Assignment scoring guide

Individual work. 100 raw points. Due December 2, 2026.

Use the assignment brief for the problem statements. The allocations below specify the evidence for full credit. Accept equivalent valid methods and justified numerical tolerances. Give partial credit for correct work and avoid repeated penalties for one arithmetic error.

1. LP certificate and sensitivity (25 points)

(a) 4 points

• Dual objective: 1 point
• Dual inequalities and multiplier domains: 2 points
• Interpretation of price units: 1 point

(b) 6 points

• Primal and dual optimal solutions: 2 points
• Both feasibility checks: 1 point
• All complementary-slackness products: 1 point
• Equal objective values: 1 point
• Reasoning for global optimality: 1 point

(c) 9 points

• All primal and value-function regimes: 3 points
• Feasible dual certificates for the regimes: 2 points
• Interval for the original prices: 1 point
• Plot and slopes: 1 point
• Endpoint and one-sided derivative interpretation: 2 points

(d) 6 points

• Both reoptimized solutions and objective values: 2 points
• Price predictions and their errors: 2 points
• Recommendation that accounts for costs and the valid regime: 2 points

2. Integer production and start-up cost (20 points)

(a) 4 points

• Implications of the binary constraint and its converse: 2 points
• Operating limit versus a bound derived from resource capacity: 2 points

(b) 5 points

• Relaxed variable domains: 1 point
• Feasible relaxation optimum and value: 2 points
• Mathematical upper-bound certificate: 2 points

(c) 4 points

• Candidate feasibility and value: 1 point
• Lower and upper bounds: 1 point
• Integer tightening of the bound: 1 point
• Explanation of the remaining optimality gap: 1 point

(d) 7 points

• Finite search bounds and enumeration method: 1 point
• Complete enumeration table for the original model: 2 points
• Optimum and exhaustive certificate: 1 point
• Optimum of the altered model: 1 point
• Diagnosis of the operating-limit violation and model change: 2 points

3. Unconstrained iterations and Armijo (20 points)

(a) 3 points

• Gradient: 1 point
• Hessian: 1 point
• Unique global minimizer with justification: 1 point

(b) 3 points

• Fixed-step point and value: 1 point
• Exact-line-search derivation and step size: 1 point
• Exact-search point and value with comparison: 1 point

(c) 4 points

• Direction and directional derivative: 1 point
• Both trial points and actual values: 1 point
• Armijo targets: 1 point
• Decisions and treatment of equality: 1 point

(d) 5 points

• Reproducible implementation with stopping and reset rules: 1 point
• Complete comparable traces: 1 point
• Labeled gradient-norm plot: 1 point
• Iteration and objective-evaluation accounting: 1 point
• Explanation of the special finite termination: 1 point

(e) 5 points

• Derivatives and raw Newton direction: 1 point
• Directional derivative and its sign: 1 point
• Argument about sufficiently small Armijo steps: 1 point
• A justified descent safeguard: 1 point
• Accepted-step calculation and nonconvex interpretation: 1 point

4. Constrained NLP and a capacity decision (35 points)

(a) 8 points

• Stationarity including bound multipliers: 2 points
• Equations for the proposed active set: 2 points
• Candidate point and objective value: 2 points
• All multipliers: 2 points

(b) 6 points

• Primal and dual feasibility: 1 point
• Complementarity: 1 point
• Stationarity: 2 points
• Convexity and uniqueness argument: 2 points

(c) 7 points

• Parameter formulas: 2 points
• Complete closed KKT interval: 2 points
• Separate coordinate and multiplier positivity ranges: 1 point
• Both endpoint explanations: 2 points

(d) 9 points

• Exact reduction and cost for the fixed offer: 1 point
• First-order estimate and error: 1 point
• Exact fixed-offer net benefit and decision: 1 point
• Continuous net-benefit function: 2 points
• Derivative, concavity and feasible maximizing quantity: 1 point
• Optimal benefit and endpoint comparison: 1 point
• Plot of exact and linear net benefit: 1 point
• Interpretation and valid sensitivity regime: 1 point

(e) 5 points

• Three starts and solver messages: 1 point
• Explicit source of multipliers: 1 point
• Four residuals with stated tolerances: 2 points
• Analytic comparison or diagnosis of failure: 1 point

Numerical answers alone earn only the numerical components. Supply the requested derivations, plots, comparisons, interpretations and independent checks. Report failed solver runs honestly and provide a diagnosis. Report length is a guide rather than a scoring criterion.