Sage Reference Manual: Numerical Optimization - Mirrors

Sage Reference Manual: Numerical Optimization, Release 6.1.1
sage: c=vector(RDF,[-4,-5])
sage: G=matrix(RDF,[[2,1],[1,2],[-1,0],[0,-1]])
sage: h=vector(RDF,[3,3,0,0])
sage: sol=linear_program(c,G,h)
sage: sol[’x’]
(0.999..., 1.000...)
Next, we maximize x + y − 50 subject to 50x + 24y ≤ 2400, 30x + 33y ≤ 2100, x ≥ 45, and y ≥ 5:
sage: v=vector([-1.0,-1.0,-1.0])
sage: m=matrix([[50.0,24.0,0.0],[30.0,33.0,0.0],[-1.0,0.0,0.0],[0.0,-1.0,0.0],[0.0,0.0,1.0],[0.0
sage: h=vector([2400.0,2100.0,-45.0,-5.0,1.0,-1.0])
sage: sol=linear_program(v,m,h)
sage: sol[’x’]
(45.000000..., 6.2499999...3, 1.00000000...)
sage: sol=linear_program(v,m,h,solver=’glpk’)
GLPK Simplex Optimizer...
OPTIMAL SOLUTION FOUND
sage: sol[’x’]
(45.0..., 6.25, 1.0...)
sage.numerical.optimize.minimize(func, x0, gradient=None, hessian=None, algorithm=’default’,
**args)
This function is an interface to a variety of algorithms for computing the minimum of a function of several
variables.
INPUT:
•func – Either a symbolic function or a Python function whose argument is a tuple with n components
•x0 – Initial point for finding minimum.
•gradient – Optional gradient function. This will be computed automatically for symbolic functions.
For Python functions, it allows the use of algorithms requiring derivatives. It should accept a tuple of
arguments and return a NumPy array containing the partial derivatives at that point.
•hessian – Optional hessian function. This will be computed automatically for symbolic functions. For
Python functions, it allows the use of algorithms requiring derivatives. It should accept a tuple of arguments
and return a NumPy array containing the second partial derivatives of the function.
•algorithm – String specifying algorithm to use. Options are ’default’ (for Python functions, the
simplex method is the default) (for symbolic functions bfgs is the default):
EXAMPLES:
–’simplex’
–’powell’
–’bfgs’ – (Broyden-Fletcher-Goldfarb-Shanno) requires gradient
–’cg’ – (conjugate-gradient) requires gradient
–’ncg’ – (newton-conjugate gradient) requires gradient and hessian
sage: vars=var(’x y z’)
sage: f=100*(y-x^2)^2+(1-x)^2+100*(z-y^2)^2+(1-y)^2
sage: minimize(f,[.1,.3,.4],disp=0)
(1.00..., 1.00..., 1.00...)
sage: minimize(f,[.1,.3,.4],algorithm="ncg",disp=0)
(0.9999999..., 0.999999..., 0.999999...)
46 Chapter 4. Numerical Root Finding and Optimization