum. A proof-and-audit route is available; implementation proficiency additionally requires your own runnable enumerator, tests and work log. Submitted text and a passing arithmetic check do not establish independent mastery.
Planning estimate: 20 hours, including analysis, writing and revision. Use the investigation and working agreement below. Keep a single coherent report across milestones rather than unrelated short answers.
Preparation and working agreement
The main model maximizes 5x+4y subject to 2x+y≤b, x+2y≤8, x≤3 and x,y≥0. Treat production amounts and resource capacities as normalized synthetic units and the objective as credits; no real factory measurements or forecast accuracy are implied. Default budget b=8. Keep model identity, row order, variable domain and objective direction with every certificate.
Use the seven milestones as sections of one report, retaining derivations, a candidate ledger, failed and repaired certificate records, a parameter analysis and reviewer revisions. Existing lesson proofs and the introductory LP explorer are preparation; the explorer’s supplied optimum is not your independent implementation. Record all public-solution/checker exposure.
The exact checker accepts quoted rational strings, numerator magnitude at most1000 and positive denominator at most1000, at most2048 input characters and12 submitted audit attempts per temporary session. It checks only the selected fixed model and supplied vectors. Save its downloadable or copyable JSON before navigating/reloading; the widget history is temporary, while project response revisions use the existing local progress system.
The independent search contract is at most10 unordered boundary-line pairs for the main continuous model, with exact determinants, every inequality checked and duplicate vertices identified. A smaller deliberately imposed pair budget must report incomplete search, not infeasibility or optimality. Checker arithmetic work, independent search work, rational reduction/bit cost and measured runtime are separate quantities.
Keep the complete report and any executable source/tests separately; record filenames, versions and reproduction instructions in the milestone responses. This page does not upload or run your program. Written answers have the existing8000-character bound. Human review must actually occur; otherwise mark it pending. The planning estimate is not a validated course workload.
Use each argument's planning fields, immutable revisions and revision-specific feedback. Save derivations, data provenance, numerical checks and a reproducible method in the report. External code or files should be identified by filename and instructions in your response; this page stores text and does not run code or upload artifacts.
Submit your own certificate for checking
This checker verifies supplied rational vectors; it does not search for a solution or generate an optimal certificate. Write your prediction and derivation in the milestone before requesting feedback. All examples are synthetic continuous linear programs unless the separate integer-transfer argument says otherwise.
Maximize 5x + 4y with x,y ≥ 0. All rows use A x ≤ b. The selected problem is Main model: budget 8.
Resource 1: 2x + y ≤ b; row coefficients (2, 1), right side 8.
Resource 2: x + 2y ≤ 8; row coefficients (1, 2), right side 8.
Quantity cap: x ≤ 3; row coefficients (1, 0), right side 3.
Optimality input: kind, a two-entry primal vector and a dual vector with 3 entries in this row order. Infeasibility input: kind and a multipliers vector with 3 entries. Every number is a quoted integer or fraction such as "8/3". Negative entries are accepted for diagnosis; feasibility still requires the specified signs.
Exact input bounds: numerator magnitude at most 1000, positive denominator at most 1000. Decimals, exponents, extra keys and incorrect vector lengths are rejected. The initial zero vectors only demonstrate the format.
0 of 12 audit records used. Failed parsing also counts as an audit attempt.
Retain the dossier before navigating or reloading
The at-most-12 submitted records are immutable within this temporary session. They are not automatically saved to your project. Download or copy the report, identify its filename in the relevant milestone, and retain both failed and repaired certificates. The ordinary project response/revision fields below are saved through the existing local progress system. Public feedback and supplied solution exposure are practice, not an untouched assessment.
Copyable full audit record (0 entries)
For integer transfer, a certified continuous branch optimum supplies a bound; proving that the branches cover every integer feasible point is your separate mathematical obligation. Reusing an old certificate after a changed row or right side requires a new audit.
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0 of 7 milestones have recorded responses. This counts submissions, not quality, independent review or project mastery.
Declare the model, decision and evidence plan
Create the report manifest before solving: write the primal and its dual with all signs, row order and units; identify the decision you will defend. State why a feasible optimum is attained for b≥0. Specify a proof-and-audit or implementation route, independent candidate-search method, stopping budget, assistance/exposure record and artifact filenames. Predict which constraints might bind at b=8 without revealing the later solutions.
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.
Construct and audit an independent candidate ledger
At b=8, inspect all unordered pairs of the five boundary lines x=0,y=0,2x+y=8,x+2y=8,x=3. Retain every determinant, candidate, failed inequality and duplicate decision; evaluate feasible vertices. Explain why the list is exhaustive for this bounded polygon. Stop once with a deliberately smaller pair budget and report what can still be concluded. For implementation evidence, retain your own exact-arithmetic code and independent tests; otherwise retain a fully reproducible hand ledger.
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.
Submit, break and repair a primal-dual certificate
Derive your own optimal primal and dual vectors at b=8, then submit them to the checker. Retain the full report. Deliberately audit both the zero vectors and primal(4,1) with your optimal dual; explain why equal objective values in a failed pair prove nothing. Derive the exact gap decomposition and identify which failed feasibility condition permits negative complementary terms. Retain failed records before submitting a repaired 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.
Analyze the budget family and its degenerate breakpoint
Derive the optimal value for every b≥0, including both breakpoints. At b=17/2 identify all active rows, exhibit two distinct standard-form bases for the same primal solution and give a nontrivial interval of optimal dual multipliers. Submit at least two distinct exact dual certificates there. Explain why primal uniqueness, basis degeneracy, dual nonuniqueness and a nonexistent two-sided sensitivity derivative are different statements. Predict what fails if a breakpoint certificate is copied to b=10 without adjustment.
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.
Diagnose an infeasible demand without inventing an optimum
At b=8 add a demand x+y≥7 as the fourth row−x−y≤−7. Derive and submit a Farkas multiplier vector using the checker’s inequality convention. Write the contradiction for an arbitrary supposedly feasible x. Compare it with a zero multiplier submission and a failed search. Then reduce demand to5 and construct a feasible witness manually; explain why the old infeasibility certificate must be rechecked. Keep the two model manifests distinct.
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.
Reproduce, budget and challenge the evidence
Give an independent reproduction plan for the complete dossier, including model identity, exact rational parsing, candidate search, certificate verification and failure tests. Reproduce at least one valid certificate without this widget and one adversarial failed certificate. For an implementation route, submit runnable source and tests with a work ledger; for a proof route, submit independently checkable arithmetic and an actual reviewer’s sampled checks when available. Explain the checker’s product counts, the 12-audit cap and why neither implies polynomial bit complexity or real-world model validity.
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.
Defend and revise one coherent decision report
Assemble the seven sections into one report with an executive decision, exact mathematical appendix, candidate/audit ledgers, source/exposure manifest and limitations. Give recommendations for the default, degenerate-budget and infeasible-demand cases. Ask a reviewer to challenge a sign convention, a changed row and a sensitivity claim; retain the actual feedback tied to a revision and respond with corrections or a reasoned defense. If no review occurs, state what remains unreviewed. Prepare for the separate integer transfer without treating prior public feedback as unseen evidence.
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, defend and revise
Ask a reviewer to examine the complete reasoning, reproduce selected results and pose an unfamiliar case. Record their feedback against the specific revision discussed. Reviewer labels are entered locally and do not verify identity or judgment. A complete set of text submissions does not certify an independently defended investigation.
