%************************************************
\chapter{Incremental Constraint Modelling}\label{ch:incremental}
%************************************************

\glsreset{rbmo}
\glsreset{incremental-rewriting}

\input{chapters/5_incremental_preamble}

\section{Modelling of Restart Based Meta-Optimisation}\label{sec:6-modelling}

This section introduces a \minizinc{} extension that enables modellers to define \gls{meta-optimisation} algorithms in \minizinc{}.
This extension is based on the construct introduced in \minisearch{} \autocite{rendl-2015-minisearch}, as summarised below.

\subsection{Meta-Optimisation in MiniSearch}\label{sec:6-minisearch}

\minisearch{} introduced a \minizinc{} extension that enables modellers to express \gls{meta-optimisation} inside a \minizinc{} model.
A \gls{meta-optimisation} algorithm in \minisearch{} typically solves a given \minizinc{} model, performs some calculations on the solution, adds new \constraints{} and then solves again.

Most \gls{meta-optimisation} definitions in \minisearch{} consist of two parts.
The first part is a declarative definition of any restriction to the \gls{search-space} that the \gls{meta-optimisation} algorithm can apply, called a \gls{neighbourhood}.
In \minisearch{} these definitions can make use of the following function.

\begin{mzn}
	function int: sol(var int: x)
\end{mzn}

It returns the value that variable \mzninline{x} was assigned to in the previous \gls{sol} (similar functions are defined for Boolean, float and set variables).
This allows the \gls{neighbourhood} to be defined in terms of the previous \gls{sol}.
In addition, a \gls{neighbourhood} definition will typically make use of the random number generators available in \minizinc{}.

\Cref{lst:6-lns-minisearch-pred} shows a simple random \gls{neighbourhood}.
For each decision variable \mzninline{x[i]}, it draws a random number from a uniform distribution. 
If it exceeds threshold \mzninline{destr_rate}, then it posts \constraints{} forcing \mzninline{x[i]} to take the same value from previous \gls{sol}.
For example, \mzninline{uniform_neighbourhood(x, 0.2)} would result in each variable in the \gls{array} \mzninline{x} having a 20\% chance of being unconstrained, and an 80\% chance of being assigned to the value it had in the previous \gls{sol}.

\begin{listing}
	\mznfile{assets/listing/6_lns_minisearch_pred.mzn}
	\caption{\label{lst:6-lns-minisearch-pred} A simple random \gls{lns} predicate implemented in \minisearch{}}
\end{listing}

\begin{listing}
	\mznfile{assets/listing/6_lns_minisearch.mzn}
	\caption{\label{lst:6-lns-minisearch} A simple \gls{lns} \gls{meta-optimisation} implemented in \minisearch{}}
\end{listing}

The second part of a \minisearch{} \gls{meta-optimisation} is the \gls{meta-optimisation} algorithm itself.
\Cref{lst:6-lns-minisearch} shows a basic \minisearch{} implementation of a basic \gls{lns}, called \mzninline{lns}.
It performs a fixed number of iterations, each invoking the \gls{neighbourhood} predicate \mzninline{uniform_neighbourhood} in a fresh scope.
This means that the \constraints{} only affect the current loop iteration.
It then searches for a solution (\mzninline{minimize_bab}) with a given timeout.
If the search does return a new solution, then it commits to that solution and it becomes available to the \mzninline{sol} function in subsequent iterations.
The \mzninline{lns} function also posts the \constraint{} \mzninline{obj < sol(obj)}, ensuring the objective value in the next iteration is strictly better than that of the current solution.

Although \minisearch{} enables the modeller to express \glspl{neighbourhood} in a declarative way, the definition of the \gls{meta-optimisation} algorithm is rather unintuitive and difficult to debug, leading to unwieldy code for defining even simple algorithm.
Furthermore, the \minisearch{} implementation requires either a close integration of the backend \solver{} into the \minisearch{} system, or it drives the solver through the regular text file based \flatzinc{} interface.
This leads to a significant communication overhead.

To address these two issues, we propose to keep modelling \glspl{neighbourhood} as predicates, but define \gls{meta-optimisation} algorithm from an imperative perspective.

We define a few additional \minizinc{} \glspl{annotation} and functions that 

\begin{itemize}
	\item allow us to express important aspects of the meta-optimisation algorithm in a more convenient way,
	\item and enable a simple \gls{rewriting} scheme that requires no additional communication with and only small, simple extensions of the target \solver{}.
\end{itemize}

\subsection{Restart Annotation}

Instead of the complex \minisearch{} definitions, we propose to add support for \glspl{meta-optimisation} that are purely based on the notion of \glspl{restart}.
A \gls{restart} happens when a solver abandons its current search efforts, returns to its initial \gls{search-space}, and begins a new exploration.
Many \gls{cp} \solvers{} already provide support for controlling their restarting behaviour.
They can periodically restart after a certain number of nodes, or restart for every \gls{sol}.
Typically, \solvers{} also support posting additional \constraints{} upon restarting that are only valid for the particular \gls{restart} (\eg{} Comet \autocite{michel-2005-comet}).
These \constraints{} are ``retracted'' for the next \gls{restart}.

In its simplest form, we can therefore implement \gls{lns} by merely specifying a \gls{neighbourhood} predicate.
We then annotate the \mzninline{solve} item with the following \gls{annotation} to indicate the predicate should be invoked upon each restart.

\begin{mzn}
	solve ::on_restart(my_neighbourhood) minimize cost;
\end{mzn}

Note that \minizinc{} currently does not support passing functions or predicates as arguments.
Calling the predicate, as in \mzninline{::on_restart(my_neighbourhood())}, would not have the correct semantics. 
The predicate needs to be called for each restart.
As a workaround, we currently pass the name of the predicate to be called for each restart as a string (see the definition of the new \mzninline{on_restart} annotation in \cref{lst:6-restart-ann}).

The second component of our \gls{lns} definition is the ``restarting strategy'', defining how much effort the \solver{} should put into each \gls{neighbourhood} (\ie{} \gls{restart}), and when to stop the overall search. We propose adding new search \glspl{annotation} to the \minizinc{} to control this behaviour.
The proposed \glspl{annotation} are shown in \cref{lst:6-restart-ann}.

The \mzninline{restart_on_solution} annotation tells the solver to restart immediately for each \gls{sol}, rather than looking for the best one in each restart.
The \mzninline{restart_without_objective} tells it not to add \gls{bnb} \constraints{} on the \gls{objective}.
The other \mzninline{restart_X} annotations define different strategies for restarting the search when no \gls{sol} is found.
The \mzninline{timeout} annotation gives an overall time limit for the search.
Finally, the \mzninline{restart_limit} stops the search after a fixed number of \glspl{restart}.

\begin{listing}
	\mznfile{assets/listing/6_restart_ann.mzn}
	\caption{\label{lst:6-restart-ann} New annotations to control the restarting
		behaviour}
\end{listing}

\subsection{Advanced Meta-Optimisation Algorithms}

Although using just a \gls{restart} \glspl{annotation} by themselves allows us to run the basic \gls{lns}, more advanced \gls{meta-optimisation} algorithms will require more than reapplying the same \gls{neighbourhood} time after time.
It is, for example, often beneficial to use several \gls{neighbourhood} definitions within the same \instance{}.
Different \glspl{neighbourhood} may be able to improve different aspects of a \gls{sol} at different phases of the search.
Adaptive \gls{lns} \autocite{ropke-2006-adaptive, pisinger-2007-heuristic} is a variant of \gls{lns} that keeps track of the \glspl{neighbourhood} that led to improvements and favours them for future iterations.
This has proven to be a highly successful approach to \gls{lns}.
However, even a simpler scheme that applies several \glspl{neighbourhood} in a round-robin fashion can be effective.

In \minisearch{}, these adaptive or round-robin approaches can be implemented using ``state variables'', which support destructive update (overwriting the value they store).
In this way, the \minisearch{} strategy can store values to be used in later iterations.
We use the ``\solver{} state'' instead.
All values are stored in normal \variables{}.
We then access them using two simple functions that reveal the solver state of the previous \gls{restart}.
This approach is sufficient for expressing many \gls{meta-optimisation} algorithms, and its implementation is much simpler.

\paragraph{State access and initialisation}

The state access functions are defined in \cref{lst:6-state-access}.
Function \mzninline{status} returns the status of the previous restart, namely: 

\begin{description}
	\item[\mzninline{START}] there has been no \gls{restart} yet;
	\item[\mzninline{UNSAT}] the \gls{search-space} of the last \gls{restart} was \gls{unsat};
	\item[\mzninline{SAT}] the last \gls{restart} found a \gls{sol};
	\item[\mzninline{OPT}] the last \gls{restart} found and proved an \gls{opt-sol} in its \gls{search-space};
	\item[\mzninline{UNKNOWN}] the last \gls{restart} did not find a solution, but did not exhaust its \gls{search-space}.
\end{description}

\noindent{}Function \mzninline{last_val} allows modellers to access the last value assigned to a \variable{}.
The value is undefined when \mzninline{status()=START}.
Like \mzninline{sol}, it has versions for all basic \variable{} types.

\begin{listing}
	\mznfile{assets/listing/6_state_access.mzn}
	\caption{\label{lst:6-state-access} Functions for accessing previous \solver{} states}
\end{listing}

In order to be able to initialise the \variables{} used for state access, we interpret \mzninline{on_restart} also before initial search using the same semantics.
As such, the predicate is also called before the first ``real'' \gls{restart}.
Any \constraint{} posted by the predicate will be retracted for the next \gls{restart}.

\paragraph{Parametric neighbourhood selection predicates}

We define standard \gls{neighbourhood} selection strategies as predicates that are parametric over the \glspl{neighbourhood} they should apply.
For example, we can define a strategy \mzninline{basic_lns} that applies an \gls{lns} \gls{neighbourhood}.
Since \mzninline{on_restart} now also includes the initial search, we apply the \gls{neighbourhood} only if the current status is not \mzninline{START}, as shown in the following predicate.

\begin{mzn}
	predicate basic_lns(var bool: nbh) = (status()!=START -> nbh);
\end{mzn}

In order to use this predicate with the \mzninline{on_restart} annotation, we cannot simply call a \gls{neighbourhood} predicate as follows.

\begin{mzn}
	basic_lns(uniform_neighbourhood(x, 0.2))
\end{mzn}

\noindent{}Calling \mzninline{uniform_neighbourhood} like this would result in a single evaluation of the predicate, since \minizinc{} employs a call-by-value evaluation strategy.
Furthermore, the \mzninline{on_restart} annotation only accepts the name of a nullary predicate.
Therefore, users have to define their overall strategy in a new predicate.
\Cref{lst:6-basic-complete} shows a complete example of a basic \gls{lns} model.

\begin{listing}
	\mznfile{assets/listing/6_basic_complete.mzn}
	\caption{\label{lst:6-basic-complete} Complete \gls{lns} example}
\end{listing}

We can also define round-robin and adaptive strategies using these primitives.
\Cref{lst:6-round-robin} defines a round-robin \gls{lns} meta-heuristic, which cycles through a list of \mzninline{N} neighbourhoods \mzninline{nbhs}.
To do this, it uses the \variable{} \mzninline{select}.
In the initialisation phase (\mzninline{status()=START}), \mzninline{select} is set to \mzninline{-1}, which means none of the \glspl{neighbourhood} is activated.
In any following \gls{restart}, \mzninline{select} is incremented modulo \mzninline{N}, by accessing the last value assigned in a previous \gls{restart}.
This will activate a different \gls{neighbourhood} for each subsequent \gls{restart} (\lref{line:6:roundrobin:post}).

\begin{listing}
	\mznfile{assets/listing/6_round_robin.mzn}
	\caption{\label{lst:6-round-robin} A predicate providing the round-robin
		meta-heuristic}
\end{listing}

%\paragraph{Adaptive \gls{lns}}

For adaptive \gls{lns}, a simple strategy is to change the size of the \gls{neighbourhood} depending on whether the previous size was successful or not.
\Cref{lst:6-adaptive} shows an adaptive version of the \mzninline{uniform_neighbourhood} that increases the number of free \variables{} when the previous \gls{restart} failed, and decreases it when it succeeded, within the bounds \([0.6,0.95]\).

\begin{listing}
	\mznfile{assets/listing/6_adaptive.mzn}
	\caption{\label{lst:6-adaptive} A simple adaptive neighbourhood}
\end{listing}

\subsection{Optimisation strategies}

The \gls{meta-optimisation} algorithms we have seen so far rely on the default behaviour of \minizinc{} \solvers{} to use \gls{bnb} for optimisation: when a new \gls{sol} is found, the \solver{} adds a \constraint{} to the remainder of the search to only accept better \glspl{sol}, as defined by the \gls{objective} in the \mzninline{minimize} or \mzninline{maximize} clause of the \mzninline{solve} item.
When combined with \glspl{restart} and \gls{lns}, this is equivalent to a simple hill-climbing meta-heuristic.

We can use the constructs introduced above to implement alternative meta-heuristics such as simulated annealing.
We use the \mzninline{restart_without_objective} \gls{annotation}, in particular, to tell the solver not to add the \gls{bnb} \constraint{} on \gls{restart}.
It will still use the declared \gls{objective} to decide whether a new solution is globally the best one seen so far, and only output those.
This maintains the convention of \minizinc{} \solvers{} that the last \gls{sol} printed at any point in time is the currently best known one.

With \mzninline{restart_without_objective}, the \gls{restart} predicate is now responsible for constraining the \gls{objective}.
Note that a simple hill-climbing (for minimisation) can still be defined easily in this context as follows.

\begin{mzn}
	predicate hill_climbing() = status() != START -> _objective < sol(_objective);
\end{mzn}

It takes advantage of the fact that the declared \gls{objective} is available through the built-in \variable{} \mzninline{_objective}.
A more interesting example is a simulated annealing strategy.
When using this strategy, the \glspl{sol} that the \solver{} finds are no longer required to steadily improve in quality.
Instead, we ask the \solver{} to find a \gls{sol} that is a significant improvement over the previous \gls{sol}.
Over time, we decrease the amount by which we require the \gls{sol} needs to improve until we are just looking for any improvements.
This \gls{meta-optimisation} can help improve the qualities of \gls{sol} quickly and thereby reaching the \gls{opt-sol} quicker.
\Cref{lst:inc-sim-ann} show how this strategy is also easy to express using \gls{rbmo}.

\begin{listing}
	\mznfile{assets/listing/inc_sim_ann.mzn}
	\caption{\label{lst:inc-sim-ann}A predicate implementing a simulated annealing search.}
\end{listing}

So far, the algorithms used have been for versions of incomplete search or we have trusted the \solver{} to know when to stop searching.
However, for the following algorithms the \solver{} will (or should) not be able to determine whether the search is complete.
Instead, we introduce the following function that can be used to signal to the solver that the search is complete.

\begin{mzn}
	function var bool: complete();
\end{mzn}

\noindent{}When the result of this function is said to be \mzninline{true}, then search is complete.
If any \gls{sol} was found, it is declared an \gls{opt-sol}.

Using the same methods it is also possible to describe optimisation strategies with multiple \glspl{objective}.
An example of such a strategy is lexicographic search.
Lexicographic search can be employed when there is a strict order between the importance of different \variables{}.
It required that, once a \gls{sol} is found, each subsequent \gls{sol} must either improve the first \gls{objective}, or have the same value for the first \gls{objective} and improve the second \gls{objective}, or have the same value for the first two \glspl{objective} and improve the third \gls{objective}, and so on.
A predicate that implements lexicographic search is shown in \cref{lst:inc-lex-min}.

\begin{listing}
	\mznfile{assets/listing/inc_lex_min.mzn}
	\caption{\label{lst:inc-lex-min}A predicate implementing lexicographic search.}
\end{listing}

The lexicographic search changes the \gls{objective} at each stage in the evaluation.
Initially, the search is in stage one: it tries to find an \gls{opt-sol} for the first \gls{objective}.
Whenever we find a \gls{sol} we stay in the same stage, but constrain the \gls{search-space} to find a better \gls{sol}.
When there exists no better \gls{sol}, we move on to the next stage, trying to find the \gls{opt-sol} for the next \gls{objective} while the earlier \glspl{objective} maintain their value.
This process is repeated until reach the final \gls{objective}.
When we reach a stage without an associated \gls{objective}, we are guaranteed to have reached the lexicographic optimal.
Our search is thus complete.

There is not always a clear order of importance for different \glspl{objective} in a problem.
We can instead look for a number of diverse \glspl{sol} and allow the user to pick the most acceptable options.
The predicate in \cref{lst:inc-pareto} shows a \gls{meta-optimisation} for the Pareto optimality of a pair of \glspl{objective}.

\begin{listing}
	\mznfile{assets/listing/inc_pareto.mzn}
	\caption{\label{lst:inc-pareto}A predicate implementing pareto optimality search for two \glspl{objective}.}
\end{listing}

In this implementation we keep track of the number of \glspl{sol} found so far using \mzninline{nsol}.
There is a maximum number we can handle, \mzninline{ms}.
At the start the number of solutions is zero.
If we find no \glspl{sol}, then we finish the entire search.
Otherwise, we record that we have one more \gls{sol}.
We store the \gls{sol} values in \mzninline{s1} and \mzninline{s2} \glspl{array}.
Before each \gls{restart} we add \constraints{} removing Pareto dominated \glspl{sol}, based on each previous \gls{sol}.

\section{Rewriting of Meta-Optimisation Algorithms}\label{sec:6-solver-extension}

The \glspl{neighbourhood} defined in the previous section can be executed with \minisearch{} by adding support for the \mzninline{status} and \mzninline{last_val} built-in functions, and by defining the main \gls{restart} loop.
The \minisearch{} evaluator will then call a \solver{} to produce a \gls{sol}, and evaluate the \gls{neighbourhood} predicate, incrementally producing new \flatzinc{} to be added to the next round of solving.

While this is a viable approach, our goal is to keep \gls{rewriting} and solving separate, by embedding the entire \gls{meta-optimisation} algorithms into the \gls{slv-mod}.

This section introduces such a \gls{rewriting} approach.
It only requires simple modifications of the \gls{rewriting} process, and the produced \gls{slv-mod} can be executed by standard \gls{cp} \solvers{} with a small set of simple extensions.

\subsection{Rewriting Overview}

The \gls{neighbourhood} definitions from the previous section have an important property that makes them easy to rewrite: they are defined in terms of standard \minizinc{} expressions, except for a few new functions.
When the \gls{neighbourhood} predicates are evaluated in the \minisearch{} way, the \minisearch{} \interpreter{} implements those functions.
It has to compute the correct value whenever a predicate is evaluated.

Instead, the \gls{rewriting} scheme presented below uses a limited form of partial evaluation: \parameters{}, known during \gls{rewriting} will be fully evaluated; those only known during the solving, such as the result of a call to any of the new functions (\mzninline{sol}, \mzninline{status}, etc.), are replaced by \variables{}.
This essentially \textbf{turns the new functions into \gls{native} \constraints{}} for the target \solver{}.
The \gls{neighbourhood} predicate can then be added as a \constraint{} to the \gls{slv-mod}.
The evaluation is performed by hijacking the solver's own capabilities: It will automatically perform the evaluation of the new functions by propagating the \gls{native} \constraints{}.

To compile a \gls{meta-optimisation} algorithms to a \gls{slv-mod}, the \gls{rewriting} process performs the following four simple steps.

\begin{enumerate}
	\item It replaces the annotation \mzninline{::on_restart("X")} with a call to predicate \mzninline{X}.
	\item Inside predicate \mzninline{X} and any other predicate called recursively from \mzninline{X}: it must treat any call to functions \mzninline{sol}, \mzninline{status}, and \mzninline{last_val} as returning a \variable{}; and rename calls to random functions, e.g., \mzninline{uniform} to \mzninline{uniform_slv}, in order to distinguish them from their standard library versions.
	\item It converts any expression containing a call from step 2 to be a \variable{} to ensure the functions are compiled as \constraints{}, rather than directly evaluated during \gls{rewriting}.
	\item It rewrites the resulting model using an extension of the \minizinc{} standard library that provides declarations for these functions, as defined in the following of subsection.
\end{enumerate}

These transformations will not change the code of many \gls{neighbourhood} definitions, since the functions are often used in positions that accept both \parameters{} and \variables{}.
For example, the \mzninline{uniform_neighbourhood} predicate from \cref{lst:6-lns-minisearch-pred} uses \mzninline{uniform(0.0, 1.0)} in an \mzninline{if} expression, and \mzninline{sol(x[i])} in an equality \constraint{}.
Both expressions can be rewritten when the functions return a \variable{}.

\subsection{Rewriting the new functions}

We can rewrite \instances{} that contain the new functions by extending the \minizinc{} standard library using the following definitions.

\paragraph{\mzninline{status}}

\Cref{lst:6-status} shows the definition of the \mzninline{status} function.
It simply replaces the functional form by a predicate \mzninline{status} (declared in \lref{line:6:status}), which constrains its local variable argument \mzninline{stat} to take the status value.

\begin{listing}
	\mznfile{assets/listing/6_status.mzn}
	\caption{\label{lst:6-status} MiniZinc definition of the \mzninline{status} function}
\end{listing}

\paragraph{\mzninline{sol} and \mzninline{last_val}}

The \mzninline{sol} function is overloaded for different types.
Our \glspl{slv-mod} does not support overloading.
Therefore, we produce type-specific \gls{native} \constraints{} for every type of \gls{native} \variable{} (\eg{} \mzninline{int_sol(x, xi)}, and \mzninline{bool_sol(x, xi)}).
The resolving of the \mzninline{sol} function into these specific \gls{native} \constraints{} is done using an overloaded definition, like the one shown in \cref{lst:6-int-sol} for integer \variables{}.
If the value of the \variable{} has become known during \gls{rewriting}, then we use its \gls{fixed} value instead.
Otherwise, the function call is rewritten into a type specific \mzninline{int_sol} \gls{native} \constraint{} that will be executed by the solver.
We use the current \domain{} of the \variable{}, \mzninline{dom(x)}, as the \domain{} the \variable{} returned by \mzninline{sol}.

\begin{listing}
	\mznfile{assets/listing/6_sol_function.mzn}
	\caption{\label{lst:6-int-sol} MiniZinc definition of the \mzninline{sol}
		function for integer variables}
\end{listing}

The compilation of \mzninline{last_val} is similar to that for \mzninline{sol}.

\paragraph{Random number functions}

Calls to the random number functions have been renamed by appending \texttt{\_slv}, so that they are not simply evaluates directly.
The definition of these new functions follows the same pattern as for \mzninline{sol}, \mzninline{status}, and \mzninline{last_val}.
The \minizinc{} definition of the \mzninline{uniform_nbh} function is shown in \Cref{lst:6-int-rnd}\footnote{Random number functions need to be marked as \mzninline{::impure} for the compiler not to apply \gls{cse} when they are called multiple times with the same arguments.}.
Note that the function accepts \variable{} as its arguments \mzninline{l} and \mzninline{u}.
As such, it can be used in combination with other functions, such as \mzninline{sol}.

\begin{listing}
	\mznfile{assets/listing/6_uniform_slv.mzn}
	\caption{\label{lst:6-int-rnd} MiniZinc definition of the
		\mzninline{uniform_nbh} function for floats}
\end{listing}

\paragraph{\mzninline{complete}}

The implementation of the \mzninline{complete} function should merely always return the same Boolean \variable{}.
However, since functions returning Boolean \variables{} are equivalent to a predicate, it is actually the \gls{reif} of \mzninline{complete} that gets used.
As such, we can define \mzninline{complete} as shown in \cref{lst:inc-complete}.
Note that the \gls{slv-mod} will actually include the \mzninline{complete_reif} \gls{native} \constraint{}.

\begin{listing}
	\mznfile{assets/listing/inc_complete.mzn}
	\caption{\label{lst:inc-complete} MiniZinc definition of the \mzninline{complete} function.}
\end{listing}

\subsection{Solver Support for Restart Based Native Constraints}

We will now show the minimal extensions required from a \solver{} to interpret the new \gls{native} \constraints{} and, consequently, to execute \gls{meta-optimisation} definitions expressed in \minizinc{}.

First, the \solver{} needs to parse and support the \gls{restart} \glspl{annotation} of \cref{lst:6-restart-ann}.
Many \solvers{} already support all this functionality.
Second, the \solver{} needs to be able to parse the new \constraints{} \mzninline{status}, and all versions of \mzninline{sol}, \mzninline{last_val}, random number functions like \mzninline{float_uniform}, and \mzninline{complete}.
In addition, for the new \constraints{} the \solver{} is extended using the following algorithms.

\begin{description}
	\item[\mzninline{status(s)}] Record the status of the previous \gls{restart}, and fix \mzninline{s} to the recorded status.
	\item[\mzninline{sol(x, sx)} (variants)] Constrain \mzninline{sx} to be equal to the value of \mzninline{x} in the incumbent solution.
		If there is no incumbent solution, it has no effect.
	\item[\mzninline{last_val(x, lx)} (variants)] Constrain \mzninline{lx} to take the last value assigned to \mzninline{x} during search.
		If no value was ever assigned, it has no effect.
		Note that many solvers (in particular \gls{sat} and \gls{lcg} solvers) already track \mzninline{last_val} for their \variables{} for use in search.
		To support \gls{rbmo} a \solver{} must at least track the last value of each of the \variables{} involved in such a \constraint{}.
		This is straightforward by using a \mzninline{last_val} \gls{propagator}.
		It wakes up whenever its argument is fixed, and updates the last value (a non-\gls{backtrack}able value).
	\item[Random number functions] Fix their \variable{} argument to a random number in the appropriate probability distribution.
	\item[\mzninline{complete_reif(c)}] Stop and mark to search as ``complete'' when its \variable{} argument is forced to be true.
\end{description}

Importantly, these \constraints{} need to be propagated in a way that their effects can be undone for the next \gls{restart}.
Typically, this means the solver must not propagate these \constraints{} until it can \gls{backtrack} it changes.

Modifying a solver to support this functionality is straightforward if it already has a mechanism for posting \constraints{} during restarts.
We have implemented these extensions for both \gls{gecode} (214 new lines of code) and \gls{chuffed} (173 new lines of code).

For example, consider the model from \cref{lst:6-basic-complete} again that uses the following \gls{meta-optimisation} algorithms.

\begin{mzn}
	basic_lns(uniform_neighbourhood(x, 0.2))}
\end{mzn}

\Cref{lst:6-flat-pred} shows a part of the resulting \gls{slv-mod}.
\Lrefrange{line:6:status:start}{line:6:status:end} define a Boolean variable \mzninline{b1} that is true if-and-only-if the status is not \mzninline{START}.
The second block of code, \lrefrange{line:6:x1:start}{line:6:x1:end}, represents the \gls{decomp} of the following expression.

\begin{mzn}
	(status() != START /\ uniform(0.0, 1.0) > 0.2) -> x[1] = sol(x[1])
\end{mzn}

It is the result of merging the implication from the \mzninline{basic_lns} predicate with the \mzninline{if} expression from \mzninline{uniformNeighbourhood}.
The code first introduces and constrains a variable for the random number, then adds two Boolean \variables{}: \mzninline{b2} is constrained to be true if-and-only-if the random number is greater than 0.2; while \mzninline{b3} is constrained to be the conjunction of the two.
\Lref{line:6:x1:sol} constrains \mzninline{x1} to be the value of \mzninline{x[1]} in the previous \gls{sol}.
Finally, the \gls{half-reified} \constraint{} in \lref{line:6:x1:end} implements the following expression.

\begin{mzn}
	b3 -> x[1] = sol(x[1])
\end{mzn}

We have omitted the similar code generated for \mzninline{x[2]} to \mzninline{x[n]}.
Note that the \flatzinc{} \gls{slv-mod} shown here has been simplified for presentation.

\begin{listing}
	\mznfile{assets/listing/6_basic_complete_transformed.mzn}
	\caption{\label{lst:6-flat-pred} \flatzinc{} that results from compiling \\
		\mzninline{basic_lns(uniformNeighbourhood(x,0.2))}.}
\end{listing}

The first time the \solver{} is invoked, it sets \mzninline{s} to one (\mzninline{START}).
\Gls{propagation} will fix \mzninline{b1} to \mzninline{false} and \mzninline{b3} to \mzninline{false}.
Therefore, the implication in \lref{line:6:x1:end} is not activated, leaving \mzninline{x[1]} unconstrained.
The \gls{neighbourhood} \constraints{} are effectively switched off.

When the \solver{} \glspl{restart}, all the special \glspl{propagator} are re-executed.
Now \mzninline{s} is not one, and \mzninline{b1} is set to \mzninline{true}.
The \mzninline{float_random} \gls{propagator} assigns \mzninline{rnd1} a new random value and, depending on whether it is greater than \mzninline{0.2}, the Boolean \variables{} \mzninline{b2}, and consequently \mzninline{b3} is assigned.
If it is \mzninline{true}, the \constraint{} in \lref{line:6:x1:end} becomes active and assigns \mzninline{x[1]} to its value in the previous \gls{sol}.

Furthermore, it is not strictly necessary to guard \mzninline{int_uniform} against being invoked before the first solution is found.
The \mzninline{sol} \constraints{} will simply not propagate anything in case no \gls{sol} has been recorded yet, but we use this simple example to illustrate how these Boolean conditions are rewritten and evaluated.

\section{An Incremental Constraint Modelling Interface}%
\label{sec:6-incremental-compilation}

A universal approach to the incremental usage of \cmls{}, and \gls{meta-optimisation}, is to allow incremental changes to an \instance{} of a \cmodel{}.
To solve these changing \instances{}, they have to be rewritten repeatedly to \glspl{slv-mod}.
In this section we extend our architecture with an incremental constraint modelling interface that allows the modeller to change an \instance{}.
For changes made using this interface, the architecture can employ \gls{incremental-rewriting} to minimise the required \gls{rewriting}.

As such, the \microzinc{} \interpreter{} is extended to be able \textbf{add} and \textbf{remove} \nanozinc{} \constraints{} from/to an existing \nanozinc{} program.
Adding new \constraints{} is straightforward.
\nanozinc{} is already processed one \constraint{} at a time, in any order.
The new \constraints{} can be added to the program, and the \gls{rewriting} can proceed as normal.

Removing a \constraint{}, however, is not so simple.
When we remove a \constraint{}, all effects the \constraint{} had on the \nanozinc{} program have to be undone.
This includes results of \gls{propagation}, \gls{cse} and other simplifications.

\begin{example}\label{ex:6-incremental}
	Consider the following \minizinc\ fragment:

	\begin{mzn}
		constraint x < 10;
		constraint y < x;
	\end{mzn}

	After evaluating the first \constraint{}, the \domain{} of \mzninline{x} is changed to be less than ten.
	Evaluating the second \constraint{} causes the \domain{} of \mzninline{y} to be less than nine.
	If we now, however, try to remove the first \constraint{}, then it is not just the direct inference on the \domain{} of \mzninline{x} that has to be undone, but also any further effects of those changes --- in this case, the changes to the \domain{} of \mzninline{y}.

\end{example}

The addition of \constraints{} that can later be removed, is similar to making \glspl{search-decision} in a \gls{cp} \solver{} (see \cref{subsec:back-cp}).
As such, we can support the removal of \constraints{} in reverse chronological order by \gls{backtrack}ing. 
The \microzinc{} \interpreter{} maintains \gls{trail} of the changes to be undone when a \constraint{} is removed.
The \gls{trail} records all changes to the \nanozinc{} program:

\begin{itemize}
	\item the addition or removal of new \variables{} or \constraints{},
	\item changes made to the \domains{} of \variables{},
	\item additions to the \gls{cse} table, and
	\item substitutions made due to equality \gls{propagation}.
\end{itemize}

These changes can be caused by the \gls{rewriting} of a \constraint{}, \gls{propagation}, or \gls{cse}.
When a \constraint{} is removed, its corresponding changes are undone by reversing all actions recorded on the \gls{trail} up to the point where the \constraint{} was added.

In order to limit the amount of \gls{trail}ing required, the programmer must create explicit ``choice points'' to which the system state can be reset.
In particular, this means that if no choice point was created before the initial \instance{} was rewritten, then this \gls{rewriting} can be performed without any \gls{trail}ing.

\begin{example}\label{ex:6-trail}

	Let us once again consider the following \minizinc{} fragment from \cref{ex:rew-absreif}.

	\begin{mzn}
		constraint abs(x) > y \/ c;
		constraint c;
	\end{mzn}

	After rewriting it results in the following \nanozinc{} program.

	\begin{nzn}
		var true: c;
		var -10..10: x;
		var -10..10: y;
	\end{nzn}

	Assume that we added a choice point before posting the \constraint{} \mzninline{c}.
	Then the \gls{trail} stores the inverse of all modifications that were made to the \nanozinc{} as a result of \mzninline{c}.
	The following code fragment illustrates the elements in this \gls{trail}.
	The reversal actions are annotated on the different elements.
	The \mzninline{attach} and \mzninline{detach} actions signal the \interpreter{} to attach/detach a \constraint{} to the argument \variable{}, or the global \cmodel{} in case the argument is \mzninline{true}.
	The \mzninline{add} action tells to \interpreter{} to add a \variable{} to the \nanozinc{} program.
	Finally, the \mzninline{set_domain} action makes the \interpreter{} restore the \domain{} using declaration to which it was attached.

	\begin{nzn}
		% Posted c
		constraint c ::detach(true); 
		% Propagated c = true
		var bool: c ::set_domain; 
		constraint c ::attach(true); 
		% Simplified bool_or(b1, true) = true
		constraint bool(b1, c) ::attach(true);
		% b became unused...
		constraint int_gt_reif(z, y, b) ::attach(b); 
		var bool: b ::add;
		% then z became unused 
		constraint int_abs(x, z) ::attach(z);
		var int: z ::add;
	\end{nzn}

	To reconstruct the \nanozinc{} program at the choice point, we simply apply the changes recorded in the \gls{trail}, in reverse order.
\end{example}

\subsection{Incremental Solving}

Ideally, the incremental changes made by the \interpreter{} would also be applied incrementally to the \solver{}.
This requires the \solver{} to support both the dynamic addition and removal of \variables{} and \constraints{}.
While some solvers can support this functionality, most \solvers{} have limitations.
We therefore define the following three interfaces, using which we can apply the changes to the \solvers{}.

\begin{itemize}

	\item Using a non-incremental interface, the \solver{} is reinitialised with the updated \nanozinc\ program every time.
		In this case, we still get a performance benefit from the improved \gls{rewriting} time, but not from incremental solving.

	\item Using a \textit{warm-starting} interface, the \solver{} is reinitialised with the updated \gls{slv-mod} as above, but it is also given a previous \gls{sol} to initialise some internal data structures.
		In particular for mathematical programming \solvers{}, this can result in dramatic performance gains compared to ``cold-starting'' the \solver{} every time.
				
	\item Using a fully incremental interface, the \solver{} is instructed to apply the changes made by the \interpreter{}.
		In this case, the \gls{trail} data structure is used to compute the set of \nanozinc{} changes since the last choice point.

\end{itemize}


\section{Experiments}\label{sec:inc-experiments}

In this section we present two experiments to test the efficiency and potency of the incremental methods introduced in this chapter.

In our first experiment, we consider the effectiveness \gls{meta-optimisation} within during solving.
In particular, we investigate a round-robin \gls{lns} implemented using \gls{rbmo}.
On three different \minizinc{} models we compare this approach with solving the \instances{} directly using 2 different \solvers{}.
For one of the \solvers{}, we also compare with an oracle approach that can directly apply the exact same \gls{neighbourhood} as our \gls{rbmo}, without the need for computation.
As such, we show that the use of \gls{rbmo} introduces a insignificant computational overhead.

Our second experiment compares the performance of using \minizinc{} incrementally. 
We compare our two methods, \gls{rbmo} and the incremental constraint modelling interface, against the baseline of continuously \gls{rewriting} and reinitialising the \solver{}.
For this comparison compare the time that is required to (repeatedly) rewrite an \instance{} and the time required by the \solver{}.
The first model contains a lexicographic objective.
The second model is shared between the two experiments and uses a round-robin \gls{lns} approach.
We compare the application of a fixed number of \glspl{neighbourhood}.

A description of the used computational environment, \minizinc{} instances, and versioned software has been included in \cref{ch:benchmarks}.

\subsection{Meta-Optimisation Solving}

We will now show that a solver that evaluates the rewritten \gls{meta-optimisation} specifications can (a) be effective and (b) incur only a small overhead compared to a dedicated implementation of the \glspl{neighbourhood}.

To measure the overhead, we implemented \gls{rbmo} in \gls{gecode} \autocite{gecode-2021-gecode}.
The resulting solver has been instrumented to also output the \domains{} of all \variables{} after propagating the new \gls{native} \constraints{}.
We implemented another extension to \gls{gecode} that simply reads the stream of \variable{} \domains{} for each \gls{restart}, essentially ``replaying'' the \gls{meta-optimisation} without incurring any overhead for evaluating the \glspl{neighbourhood} or handling the additional \variables{} and \constraints{}.
Note that this is a conservative estimate of the overhead: this extension has to perform less work than any real \gls{meta-optimisation} implementation.

In addition, we also present benchmark results for the standard release of \gls{gecode}; as well as the development version of \gls{chuffed}; and \gls{chuffed} performing \gls{rbmo}.
These experiments illustrate that the \gls{meta-optimisation} implementations indeed perform well compared to the standard \solvers{}.

We ran experiments for three models from the MiniZinc Challenge \autocite{stuckey-2010-challenge, stuckey-2014-challenge} (\texttt{gbac}, \texttt{steelmillslab}, and \texttt{rcpsp-wet}).
The \instances{} are rewritten using the \minizinc{} 2.5.5 \compiler{}, adjusted to rewrite \gls{rbmo} specifications.
For each model we use a round-robin \gls{lns} approach, shown in \cref{lst:6-round-robin}, of two \glspl{neighbourhood}: a \gls{neighbourhood} that destroys 20\% of the main \variables{} (\cref{lst:6-lns-minisearch-pred}) and a structured \gls{neighbourhood} for the model (described below).
For each solving method we show the cumulative integral of the objective values for all \instances{} of the model.
The underlying search strategy used is the fixed search strategy defined in the model.
The restart strategy uses the following \glspl{annotation}.

\begin{mzn}
	::restart_constant(250) ::restart_on_solution
\end{mzn}

The benchmarks are repeated with 10 different random seeds and the average is shown.
The overall time out for each run is 120 seconds.

\subsubsection{GBAC}%
\label{ssubsec:inc-exp-gbac1}

The \gls{gbac} problem comprises courses having a specified number of credits and lasting a certain number of periods, load limits of courses for each period, prerequisites for courses, and preferences of teaching periods for professors.
A detailed description of the problem is given in \autocite{chiarandini-2012-gbac}.
The main decisions are to assign courses to periods, which is done via the \variables{} \mzninline{period_of} in the \minizinc{} model.
\cref{lst:6-free-period} shows the neighbourhood chosen, which randomly picks one period and frees all courses that are assigned to it.

\begin{listing}[b]
	\mznfile{assets/listing/6_gbac_neighbourhood.mzn}
	\caption{\label{lst:6-free-period}\gls{gbac}: \gls{neighbourhood} freeing all courses in a period.}
\end{listing}

\begin{figure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_obj_gecode_gbac.pdf}
		\caption{\label{subfig:inc-obj-gecode-gbac}\gls{gecode}}
	\end{subfigure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_obj_chuffed_gbac.pdf}
		\caption{\label{subfig:inc-obj-chuffed-gbac}\gls{chuffed}}
	\end{subfigure}
	\caption{\label{fig:inc-obj-gbac}\gls{gbac}: integral of cumulative objective value of solving 5 \instances{}.}
\end{figure}

The result for the \gls{gbac} in \cref{fig:inc-obj-gbac} show that the overhead introduced by \gls{gecode} using \gls{rbmo} with regards to the replaying the \glspl{neighbourhood} is quite low.
The lines in the graph do not show any significant differences or delays.
Both their result are much better than the baseline \gls{gecode} \solver{}.
Since learning is not very effective for \gls{gbac}, the performance of \gls{chuffed} is similar to \gls{gecode}.
The use of \gls{lns} again significantly improves over standard \gls{chuffed}.

\subsubsection{Steel Mill Slab}

The steel mill slab design problem consists of cutting slabs into smaller ones, so that all orders are fulfilled while minimising the wastage.
The steel mill only produces slabs of certain sizes, and orders have both a size and a colour.
We have to assign orders to slabs, with at most two different colours on each slab.
The model uses the \variables{} \mzninline{assign} for deciding which order is assigned to which slab.
\Cref{lst:6-free-bin} shows a structured \gls{neighbourhood} that randomly selects a slab and frees the orders assigned to it in the incumbent \gls{sol}.
These orders can then be freely reassigned to any other slab.

\begin{listing}
	\mznfile{assets/listing/6_steelmillslab_neighbourhood.mzn}
	\caption{\label{lst:6-free-bin}Steel mill slab: \gls{neighbourhood} that frees
		all orders assigned to a selected slab.}
\end{listing}


\begin{figure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_obj_gecode_steelmillslab.pdf}
		\caption{\label{subfig:inc-obj-gecode-steelmillslab}\gls{gecode}}
	\end{subfigure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_obj_chuffed_steelmillslab.pdf}
		\caption{\label{subfig:inc-obj-chuffed-steelmillslab}\gls{chuffed}}
	\end{subfigure}
	\caption{\label{fig:inc-obj-steelmillslab}Steel mill slab: integral of cumulative objective value of solving 5 \instances{}.}
\end{figure}

\Cref{subfig:inc-obj-gecode-steelmillslab} again only show minimal overhead for the \gls{rbmo} compared to replaying the \glspl{neighbourhood}.
For this problem a solution with zero wastage is always optimal.
As such the \gls{lns} approaches are sometimes able to prove a \gls{sol} is optimal and might finish before the time out.
This is the case for \gls{chuffed} instances, where almost all \instances{} are solved using the \gls{rbmo} method.
As expected, the \gls{lns} approaches find better solutions quicker for \gls{gecode}.
However, We do see that, given enough time, baseline \gls{gecode} will eventually find better \glspl{sol}. 

% RCPSP/wet
\subsubsection{RCPSP/wet}

The \gls{rcpsp} with Weighted Earliness and Tardiness cost, is a classic scheduling problem in which tasks need to be scheduled subject to precedence \constraints{} and cumulative resource restrictions.
The objective is to find an optimal schedule that minimises the weighted cost of the earliness and tardiness for tasks that are not completed by their proposed deadline.
The \variables{} in \gls{array} \mzninline{s} represent the start times of each task in the model.
\Cref{lst:6-free-timeslot} shows our structured \gls{neighbourhood} for this model.
It randomly selects a time interval of one-tenth the length of the planning horizon and frees all tasks starting in that time interval, which allows a reshuffling of these tasks.

\begin{listing}
	\mznfile{assets/listing/6_rcpsp_neighbourhood.mzn}
	\caption{\label{lst:6-free-timeslot}\Gls{rcpsp}/wet: \gls{neighbourhood} freeing
		all tasks starting in the drawn interval.}
\end{listing}

\begin{figure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_obj_gecode_rcpspw.pdf}
		\caption{\label{subfig:inc-obj-gecode-rcpspw}\gls{gecode}}
	\end{subfigure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_obj_chuffed_rcpspw.pdf}
		\caption{\label{subfig:inc-obj-chuffed-rcpspw}\gls{chuffed}}
	\end{subfigure}
	\caption{\label{fig:inc-obj-rcpspw}\gls{rcpsp}/wet: integral of cumulative objective value of solving 5 \instances{}.}
\end{figure}

\Cref{fig:inc-obj-rcpspw} shows once more that \gls{rbmo} has little overhead over the replaying approach.
These approaches perform significantly better than the baseline \gls{gecode} \solver{}.
The baseline \gls{chuffed} \solver{} performs well and improves its \gls{sol} longer than the baseline \gls{gecode} \solver{}.
However, \gls{lns} makes \gls{chuffed} much more robust.

\subsection{Incremental Rewriting and Solving}

To demonstrate the advantage that dedicated incremental methods for \minizinc{} offer, we present a runtime evaluation of two \gls{meta-optimisation} algorithms implemented using the two methods presented in this chapter.
We consider the use of lexicographic search and round-robin \gls{lns}, each evaluated on a different \minizinc{} model.
We compare our methods against a naive system that repeatedly programmatically changes an \instance{}, rewrites the full \instances{}, and starts a new \solver{} instance.
The solving in our tests is performed by the \gls{gecode} \gls{solver}.
When using our incremental interface, it is connected using the fully incremental \gls{api}.
The naive system and incremental interface use our architecture prototype to rewrite the \instances{}.
The \gls{rbmo} \instances{} are rewritten using the \minizinc{} 2.5.5 \compiler{}, adjusted to rewrite \gls{rbmo} specifications.
We compare the time that is required to (repeatedly) rewrite an \instance{} and the time taken by the \solver{}.
Each test is performed 10 times and the average time is shown.

\begin{figure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_cmp_lex.pdf}
		\caption{\label{subfig:inc-cmp-lex}Radiation}
	\end{subfigure}
	\begin{subfigure}[b]{0.49\linewidth}
		\includegraphics[width=\columnwidth]{assets/img/inc_cmp_lns.pdf}
		\caption{\label{subfig:inc-cmp-lns}GBAC}
	\end{subfigure}
	\caption{\label{fig:inc-cmp} A comparison of the two new incremental techniques and a recompilation strategy.}
\end{figure}

\subsubsection{Radiation}

Our first model is based on a problem of planning cancer radiation therapy treatment using multi-leaf collimators \autocite{baatar-2011-radiation}.
Two characteristics mark the quality of a solution: the amount of time the patient is exposed to radiation, and the number of ``shots'' or different angles the treatment requires.
However, the first characteristic is considered more important than the second.
The problem therefore has a lexicographical objective: a solution is better if it requires a strictly shorter exposure time, or the same exposure time but a lower number of ``shots''.
\minizinc{} does not support lexicographical objectives directly, but a lexicographic search can be modelled in \minizinc{}, shown in \cref{lst:inc-lex-min}.

The results are shown in \cref{subfig:inc-cmp-lex}.
They show that both incremental methods have a clear advantage over are naive baseline approach.
Although some additional time is spend \gls{rewriting} the \gls{rbmo} models compared to the incremental, it is minimal.
The \gls{rewriting} of the \gls{rbmo} \instances{} would likely take less time when using the new prototype architecture.
Between all the methods, solving time is very similar.
\gls{rbmo} seems to have a slight advantage.
No benefit can be noticed from the use of the incremental solver \gls{api}.

\subsubsection{GBAC} 

We now revisit the model and method from \cref{ssubsec:inc-exp-gbac1} and compare the efficiency of using round-robin \gls{lns}.
Instead setting a time limit, we limit the number of \glspl{restart} that the \solver{} makes.
As such, we limit the number of \glspl{neighbourhood} that are computed and applied to the \instance{}.
It should be noted that the \gls{rbmo} method is not guaranteed to apply the exact same \glspl{neighbourhood}, due to the difference in random number generator.

\Cref{subfig:inc-cmp-lex} shows the results of applying 1000 \glspl{neighbourhood}. 
In this case there is an even more pronounced difference between the incremental methods and the naive method.
It is surprising to see that the application of 1000 \glspl{neighbourhood} using \gls{incremental-rewriting}, still performs very similar to \gls{rewriting} the \gls{rbmo} method.
Again, solving times are similar between all method.
Once again we do not see any real advantage of the use of the incremental \solver{} \gls{api}.
The advantage in solve time using \gls{rbmo} is more pronounced here, but it is hard to draw conclusions since the \glspl{neighbourhood} of this method are not exactly the same.

\subsection{Summary}

The results of our experiments show that there is a clear benefit from the use of incremental methods in \cmls{}.

The \gls{meta-optimisation} algorithms that can be applied using \gls{rbmo} show a significant improvement over the \solvers{} normal search.
It is shown this method is very efficient and does not under perform even when compared to a unrealistic version of the methods that do not require any computations.

The incremental interface offers a great alternative when \solvers{} are not extended for \gls{rbmo}.
\Gls{incremental-rewriting} saves a significant amount of time, compared to repeatedly \gls{rewriting} the full \instance{}.
In our experiments, after the initial \gls{rewriting} of the \instance{} the additional \gls{rewriting} to apply a \gls{neighbourhood} often becomes insignificant compared to the time required by the \solver{}.
However, the incremental \solver{} \gls{api}, used in this approach, did not show any benefit.
Although we are still convinced of its usefulness, it was not visible in the results of our experiments.
It might be considered whether these use cases (or its implementation) limited its effect in any way, or whether its effect will be more pronounced in other types of \solvers{}, where more information can be saved.
