linprog(method=’interior-point’)

scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='interior-point', callback=None, options={'alpha0': 0.99995, 'beta': 0.1, 'maxiter': 1000, 'disp': False, 'tol': 1e-08, '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

2-D array which, when matrix-multiplied by x, gives the values of the upper-bound inequality constraints at x.

b_ub : array_like, optional

1-D array of values representing the upper-bound of each inequality constraint (row) in A_ub.

A_eq : array_like, optional

2-D array which, when matrix-multiplied by x, gives the values of the equality constraints at x.

b_eq : array_like, optional

1-D 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 in x, defining the bounds on that parameter. Use None for one of min or max when there is no bound in that direction. By default bounds are (0, None) (non-negative). If a sequence containing a single tuple is provided, then min and max 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 = 1e-8)

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 predictor-corrector 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 predictor-corrector is not in use (uncommon).

sparse : bool (default = False)

Set to True if the problem is to be treated as sparse after presolve. If either A_eq or A_ub is a sparse matrix, this option will automatically be set True, 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 setting True or providing A_eq and A_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 left False 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 well-behaved.

pc : bool (default = True)

Leave True if the predictor-corrector 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 (set False) 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 in A_eq. Set False if A_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 or A_ub, representing trivial constraints;
• columns of zeros in A_eq and A_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 in b_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 option presolve=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 option rr=False or presolve=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 non-negative slack variables for inequality constraints, and expressing unbounded variables as the difference between two non-negative variables.

The primal-dual 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) Karush-Kuhn-Tucker conditions for the problem with a gradually reduced logarithmic barrier term added to the objective. This particular implementation uses a homogeneous self-dual 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 predictor-corrector 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 8.6-8.8). 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 using scipy.linalg.cho_factor followed by forward/backward substitutions via scipy.linalg.cho_solve. With cholesky=False and sym_pos=True, Cholesky decomposition is performed instead by scipy.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 ill-conditioned (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 with sym_pos=False then scipy.linalg.lstsq) are tried, at the cost of computational efficiency. These solvers can be used from the outset by setting the options sym_pos=False and lstsq=True, respectively.

Note that with the option sparse=True, the normal equations are solved using scipy.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 of CHOLMOD via scikit-sparse 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 4.1-4.2; 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 non-negativity 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. 197-232.

[2]	(1, 2) Andersen, Erling D. “Finding all linearly dependent rows in large-scale linear programming.” Optimization Methods and Software 6.3 (1995): 219-227.

[3]

(1, 2, 3) Freund, Robert M. “Primal-Dual Interior-Point Methods for Linear Programming based on Newton’s Method.” Unpublished Course Notes, March 2004. Available 2/25/2017 at https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf

[4]	(1, 2) Fourer, Robert. “Solving Linear Programs by Interior-Point Methods.” Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at http://www.4er.org/CourseNotes/Book%20B/B-III.pdf

[5]	(1, 2, 3, 4) Andersen, Erling D., and Knud D. Andersen. “Presolving in linear programming.” Mathematical Programming 71.2 (1995): 221-245.

[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.