# Pruning the expression tree with recursive value identification

Today was the release of SCIP.jl v0.11, the first release switching to SCIP 8. The major change in this (massive) release was the rewrite of the nonlinear optimization part, using a so-called expression framework. The rewrite of the wrapper had some fairly tedious parts, debugging C shared libraries is quickly a mess with cryptic error messages. But the nonlinear rewrite gave me the opportunity to tweak the way Julia expressions are passed to SCIP in a minor way.

# SCIP expressions

I will not go in depth into the new expression framework and will instead reference these slides but more importantly the SCIP 8 release report

The key part is that in a nonlinear expression, each operand is defined as an expression handler, and new ones can be introduced by users. Several specialized constraint types or constraint handlers in SCIP terminology were also removed, using the expression framework with a generic nonlinear constraint instead.

# The Julia wrapper initial framework

As a Lisp-inspired language, (some would even a Lisp dialect), Julia is a homoiconic language: valid Julia code can always be represented and stored in a primitive data structure. In this case, the tree-like structure is Expr with fields head and args:

julia> expr = :(3 + 1/x)
:(3 + 1 / x)

:call

julia> expr.args
3-element Vector{Any}:
:+
3
:(1 / x)


The SCIP.jl wrapper recursively destructures the Julia expression and builds up corresponding SCIP expressions, a SCIP data structure defined either as a leaf (a simple value or a variable) or as an operand and a number of subexpressions. This is done through a push_expr! function which either:

• Creates and returns a single variable expression if the expression is a variable
• Creates and returns a single value expression if the expression is a constant
• If the expression is a function f(arg1, arg2...), calls push_expr! on all arguments, and then creates and returns the SCIP expression corresponding to f.

One part remains problematic, imagine an expression like 3 * exp(x) + 0.5 * f(4.3), where f is not a primitive supported by SCIP. It should not have to be indeed, because that part of the expression could be evaluated at expression compile-time. But if one is walking down the expression sub-parts, there was no way to know that a given part is a pure value, the expression-constructing procedure would first create a SCIP expression for 4.3 and then try to find a function for f to apply with this expression pointer as argument. This was the use case initially reported in this issue at a time when SCIP did not support trigonometric functions yet.

Another motivation for solving this issue is on the computational and memory burden. Imagine your expression is now 3 * exp(x) + 0.1 * cos(0.1) + 0.2 * cos(0.2) + ... + 100.0 * cos(100.0). This will require producing 2 * 1000 expressions for a constant, declared, allocated and passed down to SCIP. The solver will then likely preprocess all constant expressions to reduce them down, so it ends up being a lot of work done on one end to undo immediately on the other.

# A lazified expression declaration

Make push_expr! return two values (scip_expr, pure_value), with the second being a Boolean for whether the expression is a pure value or not. At any leaf computing f(arg1, arg2...).

If the expression of all arguments are pure_value, do not compute the expression and just return a null pointer, pure_value is true for this expression.

If at least one of the arguments is not a pure_value, we need to compute the actual expression. None of the pure_value arguments were declared as SCIP expressions yet, we create a leaf value expression for them with Meta.eval(arg_i). The non-pure value arguments already have a correct corresponding SCIP expression pointer. pure_value is false for this expression.

Note here that we are traversing some sub-expressions twice, once when walking down the tree and once more hidden with Meta.eval(arg_i) which computes the value for said expression, where we delegate the expression value computation to Julia. An alternative would be to return a triplet from every push_expr! call (expr_pointer, pure_value, val) and evaluate at each pure_value node the value of f(args...), with the value of the arguments already computed. This would however complexity the code in the wrapper with no advantage of the runtime, the expression evaluation is not a bottleneck for expressions that can realistically be tackled by a global optimization solver like SCIP.

##### Mathieu Besançon
###### Researcher in mathematical optimization

Mathematical optimization, scientific programming and related.