gurobi basic usage

sheets/04_mip/README.md

@@ -55,6 +55,83 @@ grbgetkey XXXXXXXX-YOUR-KEY-XXXXXXXXXX
 to activate  it.
 You may have to be within the university's network (or use VPN) for this.
 
+## Basic Usage
+
+```python
+import gurobipy as gp  # API
+from gurobipy import GRB  # Symbols (e.g. GRB.BINARY)
+
+model = gp.Model("mip1")  # Create a model
+model.Params.TimeLimit = 90  # 90s time limit
+
+# Create variables x (bool), y (integer), z (fractional)
+x = model.addVar(vtype=GRB.BINARY, name="x")
+y = model.addVar(vtype=GRB.INTEGER, name="y", lb=-GRB.INFINITY)
+z = model.addVar(vtype=GRB.CONTINUOUS, name="z")
+# further options: 
+# * lb: float <= Lower Bound: Default Zero!!!
+# * ub: float <= Upper Bound
+# * obj: float <= Coefficient for the objective function.
+
+# Objective function (Maximization)
+model.setObjective(x + y + 2 * z, GRB.MAXIMIZE)
+
+# Constraints
+model.addConstr(x + 2 * y + 3 * z <= 4, name="Constraint1")
+model.addConstr(x + y >= 1, name="Constraint2")
+
+# AND GO!
+model.optimize()
+```
+
+This will give us the following log.
+```
+ Gurobi Optimizer version 9.1.2 build v9.1.2rc0 (mac64)
+Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
+Optimize a model with 2 rows, 3 columns and 5 nonzeros
+Model fingerprint: 0xd685009c
+Model has 1 quadratic objective term
+Variable types: 1 continuous, 2 integer (1 binary)
+Coefficient statistics:
+  Matrix range     [1e+00, 3e+00]
+  Objective range  [1e+00, 2e+00]
+  QObjective range [2e+00, 2e+00]
+  Bounds range     [1e+00, 1e+00]
+  RHS range        [1e+00, 4e+00]
+Found heuristic solution: objective 1.0000000
+Presolve removed 2 rows and 3 columns
+Presolve time: 0.00s
+Presolve: All rows and columns removed
+
+Explored 0 nodes (0 simplex iterations) in 0.01 seconds
+Thread count was 1 (of 8 available processors)
+
+Solution count 2: 3 1 
+
+Optimal solution found (tolerance 1.00e-04)
+Best objective 3.000000000000e+00, best bound 3.000000000000e+00, gap 0.0000%
+```
+
+```python
+if model.Status == GRB.OPTIMAL: # an optimal solution has been found
+    print("=== Optimal Solution ===")
+    print(f"x={x.X}")
+    print(f"y={y.X}")
+    print(f"z={z.X}")
+elif model.SolCount > 0:  # there exists a feasible solution
+    print("=== Suboptimal Solution ===")
+    print(f"x={x.X}")
+    print(f"y={y.X}")
+    print(f"z={z.X}")
+else:
+    print(f"Bad solution (code {model.Status}).")
+    print(
+        "See status codes on https://www.gurobi.com/documentation/9.5/refman/optimization_status_codes.html#sec:StatusCodes"
+    )
+```
+
+> Caveat: Due to the underlying technique, the resulting variables may not be rounded. A boolean variable could take the value 0.00001 instead of 0, so you have to round to check them.
+
 ## Tasks
 
 * Model the BTSP as a Mixed Integer Program on paper. Use the [constraint technique](https://en.wikipedia.org/wiki/Travelling_salesman_problem#Dantzig%E2%80%93Fulkerson%E2%80%93Johnson_formulation) already used for SAT to enforce a single tour and do not use the [technique](https://en.wikipedia.org/wiki/Travelling_salesman_problem#Miller%E2%80%93Tucker%E2%80%93Zemlin_formulation[21]) used with CP-SAT (as it performs poorly with MIP-solvers).