Improve MIP print formatting, etc.

Author: user202729

src/sage/numerical/mip.pyx

@@ -65,7 +65,7 @@ A mixed integer linear program can give you an answer:
 The following example shows all these steps::
 
     sage: p = MixedIntegerLinearProgram(maximization=False, solver='GLPK')
-    sage: w = p.new_variable(integer=True, nonnegative=True)
+    sage: w = p.new_variable(integer=True, nonnegative=True, name='w')
     sage: p.add_constraint(w[0] + w[1] + w[2] - 14*w[3] == 0)
     sage: p.add_constraint(w[1] + 2*w[2] - 8*w[3] == 0)
     sage: p.add_constraint(2*w[2] - 3*w[3] == 0)
@@ -74,18 +74,18 @@ The following example shows all these steps::
     sage: p.set_objective(w[3])
     sage: p.show()
     Minimization:
-       x_3
+      w[3]
     Constraints:
-      0.0 <= x_0 + x_1 + x_2 - 14.0 x_3 <= 0.0
-      0.0 <= x_1 + 2.0 x_2 - 8.0 x_3 <= 0.0
-      0.0 <= 2.0 x_2 - 3.0 x_3 <= 0.0
-      - x_0 + x_1 + x_2 <= 0.0
-      - x_3 <= -1.0
+      0.0 <= w[0] + w[1] + w[2] - 14.0 w[3] <= 0.0
+      0.0 <= w[1] + 2.0 w[2] - 8.0 w[3] <= 0.0
+      0.0 <= 2.0 w[2] - 3.0 w[3] <= 0.0
+      - w[0] + w[1] + w[2] <= 0.0
+      - w[3] <= -1.0
     Variables:
-      x_0 is an integer variable (min=0.0, max=+oo)
-      x_1 is an integer variable (min=0.0, max=+oo)
-      x_2 is an integer variable (min=0.0, max=+oo)
-      x_3 is an integer variable (min=0.0, max=+oo)
+      w[0] = x_0 is an integer variable (min=0.0, max=+oo)
+      w[1] = x_1 is an integer variable (min=0.0, max=+oo)
+      w[2] = x_2 is an integer variable (min=0.0, max=+oo)
+      w[3] = x_3 is an integer variable (min=0.0, max=+oo)
     sage: print('Objective Value: {}'.format(p.solve()))
     Objective Value: 2.0
     sage: for i, v in sorted(p.get_values(w, convert=ZZ, tolerance=1e-3).items()):
@@ -95,6 +95,34 @@ The following example shows all these steps::
     w_2 = 3
     w_3 = 2
 
+If your problem is already in the standard form, for example::
+
+    sage: A = matrix([[1, 2], [3, 5]])
+    sage: b = vector([7, 11])
+    sage: c = vector([5, 9])
+
+You can add the constraint by treating the variable dictionary as a vector::
+
+    sage: p = MixedIntegerLinearProgram(maximization=True, solver='GLPK')
+    sage: w = p.new_variable(integer=True, name='w')
+    sage: p.add_constraint(A * w <= b)
+    sage: p.set_objective((c.row() * w)[0])
+    sage: p.show()
+    Maximization:
+      5.0 w[0] + 9.0 w[1]
+    Constraints:
+      w[0] + 2.0 w[1] <= 7.0
+      3.0 w[0] + 5.0 w[1] <= 11.0
+    Variables:
+      w[0] = x_0 is an integer variable (min=-oo, max=+oo)
+      w[1] = x_1 is an integer variable (min=-oo, max=+oo)
+    sage: print('Objective Value: {}'.format(p.solve()))
+    Objective Value: 25.0
+    sage: for i, v in sorted(p.get_values(w, convert=ZZ, tolerance=1e-3).items()):
+    ....:     print(f'w_{i} = {v}')
+    w_0 = -13
+    w_1 = 10
+
 Different backends compute with different base fields, for example::
 
     sage: p = MixedIntegerLinearProgram(solver='GLPK')
@@ -144,8 +172,6 @@ also allowed::
     sage: a[4, 'string', QQ] - 7*b[2]
     x_2 - 7*x_3
     sage: mip.show()
-    Maximization:
-    <BLANKLINE>
     Constraints:
     Variables:
       a[1] = x_0 is a continuous variable (min=-oo, max=+oo)
@@ -778,8 +804,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: p.add_constraint(x[0] + x[3] <= 8)
             sage: p.add_constraint(y[0] >= y[1])
             sage: p.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               x_0 + x_1 <= 8.0
               - x_2 + x_3 <= 0.0
@@ -795,8 +819,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: mip.<x, y, z> = MixedIntegerLinearProgram(solver='GLPK')
             sage: mip.add_constraint(x[0] + y[1] + z[2] <= 10)
             sage: mip.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               x[0] + y[1] + z[2] <= 10.0
             Variables:
@@ -860,8 +882,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: a[0] + b[2]
             x_0 + x_1
             sage: mip.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
             Variables:
               a[0] = x_0 is a continuous variable (min=-oo, max=+oo)
@@ -1251,23 +1271,24 @@ cdef class MixedIntegerLinearProgram(SageObject):
                 varid_explainer[i] = varid_name[i] = default_name
 
         ##### Sense and objective function
-        print("Maximization:" if b.is_maximization() else "Minimization:")
-        print(" ", end=" ")
+        formula_parts = []
         first = True
         for 0<= i< b.ncols():
             c = b.objective_coefficient(i)
             if c == 0:
                 continue
-            print((("+ " if (not first and c>0) else "") +
-                   ("" if c == 1 else ("- " if c == -1 else str(c)+" "))+varid_name[i]
-                   ), end=" ")
+            formula_parts.append(("+ " if (not first and c>0) else "") +
+                         ("" if c == 1 else ("- " if c == -1 else str(c)+" "))+varid_name[i]
+                         )
             first = False
         d = b.objective_constant_term()
         if d > self._backend.zero():
-            print("+ {} ".format(d))
+            formula_parts.append("+ {}".format(d))
         elif d < self._backend.zero():
-            print("- {} ".format(-d))
-        print("\n")
+            formula_parts.append("- {}".format(-d))
+        if formula_parts:
+            print("Maximization:" if b.is_maximization() else "Minimization:")
+            print("  " + " ".join(formula_parts))
 
         ##### Constraints
         print("Constraints:")
@@ -2001,8 +2022,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: b = p.new_variable(nonnegative=True)
             sage: p.add_constraint(b[8] - b[15] <= 3*b[8] + 9)
             sage: p.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               -2.0 x_0 - x_1 <= 9.0
             Variables:
@@ -2043,8 +2062,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: for each in range(10):
             ....:     lp.add_constraint(lp[0]-lp[1], min=1)
             sage: lp.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               1.0 <= x_0 - x_1
             Variables:
@@ -2056,8 +2073,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: for each in range(10):
             ....:     lp.add_constraint(2*lp[0] - 2*lp[1], min=2)
             sage: lp.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               1.0 <= x_0 - x_1
             Variables:
@@ -2069,8 +2084,6 @@ cdef class MixedIntegerLinearProgram(SageObject):
               sage: for each in range(10):
               ....:     lp.add_constraint(-2*lp[0] + 2*lp[1], min=-2)
               sage: lp.show()
-              Maximization:
-              <BLANKLINE>
               Constraints:
                 1.0 <= x_0 - x_1
                 -2.0 <= -2.0 x_0 + 2.0 x_1
@@ -2268,17 +2281,13 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: p.add_constraint(x - y, max=0)
             sage: p.add_constraint(x, max=4)
             sage: p.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               x_0 + x_1 <= 10.0
               x_0 - x_1 <= 0.0
               x_0 <= 4.0
             ...
             sage: p.remove_constraint(1)
             sage: p.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               x_0 + x_1 <= 10.0
               x_0 <= 4.0
@@ -2306,17 +2315,13 @@ cdef class MixedIntegerLinearProgram(SageObject):
             sage: p.add_constraint(x - y, max=0)
             sage: p.add_constraint(x, max=4)
             sage: p.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               x_0 + x_1 <= 10.0
               x_0 - x_1 <= 0.0
               x_0 <= 4.0
             ...
             sage: p.remove_constraints([0, 1])
             sage: p.show()
-            Maximization:
-            <BLANKLINE>
             Constraints:
               x_0 <= 4.0
             ...
@@ -3627,8 +3632,19 @@ cdef class MIPVariable(FiniteFamily):
             (1, 1/2)*x_0 + (2/3, 3/4)*x_1
             sage: m * v
             (1, 2/3)*x_0 + (1/2, 3/4)*x_1
+
+            sage: c = vector([1, 2])
+            sage: v * c
+            x_0 + 2*x_1
+            sage: c * v
+            x_0 + 2*x_1
         """
+        from sage.structure.element import Vector
+        if isinstance(left, Vector):
+            left, right = right, left
         if isinstance(left, MIPVariable):
+            if isinstance(right, Vector):
+                return (<MIPVariable> left)._matrix_rmul_impl(right.column())[0]
             if not isinstance(right, Matrix):
                 return NotImplemented
             return (<MIPVariable> left)._matrix_rmul_impl(right)