scipy.optimize.linprog

scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='interior-point', callback=None, options=None, x0=None)

Linear programming: minimize a linear objective function subject to linear equality and inequality constraints.

Linear programming solves problems of the following form:

\[\begin{split}\min_x \ & c^T x \\ \mbox{such that} \ & A_{ub} x \leq b_{ub},\\ & A_{eq} x = b_{eq},\\ & l \leq x \leq u ,\end{split}\]

where \(x\) is a vector of decision variables; \(c\), \(b_{ub}\), \(b_{eq}\), \(l\), and \(u\) are vectors; and \(A_{ub}\) and \(A_{eq}\) are matrices.

Informally, that’s:

minimize:

c @ x

such that:

A_ub @ x <= b_ub
A_eq @ x == b_eq
lb <= x <= ub

Note that by default lb = 0 and ub = None unless specified with bounds.

Parameters

c1D array

The coefficients of the linear objective function to be minimized.

A_ub2D array, optional

The inequality constraint matrix. Each row of A_ub specifies the coefficients of a linear inequality constraint on x.

b_ub1D array, optional

The inequality constraint vector. Each element represents an upper bound on the corresponding value of A_ub @ x.

A_eq2D array, optional

The equality constraint matrix. Each row of A_eq specifies the coefficients of a linear equality constraint on x.

b_eq1D array, optional

The equality constraint vector. Each element of A_eq @ x must equal the corresponding element of b_eq.

boundssequence, optional

A sequence of (min, max) pairs for each element in x, defining the minimum and maximum values of that decision variable. Use None to indicate that there is no bound. By default, bounds are (0, None) (all decision variables are non-negative). If a single tuple (min, max) is provided, then min and max will serve as bounds for all decision variables.

method{‘interior-point’, ‘revised simplex’, ‘simplex’}, optional

The algorithm used to solve the standard form problem. ‘interior-point’ (default), ‘revised simplex’, and ‘simplex’ (legacy) are supported.

callbackcallable, optional

If a callback function is provided, it will be called at least once per iteration of the algorithm. The callback function must accept a single scipy.optimize.OptimizeResult consisting of the following fields:

x1D array

The current solution vector.

funfloat

The current value of the objective function c @ x.

successbool

True when the algorithm has completed successfully.

slack1D array

The (nominally positive) values of the slack, b_ub - A_ub @ x.

con1D array

The (nominally zero) residuals of the equality constraints, b_eq - A_eq @ x.

phaseint

The phase of the algorithm being executed.

statusint

An integer representing the status of the algorithm.

0 : Optimization proceeding nominally.

1 : Iteration limit reached.

2 : Problem appears to be infeasible.

3 : Problem appears to be unbounded.

4 : Numerical difficulties encountered.

nitint

The current iteration number.

messagestr

A string descriptor of the algorithm status.

optionsdict, optional

A dictionary of solver options. All methods accept the following options:

maxiterint

Maximum number of iterations to perform.

dispbool

Set to True to print convergence messages.

For method-specific options, see show_options('linprog').

x01D array, optional

Guess values of the decision variables, which will be refined by the optimization algorithm. This argument is currently used only by the ‘revised simplex’ method, and can only be used if x0 represents a basic feasible solution.

Returns

resOptimizeResult

A scipy.optimize.OptimizeResult consisting of the fields:

x1D array

The values of the decision variables that minimizes the objective function while satisfying the constraints.

funfloat

The optimal value of the objective function c @ x.

slack1D array

The (nominally positive) values of the slack variables, b_ub - A_ub @ x.

con1D array

The (nominally zero) residuals of the equality constraints, b_eq - A_eq @ x.

successbool

True when the algorithm succeeds in finding an optimal solution.

statusint

An integer representing the exit status of the algorithm.

0 : Optimization terminated successfully.

1 : Iteration limit reached.

2 : Problem appears to be infeasible.

3 : Problem appears to be unbounded.

4 : Numerical difficulties encountered.

nitint

The total number of iterations performed in all phases.

messagestr

A string descriptor of the exit status of the algorithm.

See also

show_options

Additional options accepted by the solvers.

Notes

This section describes the available solvers that can be selected by the ‘method’ parameter.

‘interior-point’ is the default as it is typically the fastest and most robust method. ‘revised simplex’ is more accurate for the problems it solves. ‘simplex’ is the legacy method and is included for backwards compatibility and educational purposes.

Method interior-point uses the primal-dual path following algorithm as outlined in [4]. This algorithm supports sparse constraint matrices and is typically faster than the simplex methods, especially for large, sparse problems. Note, however, that the solution returned may be slightly less accurate than those of the simplex methods and will not, in general, correspond with a vertex of the polytope defined by the constraints.

New in version 1.0.0.

Method revised simplex uses the revised simplex method as decribed in [9], except that a factorization [11] of the basis matrix, rather than its inverse, is efficiently maintained and used to solve the linear systems at each iteration of the algorithm.

New in version 1.3.0.

Method simplex uses a traditional, full-tableau implementation of Dantzig’s simplex algorithm [1], [2] (not the Nelder-Mead simplex). This algorithm is included for backwards compatibility and educational purposes.

New in version 0.15.0.

Before applying any method, a presolve procedure based on [8] 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 it is important to know whether the problem is actually infeasible, solve the problem again with option presolve=False.

If neither infeasibility nor unboundedness are detected in a single pass of the presolve, 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 prescribed 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 [8] 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 [5] 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 selected algorithm solves the standard form problem, and a postprocessing routine converts this to a solution to the original problem.

References

1(1,2)

Dantzig, George B., Linear programming and extensions. Rand Corporation Research Study Princeton Univ. Press, Princeton, NJ, 1963

2(1,2)

Hillier, S.H. and Lieberman, G.J. (1995), “Introduction to Mathematical Programming”, McGraw-Hill, Chapter 4.

3

Bland, Robert G. New finite pivoting rules for the simplex method. Mathematics of Operations Research (2), 1977: pp. 103-107.

4(1,2)

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.

5(1,2)

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

6

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

7

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

8(1,2,3)

Andersen, Erling D., and Knud D. Andersen. “Presolving in linear programming.” Mathematical Programming 71.2 (1995): 221-245.

9(1,2)

Bertsimas, Dimitris, and J. Tsitsiklis. “Introduction to linear programming.” Athena Scientific 1 (1997): 997.

10

Andersen, Erling D., et al. Implementation of interior point methods for large scale linear programming. HEC/Universite de Geneve, 1996.

11(1,2)

Bartels, Richard H. “A stabilization of the simplex method.” Journal in Numerische Mathematik 16.5 (1971): 414-434.

Examples

Consider the following problem:

\[\begin{split}\min_{x_0, x_1} \ -x_0 + 4x_1 & \\ \mbox{such that} \ -3x_0 + x_1 & \leq 6,\\ -x_0 - 2x_1 & \geq -4,\\ x_1 & \geq -3.\end{split}\]

The problem is not presented in the form accepted by linprog. This is easily remedied by converting the “greater than” inequality constraint to a “less than” inequality constraint by multiplying both sides by a factor of \(-1\). Note also that the last constraint is really the simple bound \(-3 \leq x_1 \leq \infty\). Finally, since there are no bounds on \(x_0\), we must explicitly specify the bounds \(-\infty \leq x_0 \leq \infty\), as the default is for variables to be non-negative. After collecting coeffecients into arrays and tuples, the input for this problem is:

>>> c = [-1, 4]
>>> A = [[-3, 1], [1, 2]]
>>> b = [6, 4]
>>> x0_bounds = (None, None)
>>> x1_bounds = (-3, None)
>>> from scipy.optimize import linprog
>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds])

Note that the default method for linprog is ‘interior-point’, which is approximate by nature.

>>> print(res)
     con: array([], dtype=float64)
     fun: -21.99999984082494 # may vary
 message: 'Optimization terminated successfully.'
     nit: 6 # may vary
   slack: array([3.89999997e+01, 8.46872439e-08] # may vary
  status: 0
 success: True
       x: array([ 9.99999989, -2.99999999]) # may vary

If you need greater accuracy, try ‘revised simplex’.

>>> res = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds], method='revised simplex')
>>> print(res)
     con: array([], dtype=float64)
     fun: -22.0 # may vary
 message: 'Optimization terminated successfully.'
     nit: 1 # may vary
   slack: array([39.,  0.]) # may vary
  status: 0
 success: True
       x: array([10., -3.]) # may vary