optiml.opti package

Subpackages

• optiml.opti.constrained package
  • Submodules
    • optiml.opti.constrained.active_set module
      • ActiveSet
    • optiml.opti.constrained.frank_wolfe module
      • FrankWolfe
    • optiml.opti.constrained.interior_point module
      • InteriorPoint
    • optiml.opti.constrained.projected_gradient module
      • ProjectedGradient
  • Module contents
    • BoxConstrainedQuadraticOptimizer
      • BoxConstrainedQuadraticOptimizer.f_star()
      • BoxConstrainedQuadraticOptimizer.x_star()
      • BoxConstrainedQuadraticOptimizer.callback()
      • BoxConstrainedQuadraticOptimizer.check_lagrangian_dual_conditions()
      • BoxConstrainedQuadraticOptimizer.check_lagrangian_dual_optimality()
      • BoxConstrainedQuadraticOptimizer.is_augmented_lagrangian_dual()
      • BoxConstrainedQuadraticOptimizer.is_lagrangian_dual()
      • BoxConstrainedQuadraticOptimizer.is_verbose()
      • BoxConstrainedQuadraticOptimizer.minimize()
    • LagrangianQuadratic
      • LagrangianQuadratic.f_star()
      • LagrangianQuadratic.x_star()
      • LagrangianQuadratic.constraints()
      • LagrangianQuadratic.function()
      • LagrangianQuadratic.jacobian()
      • LagrangianQuadratic.hessian()
      • LagrangianQuadratic.args()
      • LagrangianQuadratic.function_jacobian()
    • AugmentedLagrangianQuadratic
      • AugmentedLagrangianQuadratic.args()
      • AugmentedLagrangianQuadratic.f_star()
      • AugmentedLagrangianQuadratic.x_star()
      • AugmentedLagrangianQuadratic.constraints()
      • AugmentedLagrangianQuadratic.function()
      • AugmentedLagrangianQuadratic.jacobian()
      • AugmentedLagrangianQuadratic.function_jacobian()
      • AugmentedLagrangianQuadratic.hessian()
    • ProjectedGradient
      • ProjectedGradient.minimize()
      • ProjectedGradient.callback()
      • ProjectedGradient.check_lagrangian_dual_conditions()
      • ProjectedGradient.check_lagrangian_dual_optimality()
      • ProjectedGradient.f_star()
      • ProjectedGradient.is_augmented_lagrangian_dual()
      • ProjectedGradient.is_lagrangian_dual()
      • ProjectedGradient.is_verbose()
      • ProjectedGradient.x_star()
    • ActiveSet
      • ActiveSet.minimize()
      • ActiveSet.callback()
      • ActiveSet.check_lagrangian_dual_conditions()
      • ActiveSet.check_lagrangian_dual_optimality()
      • ActiveSet.f_star()
      • ActiveSet.is_augmented_lagrangian_dual()
      • ActiveSet.is_lagrangian_dual()
      • ActiveSet.is_verbose()
      • ActiveSet.x_star()
    • FrankWolfe
      • FrankWolfe.minimize()
      • FrankWolfe.callback()
      • FrankWolfe.check_lagrangian_dual_conditions()
      • FrankWolfe.check_lagrangian_dual_optimality()
      • FrankWolfe.f_star()
      • FrankWolfe.is_augmented_lagrangian_dual()
      • FrankWolfe.is_lagrangian_dual()
      • FrankWolfe.is_verbose()
      • FrankWolfe.x_star()
    • InteriorPoint
      • InteriorPoint.minimize()
      • InteriorPoint.callback()
      • InteriorPoint.check_lagrangian_dual_conditions()
      • InteriorPoint.check_lagrangian_dual_optimality()
      • InteriorPoint.f_star()
      • InteriorPoint.is_augmented_lagrangian_dual()
      • InteriorPoint.is_lagrangian_dual()
      • InteriorPoint.is_verbose()
      • InteriorPoint.x_star()
• optiml.opti.unconstrained package
  • Subpackages
    • optiml.opti.unconstrained.line_search package
      • Submodules
      • Module contents
    • optiml.opti.unconstrained.stochastic package
      • Submodules
      • Module contents
  • Submodules
    • optiml.opti.unconstrained.proximal_bundle module
      • ProximalBundle
  • Module contents
    • Rosenbrock
      • Rosenbrock.x_star()
      • Rosenbrock.f_star()
      • Rosenbrock.function()
      • Rosenbrock.args()
      • Rosenbrock.function_jacobian()
      • Rosenbrock.hessian()
      • Rosenbrock.jacobian()
    • Ackley
      • Ackley.x_star()
      • Ackley.f_star()
      • Ackley.function()
      • Ackley.args()
      • Ackley.function_jacobian()
      • Ackley.hessian()
      • Ackley.jacobian()
    • SixHumpCamel
      • SixHumpCamel.x_star()
      • SixHumpCamel.f_star()
      • SixHumpCamel.function()
      • SixHumpCamel.args()
      • SixHumpCamel.function_jacobian()
      • SixHumpCamel.hessian()
      • SixHumpCamel.jacobian()
    • ProximalBundle
      • ProximalBundle.minimize()
      • ProximalBundle.callback()
      • ProximalBundle.check_lagrangian_dual_conditions()
      • ProximalBundle.check_lagrangian_dual_optimality()
      • ProximalBundle.is_augmented_lagrangian_dual()
      • ProximalBundle.is_lagrangian_dual()
      • ProximalBundle.is_verbose()

Submodules

• optiml.opti.utils module
  • cholesky_null_space()
  • solve_lagrangian_equality_constrained_quadratic()
  • generate_box_constrained_quadratic()
  • plot_surface_contour()
  • plot_trajectory_optimization()
  • plot_surface_trajectory_optimization()

Module contents

class optiml.opti.Optimizer(f, x=None, eps=1e-06, tol=1e-08, max_iter=1000, callback=None, callback_args=(), random_state=None, verbose=False)

Bases: ABC

Parameters:

• f – the objective function.

• x – ([n x 1] real column vector): 1D array of points at which the Hessian is to be computed.

• eps – (real scalar, optional, default value 1e-6): the accuracy in the stopping criterion: the algorithm is stopped when the norm of the gradient is less than or equal to eps.

• max_iter – (integer scalar, optional, default value 1000): the maximum number of iterations.

• verbose – (boolean, optional, default value False): print details about each iteration if True, nothing otherwise.

is_lagrangian_dual()

is_augmented_lagrangian_dual()

callback(args=())

check_lagrangian_dual_optimality()

check_lagrangian_dual_conditions()

is_verbose()

minimize()

class optiml.opti.OptimizationFunction(ndim=2)

Bases: ABC

x_star()

f_star()

args()

function(x)

jacobian(x)

The Jacobian (i.e., the gradient) of the function.

Parameters:

x – 1D array of points at which the Jacobian is to be computed.

Returns:

the Jacobian of the function at x.

function_jacobian(*args, **kwargs)

hessian(x)

The Hessian matrix of the function.

Parameters:

x – 1D array of points at which the Hessian is to be computed.

Returns:

the Hessian matrix of the function at x.

class optiml.opti.Quadratic(Q, q)

Bases: OptimizationFunction

Construct a quadratic function from its linear and quadratic part defined as

\[\tfrac{1}{2} x^\top Q x + q^\top x\]

Parameters:

• Q – ([n x n] real symmetric matrix, not necessarily positive semidefinite): the Hessian (i.e., the quadratic part) of f. If it is not positive semidefinite, f(x) will be unbounded below.

• q – ([n x 1] real column vector): the linear part of f.

x_star()

f_star()

function(x)

A general quadratic function \(f(x) = \tfrac{1}{2} x^\top Q x + q^\top x\).

Parameters:

x – ([n x 1] real column vector): 1D array of points at which the Hessian is to be computed.

Returns:

the value \(\tfrac{1}{2} x^\top Q x + q^\top x\) of the general quadratic function at x.

jacobian(x)

The Jacobian (i.e., the gradient) of a general quadratic function \(J f(x) = Q x + q\).

Parameters:

x – ([n x 1] real column vector): 1D array of points at which the Jacobian is to be computed.

Returns:

the Jacobian of a general quadratic function.

hessian(x)

The Hessian matrix of a general quadratic function \(H f(x) = Q\).

Parameters:

x – 1D array of points at which the Hessian is to be computed.

Returns:

the Hessian matrix (i.e., the quadratic part) of a general quadratic function at x.

args()

function_jacobian(*args, **kwargs)