Solution 1:

No.

A workaround is to overload with another method that doesn't have out / ref parameters, and which just calls your current method.

public bool SomeMethod(out string input)
{
    ...
}

// new overload
public bool SomeMethod()
{
    string temp;
    return SomeMethod(out temp);
}

If you have C# 7.0, you can simplify:

// new overload
public bool SomeMethod()
{
    return SomeMethod(out _);    // declare out as an inline discard variable
}

(Thanks @Oskar / @Reiner for pointing this out.)

Solution 2:

As already mentioned, this is simply not allowed and I think it makes a very good sense. However, to add some more details, here is a quote from the C# 4.0 Specification, section 21.1:

Formal parameters of constructors, methods, indexers and delegate types can be declared optional:

fixed-parameter:
    attributesopt parameter-modifieropt type identifier default-argumentopt
default-argument:
    = expression

  • A fixed-parameter with a default-argument is an optional parameter, whereas a fixed-parameter without a default-argument is a required parameter.
  • A required parameter cannot appear after an optional parameter in a formal-parameter-list.
  • A ref or out parameter cannot have a default-argument.