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- __builtin__.object
-
- Bar
- BarCollection
- BarCollectionIter
- Environment
- KKT
- LPX
- Objective
- ObjectiveIter
- Tree
- TreeIter
- TreeNode
class Bar(__builtin__.object) |
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Bar objects are used to refer to a particular row or column of
a linear program. Rows and columns may be retrieved by
indexing into the rows and cols sequences of LPX instances.
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Methods defined here:
- __eq__(...)
- x.__eq__(y) <==> x==y
- __ge__(...)
- x.__ge__(y) <==> x>=y
- __gt__(...)
- x.__gt__(y) <==> x>y
- __le__(...)
- x.__le__(y) <==> x<=y
- __lt__(...)
- x.__lt__(y) <==> x<y
- __ne__(...)
- x.__ne__(y) <==> x!=y
- __repr__(...)
- x.__repr__() <==> repr(x)
- __str__(...)
- x.__str__() <==> str(x)
Data descriptors defined here:
- bounds
- The lower and upper bounds, where None signifies unboundedness.
- dual
- The dual value of this variable by the last solver.
- dual_i
- The dual value of this variable by the interior-point solver.
- dual_s
- The dual value of this variable by the simplex solver.
- index
- The index of the row or column this object refers to.
- iscol
- Whether this is a column.
- isrow
- Whether this is a row.
- kind
- Either the type 'float' if this is a continuous variable, 'int'
if this is an integer variable, or 'bool' if this is a binary
variable.
- matrix
- Non-zero constraint coefficients in this row/column vector
as a list of two-element (index, value) tuples.
- name
- Row/column symbolic name, or None if unset.
- nnz
- Number of non-zero constraint elements in this row/column.
- primal
- The primal value of this variable by the last solver.
- primal_i
- The primal value of this variable by the interior-point solver.
- primal_s
- The primal value of this variable by the simplex solver.
- scale
- The scale for the row or column. This is a factor which one may
set to improve conditioning in the problem. Most users will want
to use the LPX.scale() method rather than setting these directly.
The resulting constraint matrix is such that the entry at row i
and column j is (for the purpose of optimization) (ri)*(aij)*(sj)
where ri and sj are the row and column scaling factors, and aij
is the entry of the constraint matrix.
- status
- Row/column basis status. This is a two character string with
the following possible values:
bs -- This row/column is basic.
nl -- This row/column is non-basic.
nu -- This row/column is non-basic and set to the upper bound.
On assignment, if this row/column is not double bounded,
this is equivalent to 'nl'.
nf -- This row/column is non-basic and free.
On assignment this is equivalent to 'nl'.
ns -- This row/column is non-basic and fixed.
On assignment this is equivalent to 'nl'.
- valid
- Whether this row or column has a valid index in its LP.
- value
- The value of this variable by the last solver.
- value_m
- The value of this variable by the MIP solver.
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class BarCollection(__builtin__.object) |
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Bar collection objects. An instance is used to index into either
the rows and columns of a linear program. They exist as the 'rows'
and 'cols' attributes of LPX instances.
One accesses particular rows or columns by indexing the appropriate
bar collection object, or iterating over it. Valid indices include
particular row and column names (a user defined string) or numbers
(counting from 0), a slice specifying a range of numeric elements,
or a series of individual indices. For example, for an LPX instance
lp, we may have:
lp.rows[0] --> the first row
lp.rows[-1] --> the last row
lp.cols[:3] --> the first three columns
lp.cols[1,'name',5] --> column 1, a column named 'name', and column 5
One may also query the length of this sequence to get the number of
rows or columns, and del to get rid of rows or columns, e.g.:
len(lp.cols) --> the number of columns in the problem
del lp.rows['arow'] --> deletes a row named 'arow'
del lp.rows[-2:] --> deletes the last two rows
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Methods defined here:
- __contains__(...)
- x.__contains__(y) <==> y in x
- __delitem__(...)
- x.__delitem__(y) <==> del x[y]
- __getitem__(...)
- x.__getitem__(y) <==> x[y]
- __iter__(...)
- x.__iter__() <==> iter(x)
- __len__(...)
- x.__len__() <==> len(x)
- __setitem__(...)
- x.__setitem__(i, y) <==> x[i]=y
- __str__(...)
- x.__str__() <==> str(x)
- add(...)
- add(n)
Add n more rows (constraints) or columns (struct variables).
Returns the index of the first added entry.
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class Environment(__builtin__.object) |
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This represents the PyGLPK environment. Through this, one may control
the global behavior of the GLPK. One instance of this exists, named
env in the glpk module.
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Data descriptors defined here:
- blocks
- The number of currently allocated memory blocks.
- blocks_peak
- The peak value of the blocks attribute.
- bytes
- The number of currently allocated memory bytes.
- bytes_peak
- The peak value of the bytes attribute.
- mem_limit
- The memory limit in megabytes. None if no limit is set.
- term_hook
- Function to intercept all terminal output. This should be a
callable object that accepts a single string argument, or None
to indicate that no hook is set (e.g., all output goes to the
terminal, default behavior). Note that when the function is
called, there is no guarantee that the input string will be a
full line, or even non-empty. All exceptions thrown by the
function will go ignored and unreported.
- term_on
- Whether or not terminal output for the underlying GLPK
procedures is on or off.
- version
- Tuple holding the major version and minor version of the GLPK
that this PyGLPK module was built upon. For example, if built
against GLPK 4.31, version==(4,31).
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class KKT(__builtin__.object) |
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Karush-Kuhn-Tucker conditions. This is returned from a check on
quality of solutions. Four types of conditions are stored here:
- KKT.PE conditions are attributes prefixed by 'pe' measuring
error in the primal solution.
- KKT.PB conditions are attributes prefixed by 'pb' measuring
error in satisfying primal bound constraints, i.e., feasibility.
- KKT.DE and KKT.DB are analogous, but for the dual.
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Data descriptors defined here:
- db_ae_ind
- Index of the variable with the largest absolute error.
- db_ae_max
- Largest absolute error.
- db_quality
- Character representing the quality of primal feasibility.
'H', high, 'M', medium, 'L', low, or '?' wrong or infeasible.
- db_re_ind
- Index of the variable with the largest relative error.
- db_re_max
- Largest relative error.
- de_ae_max
- Largest absolute error.
- de_ae_row
- Index of the column with the largest absolute error.
- de_quality
- Character representing the quality of the primal solution.
'H', high, 'M', medium, 'L', low, or '?' wrong or infeasible.
- de_re_max
- Largest relative error.
- de_re_row
- Index of the column with the largest relative error.
- pb_ae_ind
- Index of the variable with the largest absolute error.
- pb_ae_max
- Largest absolute error.
- pb_quality
- Character representing the quality of primal feasibility.
'H', high, 'M', medium, 'L', low, or '?' wrong or infeasible.
- pb_re_ind
- Index of the variable with the largest relative error.
- pb_re_max
- Largest relative error.
- pe_ae_max
- Largest absolute error.
- pe_ae_row
- Index of the row with the largest absolute error.
- pe_quality
- Character representing the quality of the primal solution.
'H', high, 'M', medium, 'L', low, or '?' wrong or infeasible.
- pe_re_max
- Largest relative error.
- pe_re_row
- Index of the row with the largest relative error.
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class LPX(__builtin__.object) |
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LPX() --> Empty linear program.
LPX(gmp=filename) --> Linear program with data read from a
GNU MathProg file containing model and data.
LPX(mps=filename) --> Linear program with data read from a
datafile in fixed MPS format.
LPX(freemps=filename) --> Linear program with data read from a
datafile in free MPS format.
LPX(cpxlp=filename) --> Linear program with data read from a
datafile in fixed CPLEX LP format.
LPX(glp=filename) --> Linear program with data read from a
datafile in GNU LP format.
LPX(gmp=(model_filename,[data_filename,[output_filename]])-->
Linear program from GNU MathProg input files. The first
element is a path to the model second, the second to the
data section. If the second element is omitted or is None
then the model file is presumed to also hold the data.
The third elment holds the output data file to write
display statements to. If omitted or None, the output
is instead put through to standard output.
This represents a linear program object. It holds data and
offers methods relevant to the whole of the linear program.
There are many members in this class, but the most important
are:
obj Represents the objective function.
rows A collection over which one can access rows.
cols Same, but for columns.
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Methods defined here:
- __init__(...)
- x.__init__(...) initializes x; see x.__class__.__doc__ for signature
- __repr__(...)
- x.__repr__() <==> repr(x)
- __str__(...)
- x.__str__() <==> str(x)
- adv_basis(...)
- adv_basis()
Construct an advanced initial basis, triangular with as few
variables as possible fixed.
- cpx_basis(...)
- cpx_basis()
Construct an advanced Bixby basis.
This basis construction method is described in:
Robert E. Bixby. Implementing the Simplex Method: The Initial
Basis. ORSA Journal on Computing, Vol. 4, No. 3, 1992,
pp. 267-84.
- erase(...)
- erase()
Erase the content of this problem, restoring it to the state
it was in when it was first created.
- exact(...)
- exact()
Attempt to solve the problem using an exact simplex method.
This returns None if the problem was successfully solved.
Alternately, on failure it will return one of the following
strings to indicate failure type.
fault -- There are no rows or columns, or the initial basis
is invalid, or the initial basis matrix is singular
or ill-conditioned.
itlim -- Iteration limited exceeded.
tmlim -- Time limit exceeded.
- integer(...)
- integer()
MIP solver based on branch-and-bound.
This procedure has a great number of optional keyword arguments
to control the functioning of the solver. We list these here,
including descriptions of their legal values.
msg_lev : Controls the message level of terminal output.
LPX.MSG_OFF -- no output (default)
LPX.MSG_ERR -- error and warning messages
LPX.MSG_ON -- normal output
LPX.MSG_ALL -- full informational output
br_tech : Branching technique option.
LPX.BR_FFV -- first fractional variable
LPX.BR_LFV -- last fractional variable
LPX.BR_MFV -- most fractional variable
LPX.BR_DTH -- heuristic by Driebeck and Tomlin (default)
bt_tech : Backtracking technique option.
LPX.BT_DFS -- depth first search
LPX.BT_BFS -- breadth first search
LPX.BT_BLB -- best local bound (default)
LPX.BT_BPH -- best projection heuristic
pp_tech : Preprocessing technique option.
LPX.PP_NONE -- disable preprocessing
LPX.PP_ROOT -- perform preprocessing only on the root level
LPX.PP_ALL -- perform preprocessing on all levels (default)
gmi_cuts: Use Gomory's mixed integer cuts (default False)
mir_cuts: Use mixed integer rounding cuts (default False)
tol_int : Tolerance used to check if the optimal solution to the
current LP relaxation is integer feasible.
tol_obj : Tolerance used to check if the objective value in the
optimal solution to the current LP is not better than the best
known integer feasible solution.
tm_lim : Search time limit in milliseconds. (default is max int)
out_frq : Terminal output frequency in milliseconds. (default 5000)
out_dly : Terminal output delay in milliseconds. (default 10000)
callback: A callback object the user may use to monitor and control
the solver. During certain portions of the optimization, the
solver will call methods of callback object. (default None)
The last parameter, callback, is worth its own discussion. During
the branch-and-cut algorithm of the MIP solver, at various points
callback hooks are invoked which allow the user code to influence
the proceeding of the MIP solver. The user code may influence the
solver in the hook by modifying and operating on a Tree instance
passed to the hook. These hooks have various codes, which we list
here.
select - request for subproblem selection
prepro - request for preprocessing
rowgen - request for row generation
heur - request for heuristic solution
cutgen - request for cut generation
branch - request for branching
bingo - better integer solution found
During the invocation of a hook with a particular code, the
callback object will have a method of the same name as the hook
code called, with the Tree instance. For instance, for the
'cutgen' hook, it is equivalent to
callback.cutgen(tree)
being called with tree as the Tree instance. If the method does
not exist, then instead the method 'default' is called with the
same signature. If neither the named hook method nor the default
method exist, then the hook is ignored.
This method requires a mixed-integer problem where an optimal
solution to an LP relaxation (either through simplex() or
exact()) has already been found. Alternately, try intopt().
This returns None if the problem was successfully solved.
Alternately, on failure it will return one of the following
strings to indicate failure type.
fault -- There are no rows or columns, or it is not a MIP
problem, or integer variables have non-int bounds.
nopfs -- No primal feasible solution.
nodfs -- Relaxation has no dual feasible solution.
itlim -- Iteration limited exceeded.
tmlim -- Time limit exceeded.
sing -- Error occurred solving an LP relaxation subproblem.
- interior(...)
- interior()
Attempt to solve the problem using an interior-point method.
This returns None if the problem was successfully solved.
Alternately, on failure it will return one of the following
strings to indicate failure type.
fault -- There are no rows or columns.
nofeas -- The problem has no feasible (primal/dual) solution.
noconv -- Very slow convergence or divergence.
itlim -- Iteration limited exceeded.
instab -- Numerical instability when solving Newtonian system.
- intopt(...)
- intopt()
More advanced MIP branch-and-bound solver than integer(). This
variant does not require an existing LP relaxation.
This returns None if the problem was successfully solved.
Alternately, on failure it will return one of the following
strings to indicate failure type.
fault -- There are no rows or columns, or it is not a MIP
problem, or integer variables have non-int bounds.
nopfs -- No primal feasible solution.
nodfs -- Relaxation has no dual feasible solution.
itlim -- Iteration limited exceeded.
tmlim -- Time limit exceeded.
sing -- Error occurred solving an LP relaxation subproblem.
- kkt(...)
- kkt([scaled=False])
Return an object encapsulating the results of a check on the
Karush-Kuhn-Tucker optimality conditions for a basic (simplex)
solution. If the argument 'scaled' is true, return results
of checking the internal scaled instance of the LP instead.
- kktint(...)
- kktint()
Similar to kkt(), except analyzes solution quality of an
mixed-integer solution. Note that only the primal components
of the KKT object will have meaningful values.
- read_basis(...)
- read_basis(filename)
Reads an LP basis in the fixed MPS format from a given file.
- scale(...)
- scale([flags=LPX.SF_AUTO])
Perform automatic scaling of the problem data, in order to.
improve conditioning. The behavior is controlled by various
flags, which can be bitwise ORed to combine effects. Note
that this only affects the internal state of the LP
representation. These flags are members of the LPX class:
SF_GM -- perform geometric mean scaling
SF_EQ -- perform equilibration scaling
SF_2N -- round scale factors to the nearest power of two
SF_SKIP -- skip scaling, if the problem is well scaled
SF_AUTO -- choose scaling options automatically
- simplex(...)
- simplex([keyword arguments])
Attempt to solve the problem using a simplex method.
This procedure has a great number of optional keyword arguments
to control the functioning of the solver. We list these here,
including descriptions of their legal values.
msg_lev : Controls the message level of terminal output.
LPX.MSG_OFF -- no output (default)
LPX.MSG_ERR -- error and warning messages
LPX.MSG_ON -- normal output
LPX.MSG_ALL -- full informational output
meth : Simplex method option
LPX.PRIMAL -- use two phase primal simplex (default)
LPX.DUALP -- use two phase dual simplex, primal if that fails
pricing : Pricing technique
LPX.PT_STD -- standard textbook technique
LPX.PT_PSE -- projected steepest edge (default)
r_test : Ratio test technique
LPX.RT_STD -- standard textbook technique
LPX.RT_HAR -- Harris' two-pass ratio test (default)
tol_bnd : Tolerance used to check if the basic solution is primal
feasible. (default 1e-7)
tol_dj : Tolerance used to check if the basic solution is dual
feasible. (default 1e-7)
tol_piv : Tolerance used to choose pivotal elements of the simplex
table. (default 1e-10)
obj_ll : Lower limit of the objective function. The solver
terminates upon reaching this level. This is used only in
dual simplex optimization. (default is min float)
obj_ul : Upper limit of the objective function. The solver
terminates upon reaching this level. This is used only in
dual simplex optimization. (default is max float)
it_lim : Simplex iteration limit. (default is max int)
tm_lim : Search time limit in milliseconds. (default is max int)
out_frq : Terminal output frequency in iterations. (default 200)
out_dly : Terminal output delay in milliseconds. (default 0)
presolve: Use the LP presolver. (default False)
This returns None if the problem was successfully solved.
Alternately, on failure it will return one of the following
strings to indicate failure type.
fault -- There are no rows or columns, or the initial basis
is invalid, or the initial basis matrix is singular
or ill-conditioned.
objll -- The objective reached its lower limit.
objul -- The objective reached its upper limit.
itlim -- Iteration limited exceeded.
tmlim -- Time limit exceeded.
sing -- The basis matrix became singular or ill-conditioned.
nopfs -- No primal feasible solution. (Presolver only.)
nodfs -- No dual feasible solution. (Presolver only.)
- std_basis(...)
- std_basis()
Construct the standard trivial inital basis for this LP.
- unscale(...)
- unscale()
This unscales the problem data, essentially setting all
scale factors to 1.
- write(...)
- write(format=filename)
Output data about the linear program into a file with a given
format. What data is written, and how it is written, depends
on which of the format keywords are used. Note that one may
specify multiple format and filename pairs to write multiple
types and formats of data in one call to this function.
mps -- For problem data in the fixed MPS format.
bas -- The current LP basis in fixed MPS format.
freemps -- Problem data in the free MPS format.
cpxlp -- Problem data in the CPLEX LP format.
glp -- Problem data in the GNU LP format.
prob -- Problem data in a plain text format.
sol -- Basic solution in printable format.
sens_bnds -- Bounds sensitivity information.
ips -- Interior-point solution in printable format.
mip -- MIP solution in printable format.
Data descriptors defined here:
- cols
- Column collection. See the help on class BarCollection.
- kind
- Either the type 'float' if this is a pure linear programming
(LP) problem, or the type 'int' if this is a mixed integer
programming (MIP) problem.
- matrix
- The constraint matrix as a list of three element (row index,
column index, value) tuples across all non-zero elements of
the constraint matrix.
- name
- Problem name, or None if unset.
- nbin
- The number of binary column variables, i.e., integer with 0
to 1 bounds. Always 0 if this is not a mixed integer problem.
- nint
- The number of integer column variables. Always 0 if this is
not a mixed integer problem.
- nnz
- Number of non-zero constraint coefficients.
- obj
- Objective function object.
- ray
- A non-basic row or column the simplex solver has identified
as causing primal unboundness, or None if no such variable
has been identified.
- rows
- Row collection. See the help on class BarCollection.
- status
- The status of solution of the last solver. This takes the
form of a string with these possible values.
opt -- The solution is optimal.
undef -- The solution is undefined.
feas -- The solution is feasible, but not necessarily optimal.
infeas -- The solution is infeasible.
nofeas -- The problem has no feasible solution.
unbnd -- The problem has an unbounded solution.
- status_dual
- The status of the dual solution of the simplex solver.
Possible values are 'undef', 'feas', 'infeas', 'nofeas' in
similar meaning to the .status attribute.
- status_i
- The status of the interior point solver's solution.
- status_m
- The status of the MIP solver's solution.
- status_primal
- The status of the primal solution of the simplex solver.
Possible values are 'undef', 'feas', 'infeas', 'nofeas' in
similar meaning to the .status attribute.
- status_s
- The status of the simplex solver's solution.
Data and other attributes defined here:
- BR_DTH = 4
- BR_FFV = 1
- BR_LFV = 2
- BR_MFV = 3
- BT_BFS = 2
- BT_BLB = 3
- BT_BPH = 4
- BT_DFS = 1
- DUALP = 2
- MSG_ALL = 3
- MSG_ERR = 1
- MSG_OFF = 0
- MSG_ON = 2
- PP_ALL = 2
- PP_NONE = 0
- PP_ROOT = 1
- PRIMAL = 1
- PT_PSE = 34
- PT_STD = 17
- RT_HAR = 34
- RT_STD = 17
- __new__ = <built-in method __new__ of type object at 0x41e360>
- T.__new__(S, ...) -> a new object with type S, a subtype of T
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class Objective(__builtin__.object) |
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Objective function objects for linear programs. An instance is
used either to access objective function values for solutions,
or to access or set objective function coefficients. The same
indices valid for a BarCollection object over the columns
(that is, column numeric indices, column names, slices,
multiple values) are also valid for indexing into this object.
The special index None is used to specify the shift term. If
we have an LPX instance lp, we may have:
lp.obj[0] --> the first objective coefficient
lp.obj[None] --> the shift term
lp.obj[-3:] --> the last three objective coefficients
lp.obj[1,4] --> the objective coefficients 1, 4
When assigning objective coefficients, for single indices one
may assign a single number. For multiple indices, one may
assign a single number to make all indicated coefficients
identical, or specify an iterable of equal length to set them
all individiaully. For example:
lp.obj[0]=2.5 --> set the first objective coef to 2.5
lp.obj[-3:]=1.0 --> the last three obj coefs get 1.0
lp.obj[1,4]=-2.0,2.0 --> obj coefs 1, 4 get -2.0, 2.0
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Methods defined here:
- __delitem__(...)
- x.__delitem__(y) <==> del x[y]
- __getitem__(...)
- x.__getitem__(y) <==> x[y]
- __iter__(...)
- x.__iter__() <==> iter(x)
- __len__(...)
- x.__len__() <==> len(x)
- __setitem__(...)
- x.__setitem__(i, y) <==> x[i]=y
Data descriptors defined here:
- maximize
- True or False depending on whether we are trying to maximize
or minimize this objective function, respectively.
- name
- Objective name, or None if unset.
- shift
- The constant shift term of the objective function.
- value
- The current value of the objective function.
- value_i
- The current value of the interior point objective function.
- value_m
- The current value of the MIP objective function.
- value_s
- The current value of the simplex objective function.
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class Tree(__builtin__.object) |
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Tree instances are passed to MIP solver callback function. They
are used to indicate the state of the solver at some intermediate
point in a call to LPX.integer(). There are nodes within the
tree, instances of TreeNode, corresponding to subproblems within
the search tree. The currently active subproblem is stored in
the curr_node member of an instance.
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Methods defined here:
- __iter__(...)
- x.__iter__() <==> iter(x)
- branch_upon(...)
- branch_upon(col_index, select='N')
Given the index of a column in the LP, this will add two
new subproblems, down and up branches (in that order) to the
active list, where the down and up branches are the problems
with the column's variable set to the floor and ceil of the
value, respectively. The select parameter controls which
of the two branches is selected to next continue the search
with 'D', 'U', and 'N' corresponding to choosing the down,
up, or letting GLPK select a branch, respectively.
- can_branch(...)
- can_branch(col_index)
Given the index of a column in the LP, this will return True
if one can branch upon this column's varible, that is,
continue the search with this column's variable set as an
integer. Note that this function should be called only when
the reason member of the tree is 'branch'.
- heuristic(...)
- heuristic(values)
Provide an integer feasible solution of the primal problem,
where values is an iterable object yielding at least as many
float values as there are columns in the problem. If the
provided solution is better than the existing one, the
solution is accepted and the problem updated. This function
returns True or False depending on whether the solution was
accepted or not. Note that this function should be called
only when the reason member of the tree is 'heur'.
- select(...)
- select(node)
Selects a tree node to continue search from. Note that this
function should be called only when the reason member of the
tree is 'select'.
- terminate(...)
- terminate()
Prematurely terminate the MIP solver's search.
Data descriptors defined here:
- best_node
- The node of the current active subproblem with best local bound.
If the tree is empty, this is None.
- curr_node
- The node of the current active subproblem. If there is no current
active subproblem in the tree, this will return None.
- first_node
- The node of the first active subproblem. If there is no current
active subproblem in the tree, this is None.
- gap
- The relative MIP gap (duality gap), that is, the gap between the
best MIP solution (best_mip) and best relaxed solution (best_bnd)
given by this formula:
|best_mip - best_bnd|
gap = ---------------------
|best_mip|+epsilon
- last_node
- The node of the last active subproblem. If there is no current
active subproblem in the tree, this is None.
- lp
- Problem object used by the MIP solver.
- num_active
- The number of active nodes.
- num_all
- The number of all nodes, both active and inactive.
- num_total
- The total number of nodes, including those already removed.
- reason
- A string with the reason the callback function has been called.
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class TreeNode(__builtin__.object) |
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TreeNode instances represent specific subproblem instances in the
search Tree object used by the MIP solver.
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Methods defined here:
- __eq__(...)
- x.__eq__(y) <==> x==y
- __ge__(...)
- x.__ge__(y) <==> x>=y
- __gt__(...)
- x.__gt__(y) <==> x>y
- __le__(...)
- x.__le__(y) <==> x<=y
- __lt__(...)
- x.__lt__(y) <==> x<y
- __ne__(...)
- x.__ne__(y) <==> x!=y
- __repr__(...)
- x.__repr__() <==> repr(x)
- __str__(...)
- x.__str__() <==> str(x)
Data descriptors defined here:
- active
- Whether this node represents an active subproblem.
- bound
- The local bound for this node's subproblem.
- level
- The level of the node in the tree, with 0 if this is the root.
- next
- The next active subproblem node, None if there is no next active
subproblem, or if this is not an active subproblem.
- prev
- The previous active subproblem node, None if there is no previous
active subproblem, or if this is not an active subproblem.
- subproblem
- The reference number of the subproblem corresponding to this node.
- up
- The parent subproblem node, None if this is the root.
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