Add LP sensitivity analysis (duals + reduced costs) to the diet example

[cafzal]

LP sensitivity analysis (dual values + reduced costs) — diet example

Extends the diet LP example to read the dual information cuOpt returns for a solved LP — the economic "why" behind the optimal plan. Motivated by NVIDIA/cuopt#1393 (cuOpt's dual / sensitivity support per problem type: LP/QP, not integer models); companion to the QP efficient-frontier-with-duals example in #151 (LP here, QP there).

• diet_optimization/diet_optimization_lp.ipynb — after the solve, reads each constraint's dual value (DualValue, Slack) and each variable's reduced cost (ReducedCost) via problem.getConstraints() / problem.getVariables(), with a plain-language binding-vs-slack reading: which nutrition limits drive the cost, and how far each unused food is from entering the diet. The diet model itself is unchanged. diet_optimization/README.md updated to match.

Runs on cuOpt alone (Colab GPU), follows the repo's notebook idiom (GPU check → cuopt-cu12 install → solve), and ships output-stripped (repo convention); run evidence below.

User testing

Run end-to-end on a Colab GPU runtime (cuopt-cu12) — clean, no errors: solve is Optimal at $11.83. The binding constraints here price nonzero — min_protein +0.093, min_calories +0.003, max_sodium −0.002 — while slack ones price to ~0; every food left out of the diet shows a positive reduced cost (macaroni +1.34 the largest), the price drop before it could enter the diet.

raw cell output

Constraint duals — local marginal cost per unit of each limit (units differ per constraint):
min_calories dual=+0.0034 slack=-0.0000
max_calories dual=+0.0000 slack=400.0000
min_protein dual=+0.0930 slack=-0.0000
min_fat dual=+0.0000 slack=-59.0559
max_fat dual=+0.0000 slack=5.9441
min_sodium dual=+0.0000 slack=-1779.0000
max_sodium dual=-0.0016 slack=0.0000
Reduced costs (variable duals) — for foods at 0, ~price drop before it could enter the diet:
hamburger amount= 0.605 reduced_cost=+0.0001
chicken amount= 0.000 reduced_cost=+0.3425
hot dog amount= 0.000 reduced_cost=+0.5456
fries amount= 0.000 reduced_cost=+0.6421
macaroni amount= 0.000 reduced_cost=+1.3371
pizza amount= 0.000 reduced_cost=+0.7857
salad amount= 0.000 reduced_cost=+0.4400
milk amount= 6.970 reduced_cost=-0.0000
ice cream amount= 2.591 reduced_cost=-0.0000

Notes

• Dual values are LP/QP only — valid here because every variable is continuous; an integer model returns none.
• Output-stripped per repo convention; re-run on a GPU runtime to reproduce the numbers above.

[cafzal]

@rgsl888prabhu For visibility: Here's a small addition to showcase use of shadow price and reduced cost in an existing example.

[cafzal]

@ramakrishnap-nv this is all set on my end

[mlubin]

Do you have financial interests in the dairy industry? 😆
Why are we changing the formulation of the problem?

[cafzal]

Good point. Reverted: the diet formulation stays canonical, and the sensitivity section now reads the duals and reduced costs from the existing solve rather than adding a constraint. Also trimmed the duals write-up.