linprog(method=’interiorpoint’)¶

scipy.optimize.
linprog
(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='interiorpoint', callback=None, options={'alpha0': 0.99995, 'beta': 0.1, 'maxiter': 1000, 'disp': False, 'tol': 1e08, 'sparse': False, 'lstsq': False, 'sym_pos': True, 'cholesky': None, 'pc': True, 'ip': False, 'presolve': True, 'permc_spec': 'MMD_AT_PLUS_A', 'rr': True, '_sparse_presolve': False}) Minimize a linear objective function subject to linear equality constraints, linear inequality constraints, and simple bounds using the interior point method of [1].
Linear programming is intended to solve problems of the following form:
Minimize: c^T * x Subject to: A_ub * x <= b_ub A_eq * x == b_eq bounds[i][0] < x_i < bounds[i][1]
Parameters:  c : array_like
Coefficients of the linear objective function to be minimized.
 A_ub : array_like, optional
2D array which, when matrixmultiplied by
x
, gives the values of the upperbound inequality constraints atx
. b_ub : array_like, optional
1D array of values representing the upperbound of each inequality constraint (row) in
A_ub
. A_eq : array_like, optional
2D array which, when matrixmultiplied by
x
, gives the values of the equality constraints atx
. b_eq : array_like, optional
1D array of values representing the right hand side of each equality constraint (row) in
A_eq
. bounds : sequence, optional
(min, max)
pairs for each element inx
, defining the bounds on that parameter. UseNone
for one ofmin
ormax
when there is no bound in that direction. By default bounds are(0, None)
(nonnegative). If a sequence containing a single tuple is provided, thenmin
andmax
will be applied to all variables in the problem.
Returns:  A ``scipy.optimize.OptimizeResult`` consisting of the following fields:
 x : ndarray
The independent variable vector which optimizes the linear programming problem.
 fun : float
The optimal value of the objective function
 con : float
The residuals of the equality constraints (nominally zero).
 slack : ndarray
The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, then the corresponding constraint is active.
 success : bool
Returns True if the algorithm succeeded in finding an optimal solution.
 status : int
An integer representing the exit status of the optimization:
0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded 4 : Serious numerical difficulties encountered
 nit : int
The number of iterations performed.
 message : str
A string descriptor of the exit status of the optimization.
See also
For documentation for the rest of the parameters, see
scipy.optimize.linprog
Options:  maxiter : int (default = 1000)
The maximum number of iterations of the algorithm.
 disp : bool (default = False)
Set to
True
if indicators of optimization status are to be printed to the console each iteration. tol : float (default = 1e8)
Termination tolerance to be used for all termination criteria; see [1] Section 4.5.
 alpha0 : float (default = 0.99995)
The maximal step size for Mehrota’s predictorcorrector search direction; see \(\beta_{3}\) of [1] Table 8.1.
 beta : float (default = 0.1)
The desired reduction of the path parameter \(\mu\) (see [3]) when Mehrota’s predictorcorrector is not in use (uncommon).
 sparse : bool (default = False)
Set to
True
if the problem is to be treated as sparse after presolve. If eitherA_eq
orA_ub
is a sparse matrix, this option will automatically be setTrue
, and the problem will be treated as sparse even during presolve. If your constraint matrices contain mostly zeros and the problem is not very small (less than about 100 constraints or variables), consider settingTrue
or providingA_eq
andA_ub
as sparse matrices. lstsq : bool (default = False)
Set to
True
if the problem is expected to be very poorly conditioned. This should always be leftFalse
unless severe numerical difficulties are encountered. Leave this at the default unless you receive a warning message suggesting otherwise. sym_pos : bool (default = True)
Leave
True
if the problem is expected to yield a well conditioned symmetric positive definite normal equation matrix (almost always). Leave this at the default unless you receive a warning message suggesting otherwise. cholesky : bool (default = True)
Set to
True
if the normal equations are to be solved by explicit Cholesky decomposition followed by explicit forward/backward substitution. This is typically faster for moderate, dense problems that are numerically wellbehaved. pc : bool (default = True)
Leave
True
if the predictorcorrector method of Mehrota is to be used. This is almost always (if not always) beneficial. ip : bool (default = False)
Set to
True
if the improved initial point suggestion due to [1] Section 4.3 is desired. Whether this is beneficial or not depends on the problem. presolve : bool (default = True)
Leave
True
if presolve routine should be run. The presolve routine is almost always useful because it can detect trivial infeasibilities and unboundedness, eliminate fixed variables, and remove redundancies. One circumstance in which it might be turned off (setFalse
) is when it detects that the problem is trivially unbounded; it is possible that that the problem is truly infeasibile but this has not been detected. rr : bool (default = True)
Default
True
attempts to eliminate any redundant rows inA_eq
. SetFalse
ifA_eq
is known to be of full row rank, or if you are looking for a potential speedup (at the expense of reliability). permc_spec : str (default = ‘MMD_AT_PLUS_A’)
(Has effect only with
sparse = True
,lstsq = False
,sym_pos = True
.) A matrix is factorized in each iteration of the algorithm. This option specifies how to permute the columns of the matrix for sparsity preservation. Acceptable values are:NATURAL
: natural ordering.MMD_ATA
: minimum degree ordering on the structure of A^T A.MMD_AT_PLUS_A
: minimum degree ordering on the structure of A^T+A.COLAMD
: approximate minimum degree column ordering.
This option can impact the convergence of the interior point algorithm; test different values to determine which performs best for your problem. For more information, refer to
scipy.sparse.linalg.splu
.
Notes
This method implements the algorithm outlined in [1] with ideas from [5] and a structure inspired by the simpler methods of [3] and [4].
First, a presolve procedure based on [5] attempts to identify trivial infeasibilities, trivial unboundedness, and potential problem simplifications. Specifically, it checks for:
 rows of zeros in
A_eq
orA_ub
, representing trivial constraints;  columns of zeros in
A_eq
andA_ub
, representing unconstrained variables;  column singletons in
A_eq
, representing fixed variables; and  column singletons in
A_ub
, representing simple bounds.
If presolve reveals that the problem is unbounded (e.g. an unconstrained and unbounded variable has negative cost) or infeasible (e.g. a row of zeros in
A_eq
corresponds with a nonzero inb_eq
), the solver terminates with the appropriate status code. Note that presolve terminates as soon as any sign of unboundedness is detected; consequently, a problem may be reported as unbounded when in reality the problem is infeasible (but infeasibility has not been detected yet). Therefore, if the output message states that unboundedness is detected in presolve and it is necessary to know whether the problem is actually infeasible, set optionpresolve=False
.If neither infeasibility nor unboundedness are detected in a single pass of the presolve check, bounds are tightened where possible and fixed variables are removed from the problem. Then, linearly dependent rows of the
A_eq
matrix are removed, (unless they represent an infeasibility) to avoid numerical difficulties in the primary solve routine. Note that rows that are nearly linearly dependent (within a prescibed tolerance) may also be removed, which can change the optimal solution in rare cases. If this is a concern, eliminate redundancy from your problem formulation and run with optionrr=False
orpresolve=False
.Several potential improvements can be made here: additional presolve checks outlined in [5] should be implemented, the presolve routine should be run multiple times (until no further simplifications can be made), and more of the efficiency improvements from [2] should be implemented in the redundancy removal routines.
After presolve, the problem is transformed to standard form by converting the (tightened) simple bounds to upper bound constraints, introducing nonnegative slack variables for inequality constraints, and expressing unbounded variables as the difference between two nonnegative variables.
The primaldual path following method begins with initial ‘guesses’ of the primal and dual variables of the standard form problem and iteratively attempts to solve the (nonlinear) KarushKuhnTucker conditions for the problem with a gradually reduced logarithmic barrier term added to the objective. This particular implementation uses a homogeneous selfdual formulation, which provides certificates of infeasibility or unboundedness where applicable.
The default initial point for the primal and dual variables is that defined in [1] Section 4.4 Equation 8.22. Optionally (by setting initial point option
ip=True
), an alternate (potentially improved) starting point can be calculated according to the additional recommendations of [1] Section 4.4.A search direction is calculated using the predictorcorrector method (single correction) proposed by Mehrota and detailed in [1] Section 4.1. (A potential improvement would be to implement the method of multiple corrections described in [1] Section 4.2.) In practice, this is accomplished by solving the normal equations, [1] Section 5.1 Equations 8.31 and 8.32, derived from the Newton equations [1] Section 5 Equations 8.25 (compare to [1] Section 4 Equations 8.68.8). The advantage of solving the normal equations rather than 8.25 directly is that the matrices involved are symmetric positive definite, so Cholesky decomposition can be used rather than the more expensive LU factorization.
With the default
cholesky=True
, this is accomplished usingscipy.linalg.cho_factor
followed by forward/backward substitutions viascipy.linalg.cho_solve
. Withcholesky=False
andsym_pos=True
, Cholesky decomposition is performed instead byscipy.linalg.solve
. Based on speed tests, this also appears to retain the Cholesky decomposition of the matrix for later use, which is beneficial as the same system is solved four times with different right hand sides in each iteration of the algorithm.In problems with redundancy (e.g. if presolve is turned off with option
presolve=False
) or if the matrices become illconditioned (e.g. as the solution is approached and some decision variables approach zero), Cholesky decomposition can fail. Should this occur, successively more robust solvers (scipy.linalg.solve
withsym_pos=False
thenscipy.linalg.lstsq
) are tried, at the cost of computational efficiency. These solvers can be used from the outset by setting the optionssym_pos=False
andlstsq=True
, respectively.Note that with the option
sparse=True
, the normal equations are solved usingscipy.sparse.linalg.spsolve
. Unfortunately, this uses the more expensive LU decomposition from the outset, but for large, sparse problems, the use of sparse linear algebra techniques improves the solve speed despite the use of LU rather than Cholesky decomposition. A simple improvement would be to use the sparse Cholesky decomposition ofCHOLMOD
viascikitsparse
when available.Other potential improvements for combatting issues associated with dense columns in otherwise sparse problems are outlined in [1] Section 5.3 and [7] Section 4.14.2; the latter also discusses the alleviation of accuracy issues associated with the substitution approach to free variables.
After calculating the search direction, the maximum possible step size that does not activate the nonnegativity constraints is calculated, and the smaller of this step size and unity is applied (as in [1] Section 4.1.) [1] Section 4.3 suggests improvements for choosing the step size.
The new point is tested according to the termination conditions of [1] Section 4.5. The same tolerance, which can be set using the
tol
option, is used for all checks. (A potential improvement would be to expose the different tolerances to be set independently.) If optimality, unboundedness, or infeasibility is detected, the solve procedure terminates; otherwise it repeats.If optimality is achieved, a postsolve procedure undoes transformations associated with presolve and converting to standard form. It then calculates the residuals (equality constraint violations, which should be very small) and slacks (difference between the left and right hand sides of the upper bound constraints) of the original problem, which are returned with the solution in an
OptimizeResult
object.References
[1] (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17) Andersen, Erling D., and Knud D. Andersen. “The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm.” High performance optimization. Springer US, 2000. 197232. [2] (1, 2) Andersen, Erling D. “Finding all linearly dependent rows in largescale linear programming.” Optimization Methods and Software 6.3 (1995): 219227. [3] (1, 2, 3) Freund, Robert M. “PrimalDual InteriorPoint Methods for Linear Programming based on Newton’s Method.” Unpublished Course Notes, March 2004. Available 2/25/2017 at https://ocw.mit.edu/courses/sloanschoolofmanagement/15084jnonlinearprogrammingspring2004/lecturenotes/lec14_int_pt_mthd.pdf [4] (1, 2) Fourer, Robert. “Solving Linear Programs by InteriorPoint Methods.” Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at http://www.4er.org/CourseNotes/Book%20B/BIII.pdf [5] (1, 2, 3, 4) Andersen, Erling D., and Knud D. Andersen. “Presolving in linear programming.” Mathematical Programming 71.2 (1995): 221245. [6] Bertsimas, Dimitris, and J. Tsitsiklis. “Introduction to linear programming.” Athena Scientific 1 (1997): 997. [7] (1, 2) Andersen, Erling D., et al. Implementation of interior point methods for large scale linear programming. HEC/Universite de Geneve, 1996.