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

[cafzal]

LP sensitivity analysis (duals + 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, which clarifies cuOpt's dual / sensitivity-analysis 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 first solve, reads each constraint's dual (DualValue, Slack) and each variable's reduced cost (ReducedCost) via problem.getConstraints() / problem.getVariables(), with a plain-language binding-vs-slack reading. The "add a constraint" step then reads the dual of the added constraint (meat_constraint.DualValue) — the marginal cost of the cap. This also replaces the example's existing dairy (milk + ice cream) <= 6 cap, which is infeasible (max_sodium is binding and dairy is the only low-sodium way to meet the protein/calorie minimums, so the stock notebook's "Solution Comparison" printed "No optimal solution found"), with a feasible, binding hamburger + hot dog <= 0.4. 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 — every cuopt-examples notebook has 0 cell outputs); run evidence below.

User testing

Run end-to-end on a Colab GPU runtime (cuopt-cu12) — clean, no errors:

• First solve Optimal, $11.83. Binding constraints carry nonzero duals — 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 has a positive reduced cost (macaroni +1.34 the largest), i.e. how far its per-serving price must fall to enter the diet.
• Added-constraint dual. With hamburger + hot dog <= 0.4, the re-solve is Optimal at $11.86 (up from $11.83); the cap binds at hamburger = 0.400 with dual −0.164 (slack 0) — allowing one more serving would lower cost ~$0.16, matching the empirical $0.03 / 0.205 ≈ $0.15.

raw cell output

Constraint duals — change in cost per unit change in the bound:
  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 needed to 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

limit_meat dual: -0.1643  (cost change per +1 combined serving of hamburger/hot dog allowed)
slack on the cap: 0.0000   (0 means the cap is binding)

Notes

• Duals 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.