IbexSolve¶
This page describes IbexSolve, the plugin installed with the withsolver
option.
Getting started¶
IbexSolve is a enduser program that solves a system of nonlinear equations rigorously (that is, it does not lose any solution and return each solution under the form of a small box enclosing the true value). It resorts to a unique blackbox strategy (whatever the input problem is) and with a very limited number of parameters. Needless to say, this strategy is a kind of compromise and not the best one for a given problem.
Note that this program is based on the generic solver, a C++ class that allows to build a more customizable solver.
You can directly apply this solver on one of the benchmark problems
distributed with Ibex.
The benchmarks are all written in the Minibex syntax and stored in an arborescence under plugins/solver/benchs/
.
Open a terminal, move to the bin
subfolder and run IbexSolve with, for example, the problem named kolev36 located at the specified path:
~/ibex2.6.0$ cd bin
~/ibex2.6.0/bin$ ./ibexsolve ../plugins/solver/benchs/others/kolev36.bch
The following result should be displayed:
***************************** setup *****************************
file loaded: ../plugins/solver/benchs/others/kolev36.bch
output file: ../plugins/solver/benchs/others/kolev36.mnf
*****************************************************************
running............
solving successful!
number of inner boxes: 1
number of boundary boxes: 0
number of unknown boxes: 0
number of pending boxes: 0
cpu time used: 0.0640001s
number of cells: 25
results written in ../plugins/solver/benchs/others/kolev36.mnf
The number of “inner boxes” correspond to the number of solutions found (there is just one here). To see the solution, use the option s
.
In the report, the “number of cells” correspond to the number of hypothesis (bisections) that was required to solve the problem.
Options¶
e<float>, –epsmin=<float>  Minimal width of output boxes. This is a criterion to stop bisection: a nonvalidated box will not be larger than ‘epsmin’. Default value is 1e3. 
E<float>, –epsmax=<float>  Maximal width of output boxes. This is a criterion to force bisection: a validated box will not be larger than ‘epsmax’ (unless there is no equality and it is fully inside inequalities). Default value is +oo (none) 
t<float>, –timeout=<float>  Timeout (time in seconds). Default value is +oo (none). 
i<filename>, –input=<filename>  Manifold input file. The file contains a (intermediate) description of the manifold with boxes in the MNF (binary) format. 
o<filename>, –output=<filename>  Manifold output file. The file will contain the description of the manifold with boxes in the MNF (binary) format. 
s, –sols  Display the “solutions” (output boxes) on the standard output. 
–bfs  Perform breadthfirst search (instead of depthfirst search, by default) 
–trace  Activate trace. “Solutions” (output boxes) are displayed as and when they are found. 
–boundary=…  Boundary test strength. Possible values are:

–randomseed=<float>  Random seed (useful for reproducibility). Default value is 1. 
q, –quiet  Print no report on the standard output. 
Calling IbexSolve from C++¶
You can call IbexSolve (the default solver) and get the solutions from C++.
Two objects must be built: the first represents the problem (namely, a system), the second the solver itself. Then, we just run the solver. Here is a simple example:
/* Build a system of equations from the file */
System system(IBEX_BENCHS_DIR "/others/kolev36.bch");
/* Build a default solver for the system and with a precision set to 1e07 */
DefaultSolver solver(system,1e07);
solver.solve(system.box); // Run the solver
/* Display the solutions. */
output << solver.get_manifold() << endl;
The output is:
sol n°0 = ([0.1173165676349099, 0.1173165676349106] ; [0.4999999999999985, 0.5000000000000013] ; [0.8826834323650885, 0.8826834323650913] ; [0.2071067811865487, 0.2071067811865464] ; [1.207106781186543, 1.207106781186552] ; [2.000000000000004, 1.999999999999996]) [inner]
The generic solver¶
The generic solver is the main C++ class behind the implementation of ibexsolve
.
It is a classical branch and prune algorithm that interleaves contraction and branching (bisection) until
boxes get sufficiently small. However, it performs a more general task that just finding solution points of square
systems of equations: it also knows how to deal with underconstrained systems and handle manifolds.
Note
A more detailed documentation about underconstrained systems will be available soon.
Compared to ibexsolve
, the generic solver allows the following additional operators as inputs:
a contractor
Operator that contracts boxes by removing nonsolution points. The contraction operator must be compatible with the system given (equations/inequalities). The solver performs no check (it is the user responsability). See Contractors.
a bisector
Operator that splits a box. Note that some bisectors have a precision parameter: the box is bisected providing it is large enough. But this precision is not directly seen by the solver which has its own precision variables (see
e`̀ and ``E
). If however the bisector does not split a box, this will generate an exception caught by the solver, which will not continue the search and backtrack. So fixing the bisector internal precision gives basically the same effect as fixing it withe
. See Bisectors for more details.a cell buffer
Operator that manages the list of pending boxes (a cell is a box with a little bit of extra information used by the search). See Cell buffers for more details.
Our next example creates a solver for the intersection of two circles of radius \(d\), one centered on \((0,0)\) and the other in \((1,0)\).
To this end we first create a vectorvalued function:
Then, we build two contractors; a forwardbacwkard contractor and (because the system is square), an interval Newton contractor.
We chose as bisection operator the roundrobin operator, that splits each component in turn. The precision of the solver is set to 1e7.
Finally, the cell buffer is a stack, which leads to a depthfirst search.
/* Create the function (x,y)>( (x,y)d, (x,y)(1,0)d ) */
Variable x,y;
double d=1.0;
Function f(x,y,Return(sqrt(sqr(x)+sqr(y))d,
sqrt(sqr(x1.0)+sqr(y))d));
/* Create the system f(x,y)=0. */
SystemFactory factory;
factory.add_var(x);
factory.add_var(y);
factory.add_ctr(f(x,y)=0);
System system(factory);
/* Create the domain of variables */
double init_box[][2] = { {10,10},{10,10} };
IntervalVector box(2,init_box);
/* Create a first contractor w.r.t f(x,y)=0 (forwardbackward) */
CtcFwdBwd fwdBwd(f);
/* Create a second contractor (interval Newton) */
CtcNewton newton(f);
/* Compose the two contractors */
CtcCompo compo(fwdBwd,newton);
/* Create a roundrobin bisection heuristic and set the
* precision of boxes to 0. */
RoundRobin bisector(0);
/* Create a "stack of boxes" (CellStack), which has the effect of
* performing a depthfirst search. */
CellStack buff;
/* Vector precisions required on variables */
Vector prec(6, 1e07);
/* Create a solver with the previous objects */
Solver s(system, compo, bisector, buff, prec, prec);
/* Run the solver */
s.solve(box);
/* Display the solutions */
output << s.get_manifold() << endl;
The output is:
sol n°0 = ([0.4999999999999998, 0.5000000000000005] ; [0.8660254037844383, 0.8660254037844389]) [inner]
sol n°1 = ([0.4999999999999996, 0.5000000000000003] ; [0.8660254037844389, 0.8660254037844383]) [inner]
Implementing IbexSolve (the default solver)¶
IbexSolve is an instance of the generic solver with (almost) all parameters set by default.
We already showed how to Calling IbexSolve from C++. To give a further insight into the generic solver and its possible settings, we explain now how to recreate the default solver by yourself.
The contractor of the default solver is obtained with the following receipe. This is a composition of
 HC4
 ACID
 Interval Newton (only if it is a square system of equations)
 A fixpoint of the Polytope Hull of two linear relaxations combined:
 the relaxation called XTaylor;
 the relaxation generated by affine arithmetic. See Linearizations.
The bisector is based on the The Smear Function with maximal relative impact.
So the following program exactly reproduces the default solver.
System system(IBEX_BENCHS_DIR "/others/kolev36.bch");
/* ============================ building contractors ========================= */
CtcHC4 hc4(system,0.01);
CtcHC4 hc4_2(system,0.1,true);
CtcAcid acid(system, hc4_2);
CtcNewton newton(system.f_ctrs, 5e+08, 1e07, 1e04);
LinearizerCombo linear_relax(system,LinearizerCombo::XNEWTON);
CtcPolytopeHull polytope(linear_relax);
CtcCompo polytope_hc4(polytope, hc4);
CtcFixPoint fixpoint(polytope_hc4);
CtcCompo compo(hc4,acid,newton,fixpoint);
/* =========================================================================== */
/* Create a smearfunction bisection heuristic. */
SmearSumRelative bisector(system, 1e07);
/* Create a "stack of boxes" (CellStack) (depthfirst search). */
CellStack buff;
/* Vector precisions required on variables */
Vector prec(6, 1e07);
/* Create a solver with the previous objects */
Solver s(system, compo, bisector, buff, prec, prec);
/* Run the solver */
s.solve(system.box);
/* Display the solutions */
output << s.get_manifold() << endl;
/* Report performances */
output << "cpu time used=" << s.get_time() << "s."<< endl;
output << "number of cells=" << s.get_nb_cells() << endl;
Parallelizing search¶
It is possible to parallelize the search by running (in parallel) solvers for different subboxes of the initial box.
Be aware however that Ibex has not been designed (so far) to be parallelized and the following lines only reports our preliminary experiments.
Here are the important observations:
The sublibrary gaol is not threadsafe. You must compile Ibex with filib which seems to be OK (see Configuration options).
The linear solver Soplex (we have not tested yet with Cplex) seems to be threadsafe but sometimes generates error messages on the console like:
ISOLVE56 stop: 0, basis status: PRIMAL (2), solver status: RUNNING (1)
So, calling Soplex several times simultaneously seems not to be allowed, but Soplex at least manages the case properly, that is, stops. As far as we have observed, we don’t lose solutions even when this kind of message appear.
Ibex objects are not threadsafe which means that the solvers run in parallel must share no information. In particular, each solver must have its own copy of the system.
Here is an example:
// Get the system
System sys1(IBEX_BENCHS_DIR "/polynom/pontsgeo.bch");
// Create a copy for the second solver
System sys2(sys1,System::COPY);
// Precision of the solution boxes
double prec=1e08;
// Create two solvers
DefaultSolver solver1(sys1,prec);
DefaultSolver solver2(sys2,prec);
// Create a partition of the initial box into two subboxes,
// by bisecting any variable (here, n°4)
pair<IntervalVector,IntervalVector> pair=sys1.box.bisect(4);
// =======================================================
// Run the solvers in parallel
// =======================================================
#pragma omp parallel sections
{
solver1.solve(pair.first);
#pragma omp section
solver2.solve(pair.second);
}
// =======================================================
output << "solver #1 found " << solver1.get_manifold().size() << endl;
output << "solver #2 found " << solver2.get_manifold().size() << endl;
If I remove the #pragma
the program displays:
solver #1 found 64
solver #2 found 64
real 0m5.121s // < total time
user 0m5.088s
With the #pragma
, I obtain:
solver #1 found 64
solver #2 found 64
real 0m2.902s // < total time
user 0m5.468s
Note: It is pure luck that by bisecting the 4th variable, we obtain exactly half of the solutions on each subbox. Also, looking for the 64 first solutions takes here around the same time than looking for the 64 subsequent ones, which is particular to this example. So, contrary to what this example seems to prove, splitting the box in two subboxes does not divide the running time by two in general. Of course :)
If you are afraid about the messages of the linear solver, you can replace the DefaultSolver
by your own dedicated solver
that does not resort to the simplex, ex:
Vector eps_min(sys1.nb_var,prec);
Vector eps_max(sys1.nb_var,POS_INFINITY);
Solver solver1(sys1,*new CtcCompo(*new CtcHC4(sys1),*new CtcNewton(sys1.f_ctrs)), *new RoundRobin(prec), *new CellStack(), eps_min, eps_max);
Solver solver2(sys2,*new CtcCompo(*new CtcHC4(sys2),*new CtcNewton(sys2.f_ctrs)), *new RoundRobin(prec), *new CellStack(), eps_min, eps_max);