linprog(method=’highs’)¶
- scipy.optimize.linprog(c, A_ub=None, b_ub=None, A_eq=None, b_eq=None, bounds=None, method='highs', callback=None, options={'maxiter': None, 'disp': False, 'presolve': True, 'time_limit': None, 'dual_feasibility_tolerance': None, 'primal_feasibility_tolerance': None, 'ipm_optimality_tolerance': None, 'simplex_dual_edge_weight_strategy': None}, x0=None)
Linear programming: minimize a linear objective function subject to linear equality and inequality constraints using one of the HiGHS solvers.
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.
Alternatively, 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
andub = None
unless specified withbounds
.- Parameters
- c1-D array
The coefficients of the linear objective function to be minimized.
- A_ub2-D array, optional
The inequality constraint matrix. Each row of
A_ub
specifies the coefficients of a linear inequality constraint onx
.- b_ub1-D array, optional
The inequality constraint vector. Each element represents an upper bound on the corresponding value of
A_ub @ x
.- A_eq2-D array, optional
The equality constraint matrix. Each row of
A_eq
specifies the coefficients of a linear equality constraint onx
.- b_eq1-D array, optional
The equality constraint vector. Each element of
A_eq @ x
must equal the corresponding element ofb_eq
.- boundssequence, optional
A sequence of
(min, max)
pairs for each element inx
, defining the minimum and maximum values of that decision variable. UseNone
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, thenmin
andmax
will serve as bounds for all decision variables.- methodstr
This is the method-specific documentation for ‘highs’, which chooses automatically between ‘highs-ds’ and ‘highs-ipm’. ‘interior-point’ (default), ‘revised simplex’, and ‘simplex’ (legacy) are also available.
- 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,
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 or time limit reached.2
: Problem appears to be infeasible.3
: Problem appears to be unbounded.4
: The HiGHS solver ran into a problem.- messagestr
A string descriptor of the exit status of the algorithm.
- nitint
The total number of iterations performed. For the HiGHS simplex method, this includes iterations in all phases. For the HiGHS interior-point method, this does not include crossover iterations.
- crossover_nitint
The number of primal/dual pushes performed during the crossover routine for the HiGHS interior-point method. This is
0
for the HiGHS simplex method.- ineqlinOptimizeResult
Solution and sensitivity information corresponding to the inequality constraints, b_ub. A dictionary consisting of the fields:
- residualnp.ndnarray
The (nominally positive) values of the slack variables,
b_ub - A_ub @ x
. This quantity is also commonly referred to as “slack”.- marginalsnp.ndarray
The sensitivity (partial derivative) of the objective function with respect to the right-hand side of the inequality constraints, b_ub.
- eqlinOptimizeResult
Solution and sensitivity information corresponding to the equality constraints, b_eq. A dictionary consisting of the fields:
- residualnp.ndarray
The (nominally zero) residuals of the equality constraints,
b_eq - A_eq @ x
.- marginalsnp.ndarray
The sensitivity (partial derivative) of the objective function with respect to the right-hand side of the equality constraints, b_eq.
- lower, upperOptimizeResult
Solution and sensitivity information corresponding to the lower and upper bounds on decision variables, bounds.
- residualnp.ndarray
The (nominally positive) values of the quantity
x - lb
(lower) orub - x
(upper).- marginalsnp.ndarray
The sensitivity (partial derivative) of the objective function with respect to the lower and upper bounds.
See also
For documentation for the rest of the parameters, see
scipy.optimize.linprog
- Options
- maxiterint
The maximum number of iterations to perform in either phase. For ‘highs-ipm’, this does not include the number of crossover iterations. Default is the largest possible value for an
int
on the platform.- dispbool (default:
False
) Set to
True
if indicators of optimization status are to be printed to the console during optimization.- presolvebool (default:
True
) Presolve attempts to identify trivial infeasibilities, identify trivial unboundedness, and simplify the problem before sending it to the main solver. It is generally recommended to keep the default setting
True
; set toFalse
if presolve is to be disabled.- time_limitfloat
The maximum time in seconds allotted to solve the problem; default is the largest possible value for a
double
on the platform.- dual_feasibility_tolerancedouble (default: 1e-07)
Dual feasibility tolerance for ‘highs-ds’. The minimum of this and
primal_feasibility_tolerance
is used for the feasibility tolerance of ‘highs-ipm’.- primal_feasibility_tolerancedouble (default: 1e-07)
Primal feasibility tolerance for ‘highs-ds’. The minimum of this and
dual_feasibility_tolerance
is used for the feasibility tolerance of ‘highs-ipm’.- ipm_optimality_tolerancedouble (default:
1e-08
) Optimality tolerance for ‘highs-ipm’. Minimum allowable value is 1e-12.
- simplex_dual_edge_weight_strategystr (default: None)
Strategy for simplex dual edge weights. The default,
None
, automatically selects one of the following.'dantzig'
uses Dantzig’s original strategy of choosing the most negative reduced cost.'devex'
uses the strategy described in [15].steepest
uses the exact steepest edge strategy as described in [16].'steepest-devex'
begins with the exact steepest edge strategy until the computation is too costly or inexact and then switches to the devex method.Curently,
None
always selects'steepest-devex'
, but this may change as new options become available.- unknown_optionsdict
Optional arguments not used by this particular solver. If
unknown_options
is non-empty, a warning is issued listing all unused options.
Notes
Method ‘highs-ds’ is a wrapper of the C++ high performance dual revised simplex implementation (HSOL) [13], [14]. Method ‘highs-ipm’ is a wrapper of a C++ implementation of an interior-point method [13]; it features a crossover routine, so it is as accurate as a simplex solver. Method ‘highs’ chooses between the two automatically. For new code involving
linprog
, we recommend explicitly choosing one of these three method values instead of ‘interior-point’ (default), ‘revised simplex’, and ‘simplex’ (legacy).The result fields ineqlin, eqlin, lower, and upper all contain marginals, or partial derivatives of the objective function with respect to the right-hand side of each constraint. These partial derivatives are also referred to as “Lagrange multipliers”, “dual values”, and “shadow prices”. The sign convention of marginals is opposite that of Lagrange multipliers produced by many nonlinear solvers.
References
- 13(1,2)
Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J. “HiGHS - high performance software for linear optimization.” Accessed 4/16/2020 at https://www.maths.ed.ac.uk/hall/HiGHS/#guide
- 14
Huangfu, Q. and Hall, J. A. J. “Parallelizing the dual revised simplex method.” Mathematical Programming Computation, 10 (1), 119-142, 2018. DOI: 10.1007/s12532-017-0130-5
- 15
Harris, Paula MJ. “Pivot selection methods of the Devex LP code.” Mathematical programming 5.1 (1973): 1-28.
- 16
Goldfarb, Donald, and John Ker Reid. “A practicable steepest-edge simplex algorithm.” Mathematical Programming 12.1 (1977): 361-371.