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bb_min(+Goal, ?Cost, +Options)
Find one or all minimal solutions using the branch-and-bound method
- Goal
- The (nondeterministic) search goal
- Cost
- A (usually numeric domain) variable representing the cost
- Options
- A bb_options structure or variable
Description
	A solution of the goal Goal is found that minimizes
	the value of Cost.  Cost should be a
	variable that is affected, and eventually instantiated, by
	Goal.  Usually, Goal is the search procedure
	of a constraint problem and Cost is the variable
	representing the cost.  The solution is found using the branch
	and bound method:  as soon as a solution is found, it gets
	remembered and the search is continued or restarted with an
	additional constraint on the Cost variable which
	requires the next solution to be better than the previous one. 
	Iterating this process finally yields an optimal solution.
	The possible options are
	
	- strategy:
- 
	    
	    - continue (default)
- after finding a solution, continue search with the newly
		found bound imposed on Cost
- restart
- after finding a solution, restart the whole search with
		the newly found bound imposed on Cost
- step
- a synonym for 'restart'
- dichotomic
- after finding a solution, split the remaining cost range
		and restart search to find a solution in the lower sub-range.
		If that fails, assume the upper sub-range as the remaining
		cost range and split again.
 The new bound (or the split point, respectively), is computed
	    from the current best solution, taking into account the
	    parameters delta and factor.
- from:
- number - an initial lower bound for the cost (default -1.0Inf).
	    Only useful if Cost is not a domain variable.
- to:
- number - an initial upper bound for the cost (default +1.0Inf).
	    Only useful if Cost is not a domain variable.
- delta:
- number - minimal absolute improvement required for each step
	    (applies to all strategies). The default value of 1.0 is
	    appropriate for integral costs.  Any solution that improves on
	    the best solution by less than this value will be missed.
- factor:
- number - minimal improvement ratio (with respect to the lower
	    cost bound) for strategies 'continue' and 'restart' (default 1.0),
	    or split factor for strategy 'dichotomic' (default 0.5)
- solutions:
- 
	    
	    - one (default)
- 
		Compute one (of possibly multiple) optimal solutions.
- all
- 
		Nondeterministically compute all optimal solutions.
		This has a performance penalty, as the search is restarted
		one more time after the optimum has been determined.
 Note the dependence on the delta-parameter: the costs of these
	    solutions may deviate by less than delta from the true optimum.
- timeout:
- number - maximum seconds of cpu time to spend (default: no limit)
- report_success:
- GoalPrefix/N - this specifies a goal to be invoked whenever
	    the branch-and-bound process finds a better solution.  GoalPrefix
	    is a callable term (atom or compound) and N is an integer between
	    0 and 3.  The invoked goal is constructed by adding N optional
	    arguments to GoalPrefix: Cost, Handle and Module.  Cost is
	    a float number representing the cost of the solution found,
	    Handle is a handle as accepted by bb_cost/2 or bb_solution/2,
	    and Module is the context module of the minimisation.  
	    To disable any reporting, choose report_success:true/0.
	    The default handler prints a message to log_output.
- report_failure:
- GoalPrefix/N - this specifies a goal to be invoked whenever
	    the branch-and-bound process cannot find a solution in a cost
	    range.  GoalPrefix is a callable term (atom or compound) and
	    N is an integer between 0 and 3.  The invoked goal is
	    constructed by adding N optional arguments to GoalPrefix:
	    Cost, Handle and Module.   Cost is a From..To structure
	    representing the range of cost in which no solution could be found,
	    Handle is a handle as accepted by bb_cost/2 or bb_solution/2,
	    and Module is the context module of the minimisation.
	    To disable any reporting, choose report_failure:true/0.
	    The default handler prints a message to log_output.
- report_timeout:
- GoalPrefix/N - this specifies a goal to be invoked when the
	    branch-and-bound process times out.  GoalPrefix is a callable
	    term (atom or compound) and N is an integer between 0 and 3.
	    The invoked goal is constructed by adding N optional arguments
	    to GoalPrefix: Cost, Handle and Module.  Cost is a float number
	    representing the cost of the best solution found, Handle
	    is a handle as accepted by bb_cost/2 or bb_solution/2,
	    and Module is the context module of the minimisation.
	    To disable any reporting, choose report_timeout:true/0.
	    The default handler prints a message to log_output.
The default options can be selected by passing a free variable as
	the Options-argument. To specify other options, pass a bb_options-
	structure in struct-syntax, e.g.
	    bb_min(..., ..., bb_options{strategy:dichotomic, timeout:60})
	
	In order to maximize instead of minimizing, introduce a negated
	cost variable in your model and minimize that instead, e.g.
	
	    % maximize Profit
	    Cost #= -Profit,
	    bb_min(search(...), Cost, bb_options{}),
	
Modes and Determinism
- bb_min(+, ?, +) is nondet
Modules
This predicate is sensitive to its module context (tool predicate, see @/2).
Fail Conditions
Goal has no solutions
Examples
% simple minimization with default options
    ?- bb_min(member(X,[9,6,8,4,7,2,4,7]), X, Options).
    Found a solution with cost 9
    Found a solution with cost 6
    Found a solution with cost 4
    Found a solution with cost 2
    Found no solution with cost -1.0Inf .. 1
    X = 2
    Options = bb_options(continue, -1.0Inf, 1.0Inf, 1, 1, 0, 0, _, _)
    yes.
% coarser granularity: faster, but missing the optimum
    ?- bb_min(member(X,[9,6,8,4,7,2,4,7]), X, bb_options{delta:4}).
    Found a solution with cost 9
    Found a solution with cost 4
    Found no solution with cost -1.0Inf .. 0
    X = 4
    yes.
% alternative strategy based on bisecting the cost space
    ?- bb_min(member(X,[99,60,80,40,70,30,70]), X,
	    bb_options{strategy:dichotomic, from:0}).
    Found a solution with cost 99
    Found a solution with cost 40
    Found no solution with cost 0.0 .. 20.0
    Found a solution with cost 30
    Found no solution with cost 20.0 .. 25.0
    Found no solution with cost 25.0 .. 27.5
    Found no solution with cost 27.5 .. 28.75
    Found no solution with cost 28.75 .. 29.0
    X = 30
    yes.
% examples with library(ic) constraints
    ?- [X,Y,Z] :: 1..5,                    % constraints (model)
       X+Z #>=Y,
       C #= 3*X - 5*Y + 7*Z,               % objective function
       bb_min(labeling([X,Y,Z]), C, _).    % nondet search + b&b
    Found a solution with cost 5
    Found a solution with cost 0
    Found a solution with cost -2
    Found a solution with cost -4
    Found a solution with cost -6
    Found no solution with cost -15.0 .. -7.0
    X = 4
    Y = 5
    Z = 1
    C = -6
    Yes (0.00s cpu)
    ?- [X,Y,Z] :: 1..5,
       X+Z #>=Y,
       C #= 3*X - 5*Y + 7*Z,
       bb_min(search([X,Y,Z],0,input_order,indomain_middle,complete,[]), C, _).
    Found a solution with cost 15
    Found a solution with cost 8
    Found a solution with cost 1
    Found a solution with cost -4
    Found a solution with cost -6
    Found no solution with cost -15.0 .. -7.0
    X = 4
    Y = 5
    Z = 1
    C = -6
    Yes (0.00s cpu)
See Also
bb_min / 6