bundles / scipy 1.17.1 / scipy / optimize / _linprog_ip / _ip_hsd

function

`scipy.optimize._linprog_ip:_ip_hsd`

source: /scipy/optimize/_linprog_ip.py :569

Signature

   def     _ip_hsd (    A    ,    b    ,    c    ,    c0    ,    alpha0    ,    beta    ,    maxiter    ,    disp    ,    tol    ,    sparse    ,    lstsq    ,    sym_pos    ,    cholesky    ,    pc    ,    ip    ,    permc_spec    ,    callback    ,    postsolve_args      )

Summary

Solve a linear programming problem in standard form:

Extended Summary

Minimize

c @ x

Subject to

A @ x == b
    x >= 0

using the interior point method of [4].

Parameters

A : 2-D array

2-D array such that A @ x, gives the values of the equality constraints at x.

b : 1-D array

1-D array of values representing the RHS of each equality constraint (row) in A (for standard form problem).

c : 1-D array

Coefficients of the linear objective function to be minimized (for standard form problem).

c0 : float

Constant term in objective function due to fixed (and eliminated) variables. (Purely for display.)

alpha0 : float

The maximal step size for Mehrota's predictor-corrector search direction; see :math:`beta_3`of [4] Table 8.1

beta : float

The desired reduction of the path parameter $μ$ (see ^[6])

maxiter : int

The maximum number of iterations of the algorithm.

disp : bool

Set to True if indicators of optimization status are to be printed to the console each iteration.

tol : float

Termination tolerance; see ^[4] Section 4.5.

sparse : bool

Set to True if the problem is to be treated as sparse. However, the inputs A_eq and A_ub should nonetheless be provided as (dense) arrays rather than sparse matrices.

lstsq : bool

Set to True if the problem is expected to be very poorly conditioned. This should always be left as False unless severe numerical difficulties are frequently encountered, and a better option would be to improve the formulation of the problem.

sym_pos : bool

Leave True if the problem is expected to yield a well conditioned symmetric positive definite normal equation matrix (almost always).

cholesky : bool

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

Leave True if the predictor-corrector method of Mehrota is to be used. This is almost always (if not always) beneficial.

ip : bool

Set to True if the improved initial point suggestion due to ^[4] Section 4.3 is desired. It's unclear whether this is beneficial.

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.

callback : callable, optional

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

x
x
fun
fun
success
success
slack
slack
con
con
phase
phase
status
status
nit
nit
message
message

postsolve_args : tuple

Data needed by _postsolve to convert the solution to the standard-form problem into the solution to the original problem.

Returns

x_hat : float

Solution vector (for standard form problem).

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

message : str

A string descriptor of the exit status of the optimization.

iteration : int

The number of iterations taken to solve the problem

Aliases

scipy.optimize._linprog_ip._ip_hsd