C# operator overload for `+=`?

Overloadable Operators, from MSDN:

Assignment operators cannot be overloaded, but +=, for example, is evaluated using +, which can be overloaded.

Even more, none of assignment operators can be overloaded. I think this is because there will be an effect for the Garbage collection and memory management, which is a potential security hole in CLR strong typed world.

Nevertheless, let's see what exactly an operator is. According to the famous Jeffrey Richter's book, each programming language has its own operators list, which are compiled in a special method calls, and CLR itself doesn't know anything about operators. So let's see what exactly stays behind the + and += operators.

See this simple code:

Decimal d = 10M;
d = d + 10M;
Console.WriteLine(d);

Let view the IL-code for this instructions:

  IL_0000:  nop
  IL_0001:  ldc.i4.s   10
  IL_0003:  newobj     instance void [mscorlib]System.Decimal::.ctor(int32)
  IL_0008:  stloc.0
  IL_0009:  ldloc.0
  IL_000a:  ldc.i4.s   10
  IL_000c:  newobj     instance void [mscorlib]System.Decimal::.ctor(int32)
  IL_0011:  call       valuetype [mscorlib]System.Decimal [mscorlib]System.Decimal::op_Addition(valuetype [mscorlib]System.Decimal,
                                                                                                valuetype [mscorlib]System.Decimal)
  IL_0016:  stloc.0

Now lets see this code:

Decimal d1 = 10M;
d1 += 10M;
Console.WriteLine(d1);

And IL-code for this:

  IL_0000:  nop
  IL_0001:  ldc.i4.s   10
  IL_0003:  newobj     instance void [mscorlib]System.Decimal::.ctor(int32)
  IL_0008:  stloc.0
  IL_0009:  ldloc.0
  IL_000a:  ldc.i4.s   10
  IL_000c:  newobj     instance void [mscorlib]System.Decimal::.ctor(int32)
  IL_0011:  call       valuetype [mscorlib]System.Decimal [mscorlib]System.Decimal::op_Addition(valuetype [mscorlib]System.Decimal,
                                                                                                valuetype [mscorlib]System.Decimal)
  IL_0016:  stloc.0

They are equal! So the += operator is just syntactic sugar for your program in C#, and you can simply overload + operator.

For example:

class Foo
{
    private int c1;

    public Foo(int c11)
    {
        c1 = c11;
    }

    public static Foo operator +(Foo c1, Foo x)
    {
        return new Foo(c1.c1 + x.c1);
    }
}

static void Main(string[] args)
{
    Foo d1 =  new Foo (10);
    Foo d2 = new Foo(11);
    d2 += d1;
}

This code will be compiled and successfully run as:

  IL_0000:  nop
  IL_0001:  ldc.i4.s   10
  IL_0003:  newobj     instance void ConsoleApplication2.Program/Foo::.ctor(int32)
  IL_0008:  stloc.0
  IL_0009:  ldc.i4.s   11
  IL_000b:  newobj     instance void ConsoleApplication2.Program/Foo::.ctor(int32)
  IL_0010:  stloc.1
  IL_0011:  ldloc.1
  IL_0012:  ldloc.0
  IL_0013:  call       class ConsoleApplication2.Program/Foo ConsoleApplication2.Program/Foo::op_Addition(class ConsoleApplication2.Program/Foo,
                                                                                                          class ConsoleApplication2.Program/Foo)
  IL_0018:  stloc.1

Update:

According to your Update - as the @EricLippert says, you really should have the vectors as an immutable object. Result of adding of the two vectors is a new vector, not the first one with different sizes.

If, for some reason you need to change first vector, you can use this overload (but as for me, this is very strange behaviour):

public static Vector operator +(Vector left, Vector right)
{
    left.x += right.x;
    left.y += right.y;
    return left;
}

This is because of the same reason that the assignment operator cannot be overloaded. You cannot write code that would perform the assignment correctly.

class Foo
{
   // Won't compile.
   public static Foo operator= (Foo c1, int x)
   {
       // duh... what do I do here?  I can't change the reference of c1.
   }
}

Assignment operators cannot be overloaded, but +=, for example, is evaluated using +, which can be overloaded.

From MSDN.

You can't overload += because it isn't really a unique operator, it's just syntactic sugar. x += y is just a shorthand way of writing x = x + y. Because += is defined in terms of the + and = operators, allowing you to override it separately could create problems, in cases where x += y and x = x + y didn't behave in the exact same way.

At a lower level, it's very likely that the C# compiler compiles both expressions down to the same bytecode, meaning that it's very likely the the runtime can't treat them differently during program execution.

I can understand that you might want to treat it like a separate operation: in an statement like x += 10 you know that you can mutate the x object in place and perhaps save some time/memory, rather than creating a new object x + 10 before assigning it over the old reference.

But consider this code:

a = ...
b = a;
a += 10;

Should a == b at the end? For most types, no, a is 10 more than b. But if you could overload the += operator to mutate in place, then yes. Now consider that a and b could get passed around to distant parts of the program. Your possible optimization could create confusing bugs if your object starts to change where code doesn't expect it to.

In other words, if performance is that important, it's not too hard to replace x += 10 with a method call like x.increaseBy(10), and it's a lot clearer for everyone involved.