AMPL (A Mathematical Programming Language) is a high-level language for describing mathematical
programs. AMPL allows a mathematical programming model to be specified independently of the data
used for a specific instance of the model. AMPL's language for describing mathematical programs closely
follows that used by humans to describe mathematical programs to each other. For this reason, modellers
may spend more time improving the model and less time on the tedious details of data manipulation and
problem solution.
To start, AMPL needs a mathematical
programming model, which describes variables, objectives and relationships without referring to specific data.
AMPL also needs an instance of the data, or a particular data set. The model and one (or more) data files are
fed into the AMPL program. AMPL works like a compiler: the model and input are put into an intermediate
file that can be read by a solver. The solver actually finds an optimal solution to the problem by reading
in the intermediate file produced by AMPL and applying an appropriate algorithm. The solver outputs the
solution as a text file, which can be viewed directly and cross-referenced with the variables and constraints
specified in the model file.
We will not discuss the specifics of AMPL here but instead refer the reader to AMPL (A Mathematical Programming Language) at the University of Michigan Documentation and the AMPL website http://www.ampl.com.
One of the possible solvers that can be used by AMPL is lpsolve. However note that AMPL allows also defining non-linear models. These are not solvable by lpsolve because lpsolve can only handle MILP models. A message will be given if the AMPL model results in a model that cannot be handled by lpsolve.
To make this possible, a driver program is needed: lpsolve(.exe). This program must be put in the AMPL directory and AMPL can call the lpsolve solver. The newer versions of the lpsolve(.exe) driver call lpsolve via the shared library (lpsolve51.dll under Windows and liblpsolve51.so under Unix/Linux). This has the advantage that the lpsolve driver program doesn't have to be recompiled when an update of lpsolve is provided. The shared library must be somewhere in the path. That is all.
In the following text, ampl: before the AMPL commands is the AMPL prompt. Only the text after the : must be entered.
To select the lpsolve solver, the following command must be executed in AMPL:
ampl: option solver lpsolve;
To solve the model, the following command must be executed in AMPL:
ampl: solve;
Options can be passed to lpsolve. For example to enable scaling, the following command must be executed in AMPL:
ampl: option lpsolve_options 'scale';
Multiple options can be specified by separating them with at least one space:
ampl: option lpsolve_options 'scale scalemode=7 verbose';
A list of all options is given at the end of this document.
ampl: model diet.mod; ampl: data diet.dat; ampl: option solver lpsolve; ampl: option lpsolve_options 'scale scalemode=7'; ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: scale scalemode=7 LP_SOLVE 5.1.1.3: optimal, objective 88.2 1 simplex iterations
AMPL has a parameter that is used to indicate the outcome of the optimisation process. It is used as follows
ampl: display solve_result_num
solve_result_num can take the values shown in following table which also presents a short explanation for each value.
|
ampl: model diet.mod; ampl: data diet.dat; ampl: option solver lpsolve; ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: optimal, objective 88.2 3 simplex iterations
Note that 3 iterations were needed to solve the model. Now solve the model again:
ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: optimal, objective 88.2 0 simplex iterations
Note now that no iterations are needed to solve the model. This because the result of the previous solve is used as a starting point for the new solve. It is even allowed to add or deleted variables and constraints in the model, lpsolve will still try to start from the last result to continue.
If, for some reason, you don't want this and let lpsolve resolve the model from scratch, there are two possibilities to let lpsolve ignore the starting base. Either via an lpsolve option or an AMPL option.
The lpsolve option that can be set is:
ampl: option lpsolve_options 'coldstart';
The AMPL option that can be set is:
ampl: option send_statuses 0;
ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: coldstart LP_SOLVE 5.1.1.3: optimal, objective 88.2 3 simplex iterations
Special considerations for integer models. From the moment that a model contains integer variables,
the B&B algorithm must be used to solve this. This algorithm must go trough a tree of possible solutions
each time a solve is done. The last base of the best integer solution can not be used as a starting base for
a resolve. This is not an lpsolve limitation, but a B&B algorithm property...
For this reason, by default, AMPL doesn't provide a starting to the solver and solve is done from scratch.
By default... However, a compromise can be implemented by the solver. And this is also done in lpsolve. Solving
integer models is always started with solving the non-integer model. When this is done, the non-integer variables
are made integer via the B&B algorithm. lpsolve returns the base of the non-integer model and this base is
used as a starting base for the model at the next solve. As said, this is a compromise because it results in
a fast solve of the non-integer model, but the B&B algorithm must still be executed. So even if no
modifications are done to the model and solve is done again, there will be iterations needed by the B&B
algorithm to solve the integers. Because this must be explicitly implemented in the solver, AMPL doesn't
provide the starting base by default for integer models. It must explicitly be activated:
ampl: option send_statuses 2;
Only then, lpsolve will get a starting base when there are integers.
Example:
ampl: model multmip3.mod; ampl: data multmip3.dat; ampl: option solver lpsolve; ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: optimal, objective 235625 1986 simplex iterations 592 branch & bound nodes: depth 21
Note that 1986 iterations were needed to solve the model. Now solve the model again:
ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: optimal, objective 235625 1986 simplex iterations 592 branch & bound nodes: depth 21
The same as above. So no restart was done. Now activate the starting base for integer models and solve again:
ampl: option send_statuses 2; ampl: solve;
This gives as result:
LP_SOLVE 5.1.1.3: optimal, objective 235625 1922 simplex iterations 592 branch & bound nodes: depth 21
Note that less iterations are needed, but not 0 as in non-integer models. These are the iterations needed for the B&B algorithm.
lpsolve accepts a lot of options. Some options are just on/off switches and for there you just specify the option keyword (for example scale). Other options need a value and this is specified by an equal (=) sign after the option keyword and then the value (for example scalemode=7). Multiple options can be specified by separating them by a space (for example scale scalemode=7).
Here is a list of the possible lpsolve options:
bb=... branch-and-bound rule: one of
0 (default) for lowest indexed non-integer variable
1 for selection based on distance from the current bounds
2 for selection based on the largest current bound
3 for selection based on largest fractional value
4 for simple, unweighted pseudo-cost of a variable
5 this is an extended pseudo-costing strategy based on
minimizing the number of integer infeasibilities
6 this is an extended pseudo-costing strategy based on
maximizing the normal pseudo-cost divided by the
number of infeasibilities.
plus
32 for greedy mode
64 for pseudo cost mode
128 select the node that has already been selected before
the most number of times
4096 select the node that has been selected before the
fewest number of times or not at all
bfp=... set basis factorization package
cauto in IPs, algorithm decides which branch being taken first
cfirst in IPs, take ceiling branch first
coldstart ignore starting base
crash=... determines a starting base: one of
0 (default) none
2 most feasible basis
debug debug mode
degen perturb degeneracies
degenx=... anti-degen handling: one of
0 (default) no anti-degeneracy handling
1 check if there are equality slacks in the basis and
try to drive them out in order to reduce chance of
degeneracy in Phase 1
2 ColumnCheck
4 Stalling
8 NumFailure
16 LostFeas
32 Infeasible
64 Dynamic
128 During BB
depth=... set branch-and-bound depth limit
dual prefer the dual simplex for both phases
eps=... tolerance for rounding to integer
epsb=... minimum tolerance for the RHS
epsd=... minimum tolerance for reduced costs
epsel=... minimum tolerance for rounding values to zero
epsp=... value that is used as perturbation scalar for
degenerative problems
f specifies that branch-and-bound algorithm stops at first
found solution
ga=... specifies the absolute MIP gap for branch-and-bound
gr=... specifies the relative MIP gap for branch-and-bound
improve=... the iterative improvement level: one of
0 (default) improve none
1 improve FTRAN
2 improve BTRAN
3 improve FTRAN + BTRAN
4 triggers automatic inverse accuracy control in the dual
simplex, and when an error gap is exceeded the basis is
reinverted
n=... specify which solution number to return
o=... specifies that branch-and-bound algorithm stops when
objective value is better than value
objno=... objective number: 0 = none, 1 (default) = first
obound=... a lower bound for the objective function.
may speed up the calculations
parse_only parse input file but do not solve
piv=... simplex pivot rule: one of
0 select first
1 select according to Dantzig
2 (default) select Devex pricing from Paula Harris
3 select steepest edge
piva temporarily use first index if cycling is detected
pivf in case of Steepest Edge, fall back to DEVEX in primal
pivla scan entering/leaving columns alternatingly left/right
pivll scan entering/leaving columns left rather than right
pivm multiple pricing
pivr adds a small randomization effect to the selected pricer
presolve presolve problem before start optimizing
presolvel also eliminate linearly dependent rows
presolver if the phase 1 solution process finds that a constraint is
redundant then this constraint is deleted
prim prefer the primal simplex for both phases
printsol=... print solution: one of
0 (default) print nothing
1 only objective value
2 obj value+variables
3 obj value+variables+constraints
4 obj value+variables+constraints+duals
5 obj value+variables+constraints+duals+lp model
6 obj value+variables+constraints+duals+lp model+scales
7 obj value+variables+constraints+duals+lp model+scales+
lp tableau
prlp print the LP
psols print (intermediate) feasible solutions
psolsa print (intermediate) feasible solutions (non-zeros)
r=... max nbr of pivots between a re-inversion of the matrix
scale scale the problem
scalemode=... scale mode: one of
1 for scale to convergence using largest absolute value,
2 for scale based on the simple numerical range, or
3 (default) for numerical range-based scaling, or
4 for geometric scaling, or
7 for Curtis & Reid scaling
plus
16 for scale to convergence using logarithmic mean
of all values
32 for also do Power scaling
64 to make sure that no scaled number is above 1
128 to scale integer variables
simplexdd set Phase1 Dual, Phase2 Dual
simplexdp set Phase1 Dual, Phase2 Primal
simplexpd set Phase1 Primal, Phase2 Dual
simplexpp set Phase1 Primal, Phase2 Primal
timeout=... timeout after sec seconds when not solution found
trace trace pivot selections
trej=... minimum pivot value
verbose verbose mode
version report version details
wafter write model after solve (useful if presolve used)
wantsol=... solution report without -AMPL: sum of
1 ==> write .sol file
2 ==> print primal variable values
4 ==> print dual variable values
8 ==> do not print solution message
wfmps=... write to MPS file in free format
wlp=... write to LP filename
wmps=... write to MPS file in fixed format
wxli=... write file with xli library
wxliname=... xli library
wxliopt=... options for xli library
Normally, the lpsolve program is called from AMPL. However the program can also be called stand-alone. When the command is invoked without an option or with the option -?, a list of the possible options is shown:
usage: lpsolve [options] stub [-AMPL] [<assignment> ...]
Options:
-- {end of options}
-= {show name= possibilities}
-? {show usage}
-e {suppress echoing of assignments}
-s {write .sol file (without -AMPL)}
-v {just show version}
stub and the -AMPL option is used when the program is called from AMPL.
The -v option shows the version of lpsolve.
The -= option shows all the options that can be passed to lpsolve from within AMPL.
These options can also be specified in an environment variable lpsolve_options.
To call the lpsolve command to solve a model, you first must have a stub file that can be read by lpsolve.
This stub file can be created by AMPL as follows:
ampl: model models\diet.mod; ampl: data models\diet.dat; ampl: write bdiet; ampl: quit
This created a binary file diet.nl. As an alternative the command write gdiet; can be used. This creates an ascii file that also can be read. However it is somewhat slower to read, especially when the models are larger.
This creates a file diet.nl in the current directory. This file is the stub file needed by lpsolve:
lpsolve lpsolve diet.nl
This gives:
LP_SOLVE 5.1.1.3: optimal, objective 88.19999999999999 3 simplex iterations
Options can be passed to lpsolve via the environment variable lpsolve_options:
set lpsolve_options=scale scalemode=7 verbose
Then the result of the previous lpsolve command is:
scale
scalemode=7
verbose
Model name: '' - run #1
Objective: Minimize(R0)
Submitted:
Model size: 4 constraints, 8 variables, 39 non-zeros.
Constraints: 0 equality, 0 GUB, 0 SOS.
Variables: 0 integer, 0 semi-cont., 0 SOS.
Using DUAL simplex for phase 1 and PRIMAL simplex for phase 2.
Optimal solution with dual simplex at iteration 1
lp_solve solution 88.2 final at iteration 1, 0 nodes explor
ed
Excellent numeric accuracy ||*|| = 1.13687e-013
Memo: Largest [etaPFI v1.0] inv(B) had 0 NZ entries, 0.0x largest basis.
In the total iteration count 1, 0 (0.0%) were minor/bound swaps.
There were 0 refactorizations, 0 triggered by time and 0 by density.
... on average 1.0 major pivots per refactorization.
Total solver time was 0.000 seconds.
LP_SOLVE 5.1.1.3: optimal, objective 88.19999999999999
1 simplex iterations
In this mode, the lpsolve option wantsol shows the solution:
set lpsolve_options=wantsol=2 lpsolve diet.nl
This gives:
wantsol=2 LP_SOLVE 5.1.1.3: optimal, objective 88.19999999999999 3 simplex iterations variable value _svar[1] 0 _svar[2] 0 _svar[3] 0 _svar[4] 0 _svar[5] 46.666666666666664 _svar[6] 0 _svar[7] -3.552713678800501e-15 _svar[8] 0
The -e option results in not echoing the options passed to lpsolve:
set lpsolve_options=wantsol=2 lpsolve -e diet.nl
This gives:
LP_SOLVE 5.1.1.3: optimal, objective 88.19999999999999 3 simplex iterations variable value _svar[1] 0 _svar[2] 0 _svar[3] 0 _svar[4] 0 _svar[5] 46.666666666666664 _svar[6] 0 _svar[7] -3.552713678800501e-15 _svar[8] 0
lpsolve options can also be passed at the command line, they don't overrule the lpsolve_options environment variable, they are added:
set lpsolve_options=wantsol=2 lpsolve diet.nl "verbose scale"
This gives:
wantsol=2
verbose
scale
Model name: '' - run #1
Objective: Minimize(R0)
Submitted:
Model size: 4 constraints, 8 variables, 39 non-zeros.
Constraints: 0 equality, 0 GUB, 0 SOS.
Variables: 0 integer, 0 semi-cont., 0 SOS.
Using DUAL simplex for phase 1 and PRIMAL simplex for phase 2.
Optimal solution with dual simplex at iteration 1
lp_solve solution 88.2 final at iteration 1, 0 nodes explor
ed
Excellent numeric accuracy ||*|| = 1.13687e-013
Memo: Largest [etaPFI v1.0] inv(B) had 0 NZ entries, 0.0x largest basis.
In the total iteration count 1, 0 (0.0%) were minor/bound swaps.
There were 0 refactorizations, 0 triggered by time and 0 by density.
... on average 1.0 major pivots per refactorization.
Total solver time was 0.000 seconds.
LP_SOLVE 5.1.1.3: optimal, objective 88.19999999999997
1 simplex iterations
variable value
_svar[1] 0
_svar[2] 0
_svar[3] 0
_svar[4] 0
_svar[5] 46.66666666666666
_svar[6] 0
_svar[7] 0
_svar[8] 0