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

diet_optimization/README.md

@@ -12,6 +12,11 @@ The diet optimization notebook solves a linear programming problem where:
 - The diet is a mix of different foods.
 - The foods have different prices and nutritional values.
 
+The notebook also demonstrates **sensitivity analysis** on the solved LP: reading constraint
+**dual values** (`DualValue`) and variable **reduced costs** (`ReducedCost`) to see which
+nutritional requirements drive the cost at the margin and roughly how far each unused food is
+from entering the diet (local rates — see the notebook's degeneracy and units caveats).
+
 
 ### 2. Diet Optimization (MILP)
 

diet_optimization/diet_optimization_lp.ipynb

@@ -384,48 +384,14 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Adding Additional Constraints\n",
+    "## Sensitivity Analysis: Dual Values and Reduced Costs\n",
     "\n",
-    "Now let's demonstrate how to add additional constraints to the existing model. We'll add a constraint to limit dairy servings to at most 6.\n"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# Create LinearExpression for dairy constraint\n",
-    "dairy_expr = buy_vars[\"milk\"] + buy_vars[\"ice cream\"]\n",
-    "dairy_constraint = problem.addConstraint(dairy_expr <= 6, name=\"limit_dairy\")"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "# Solve the problem again with the new constraint\n",
-    "print(\"\\nSolving with dairy constraint...\")\n",
-    "print(f\"Problem now has {problem.NumVariables} variables and {problem.NumConstraints} constraints\")\n",
+    "Every variable here is continuous, so cuOpt returns dual information at the optimum — the economic read behind the plan:\n",
     "\n",
-    "start_time = time.time()\n",
-    "problem.solve(settings)\n",
-    "solve_time = time.time() - start_time\n",
-    "\n",
-    "print(f\"\\nSolve completed in {solve_time:.3f} seconds\")\n",
-    "print(f\"Solver status: {problem.Status.name}\")\n",
-    "print(f\"Objective value: ${problem.ObjValue:.2f}\")\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Solution Comparison\n",
+    "- Each nutrition limit carries a **dual value** — at a non-degenerate optimum, how much total cost moves per unit you tighten or relax that limit. The implication runs one way: a limit with slack prices to ~0, while a binding limit (slack ≈ 0) *can* carry a nonzero dual but need not (a binding limit with a zero dual is a form of degeneracy). And a dual is $ per *that constraint's own unit* — per kcal for calories, per mg for sodium — so raw magnitudes aren't comparable across limits; to find where renegotiating pays off most, compare the value of, say, a 1% relaxation of each limit rather than ranking raw duals.\n",
+    "- A food left at 0 carries a **reduced cost** — roughly how far its per-serving price must fall before it *could* enter the diet without raising total cost. Reduced costs are per serving and serving sizes are arbitrary, so compare each as a fraction of the food's own price (\"needs a 5% price cut\" vs. \"a 40% one\") rather than sorting raw values.\n",
     "\n",
-    "Let's compare the solutions before and after adding the dairy constraint to see the impact.\n"
+    "Two caveats keep the read honest: under degeneracy a dual is a local, one-sided rate — confirm any rate you quote with a one-unit re-solve — and cuOpt's default first-order (PDLP) path can return duals without a simplex basis, accurate only to the convergence tolerance, so tiny near-zero values are noise, not signal."
    ]
   },
   {
@@ -434,8 +400,17 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "# Display the new solution\n",
-    "print_solution()"
+    "# Sensitivity analysis — read the LP duals at the optimum\n",
+    "if problem.Status.name == \"Optimal\":\n",
+    "    print(\"Constraint duals — local marginal cost per unit of each limit (units differ per constraint):\")\n",
+    "    for c in problem.getConstraints():\n",
+    "        print(f\"  {c.ConstraintName:14s} dual={c.DualValue:+.4f}  slack={c.Slack:.4f}\")\n",
+    "\n",
+    "    print(\"\\nReduced costs (variable duals) — for foods at 0, ~price drop before it could enter the diet:\")\n",
+    "    for v in problem.getVariables():\n",
+    "        print(f\"  {v.VariableName:12s} amount={v.getValue():7.3f}  reduced_cost={v.ReducedCost:+.4f}\")\n",
+    "else:\n",
+    "    print(f\"No duals available — solver status is {problem.Status.name}.\")"
    ]
   },
   {
@@ -451,8 +426,7 @@
     "3. **Define an objective function** to minimize total cost\n",
     "4. **Add nutritional constraints** with both lower and upper bounds\n",
     "5. **Solve the optimization problem** using cuOpt's high-performance solver\n",
-    "6. **Add additional constraints** to the existing model\n",
-    "7. **Analyze and compare solutions** before and after constraint modifications\n",
+    "6. **Read dual values and reduced costs** for a local sensitivity read — which limits drive cost at the margin, and which unused foods sit closest to entering\n",
     "\n",
     "The cuOpt Python API provides a clean, intuitive interface for building and solving optimization problems, making it easy to model complex real-world scenarios like diet optimization.\n",
     "\n",
@@ -468,7 +442,7 @@
    "metadata": {},
    "source": [
     "\n",
-    "SPDX-FileCopyrightText: Copyright (c) 2025 NVIDIA CORPORATION & AFFILIATES. All rights reserved.\n",
+    "SPDX-FileCopyrightText: Copyright (c) 2026 NVIDIA CORPORATION & AFFILIATES. All rights reserved.\n",
     "\n",
     "SPDX-License-Identifier: Apache-2.0\n",
     "\n",