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
+**shadow prices** (`DualValue`) and variable **reduced costs** (`ReducedCost`) to see which
+nutritional requirements drive the cost and how far each unused food is from entering the diet,
+then adding a constraint and reading *its* shadow price (the marginal cost of the cap).
+
 
 ### 2. Diet Optimization (MILP)
 

diet_optimization/diet_optimization_lp.ipynb

@@ -13,7 +13,7 @@
     "We need to select quantities of different foods to:\n",
     "- Meet minimum and maximum nutritional requirements\n",
     "- Minimize total cost\n",
-    "- Satisfy additional constraints (like limiting dairy servings)\n",
+    "- Satisfy additional constraints (like limiting red-meat servings)\n",
     "\n",
     "The nutrition guidelines are based on USDA Dietary Guidelines for Americans, 2005.\n"
    ]
@@ -380,13 +380,44 @@
     "print_solution()\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Sensitivity Analysis: Duals and Reduced Costs\n",
+    "\n",
+    "Because every variable here is continuous, this is a linear program, and cuOpt returns dual information at the optimum — the economic \"why\" behind the plan:\n",
+    "\n",
+    "- **Constraint dual** — the sensitivity of the optimal cost to a constraint's bound: how much the minimum cost changes per unit change in that bound (traditionally called the *shadow price*). A *binding* constraint has a nonzero dual; a slack one is approximately zero.\n",
+    "- **Reduced cost (variable dual)** — for a food left out of the diet (amount 0), how much its per-serving price would have to fall before buying it could lower total cost."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Sensitivity analysis — read the LP duals at the optimum\n",
+    "if problem.Status.name == \"Optimal\":\n",
+    "    print(\"Constraint duals — change in cost per unit change in the bound:\")\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 needed to 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}.\")"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## Adding Additional Constraints\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"
+    "Now let's add a constraint to the existing model and read its **dual** — the sensitivity of cost to that constraint. We'll cap **hamburger + hot dog at 0.4 servings** combined — say, a red-meat preference — and re-solve."
    ]
   },
   {
@@ -395,9 +426,9 @@
    "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\")"
+    "# Limit combined hamburger + hot dog servings (a red-meat preference)\n",
+    "meat_expr = buy_vars[\"hamburger\"] + buy_vars[\"hot dog\"]\n",
+    "meat_constraint = problem.addConstraint(meat_expr <= 0.4, name=\"limit_meat\")"
    ]
   },
   {
@@ -407,7 +438,7 @@
    "outputs": [],
    "source": [
     "# Solve the problem again with the new constraint\n",
-    "print(\"\\nSolving with dairy constraint...\")\n",
+    "print(\"\\nSolving with the meat constraint...\")\n",
     "print(f\"Problem now has {problem.NumVariables} variables and {problem.NumConstraints} constraints\")\n",
     "\n",
     "start_time = time.time()\n",
@@ -419,13 +450,35 @@
     "print(f\"Objective value: ${problem.ObjValue:.2f}\")\n"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### The Dual of the Meat Cap\n",
+    "\n",
+    "Capping hamburger and hot dog tightens the model and nudges the cost up. The cap's **dual** gives the marginal cost of that limit directly — how much total cost changes per additional combined serving you allow — without re-solving at every level."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Marginal cost of the meat cap, read straight off its dual\n",
+    "if problem.Status.name == \"Optimal\":\n",
+    "    print(f\"limit_meat dual: {meat_constraint.DualValue:+.4f}  \"\n",
+    "          f\"(cost change per +1 combined serving of hamburger/hot dog allowed)\")\n",
+    "    print(f\"slack on the cap: {meat_constraint.Slack:.4f}   (0 means the cap is binding)\")"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## Solution Comparison\n",
     "\n",
-    "Let's compare the solutions before and after adding the dairy constraint to see the impact.\n"
+    "Let's compare the solutions before and after adding the meat cap to see the impact."
    ]
   },
   {
@@ -468,7 +521,7 @@
    "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",