"""This module implements the Sequential Least SQuares Programming optimization algorithm (SLSQP), orginally developed by Dieter Kraft. See http://www.netlib.org/toms/733 """ __all__ = ['approx_jacobian','fmin_slsqp'] from _slsqp import slsqp from numpy import zeros, array, linalg, append, asfarray, concatenate, finfo, \ sqrt, vstack from optimize import approx_fprime, wrap_function __docformat__ = "restructuredtext en" _epsilon = sqrt(finfo(float).eps) def approx_jacobian(x,func,epsilon,*args): """ Approximate the Jacobian matrix of a callable function. Parameters ---------- x : array_like The state vector at which to compute the Jacobian matrix. func : callable f(x,*args) The vector-valued function. epsilon : float The perturbation used to determine the partial derivatives. args : sequence Additional arguments passed to func. Returns ------- An array of dimensions ``(lenf, lenx)`` where ``lenf`` is the length of the outputs of `func`, and ``lenx`` is the number of elements in `x`. Notes ----- The approximation is done using forward differences. """ x0 = asfarray(x) f0 = func(*((x0,)+args)) jac = zeros([len(x0),len(f0)]) dx = zeros(len(x0)) for i in range(len(x0)): dx[i] = epsilon jac[i] = (func(*((x0+dx,)+args)) - f0)/epsilon dx[i] = 0.0 return jac.transpose() def fmin_slsqp( func, x0 , eqcons=[], f_eqcons=None, ieqcons=[], f_ieqcons=None, bounds = [], fprime = None, fprime_eqcons=None, fprime_ieqcons=None, args = (), iter = 100, acc = 1.0E-6, iprint = 1, disp = None, full_output = 0, epsilon = _epsilon ): """ Minimize a function using Sequential Least SQuares Programming Python interface function for the SLSQP Optimization subroutine originally implemented by Dieter Kraft. Parameters ---------- func : callable f(x,*args) Objective function. x0 : 1-D ndarray of float Initial guess for the independent variable(s). eqcons : list A list of functions of length n such that eqcons[j](x,*args) == 0.0 in a successfully optimized problem. f_eqcons : callable f(x,*args) Returns a 1-D array in which each element must equal 0.0 in a successfully optimized problem. If f_eqcons is specified, eqcons is ignored. ieqcons : list A list of functions of length n such that ieqcons[j](x,*args) >= 0.0 in a successfully optimized problem. f_ieqcons : callable f(x,*args) Returns a 1-D ndarray in which each element must be greater or equal to 0.0 in a successfully optimized problem. If f_ieqcons is specified, ieqcons is ignored. bounds : list A list of tuples specifying the lower and upper bound for each independent variable [(xl0, xu0),(xl1, xu1),...] fprime : callable `f(x,*args)` A function that evaluates the partial derivatives of func. fprime_eqcons : callable `f(x,*args)` A function of the form `f(x, *args)` that returns the m by n array of equality constraint normals. If not provided, the normals will be approximated. The array returned by fprime_eqcons should be sized as ( len(eqcons), len(x0) ). fprime_ieqcons : callable `f(x,*args)` A function of the form `f(x, *args)` that returns the m by n array of inequality constraint normals. If not provided, the normals will be approximated. The array returned by fprime_ieqcons should be sized as ( len(ieqcons), len(x0) ). args : sequence Additional arguments passed to func and fprime. iter : int The maximum number of iterations. acc : float Requested accuracy. iprint : int The verbosity of fmin_slsqp : * iprint <= 0 : Silent operation * iprint == 1 : Print summary upon completion (default) * iprint >= 2 : Print status of each iterate and summary disp : int Over-rides the iprint interface (preferred). full_output : bool If False, return only the minimizer of func (default). Otherwise, output final objective function and summary information. epsilon : float The step size for finite-difference derivative estimates. Returns ------- out : ndarray of float The final minimizer of func. fx : ndarray of float, if full_output is true The final value of the objective function. its : int, if full_output is true The number of iterations. imode : int, if full_output is true The exit mode from the optimizer (see below). smode : string, if full_output is true Message describing the exit mode from the optimizer. Notes ----- Exit modes are defined as follows :: -1 : Gradient evaluation required (g & a) 0 : Optimization terminated successfully. 1 : Function evaluation required (f & c) 2 : More equality constraints than independent variables 3 : More than 3*n iterations in LSQ subproblem 4 : Inequality constraints incompatible 5 : Singular matrix E in LSQ subproblem 6 : Singular matrix C in LSQ subproblem 7 : Rank-deficient equality constraint subproblem HFTI 8 : Positive directional derivative for linesearch 9 : Iteration limit exceeded Examples -------- Examples are given :ref:`in the tutorial `. """ exit_modes = { -1 : "Gradient evaluation required (g & a)", 0 : "Optimization terminated successfully.", 1 : "Function evaluation required (f & c)", 2 : "More equality constraints than independent variables", 3 : "More than 3*n iterations in LSQ subproblem", 4 : "Inequality constraints incompatible", 5 : "Singular matrix E in LSQ subproblem", 6 : "Singular matrix C in LSQ subproblem", 7 : "Rank-deficient equality constraint subproblem HFTI", 8 : "Positive directional derivative for linesearch", 9 : "Iteration limit exceeded" } if disp is not None: iprint = disp # Now do a lot of function wrapping # Wrap func feval, func = wrap_function(func, args) # Wrap fprime, if provided, or approx_fprime if not if fprime: geval, fprime = wrap_function(fprime,args) else: geval, fprime = wrap_function(approx_fprime,(func,epsilon)) if f_eqcons: # Equality constraints provided via f_eqcons ceval, f_eqcons = wrap_function(f_eqcons,args) if fprime_eqcons: # Wrap fprime_eqcons geval, fprime_eqcons = wrap_function(fprime_eqcons,args) else: # Wrap approx_jacobian geval, fprime_eqcons = wrap_function(approx_jacobian, (f_eqcons,epsilon)) else: # Equality constraints provided via eqcons[] eqcons_prime = [] for i in range(len(eqcons)): eqcons_prime.append(None) if eqcons[i]: # Wrap eqcons and eqcons_prime ceval, eqcons[i] = wrap_function(eqcons[i],args) geval, eqcons_prime[i] = wrap_function(approx_fprime, (eqcons[i],epsilon)) if f_ieqcons: # Inequality constraints provided via f_ieqcons ceval, f_ieqcons = wrap_function(f_ieqcons,args) if fprime_ieqcons: # Wrap fprime_ieqcons geval, fprime_ieqcons = wrap_function(fprime_ieqcons,args) else: # Wrap approx_jacobian geval, fprime_ieqcons = wrap_function(approx_jacobian, (f_ieqcons,epsilon)) else: # Inequality constraints provided via ieqcons[] ieqcons_prime = [] for i in range(len(ieqcons)): ieqcons_prime.append(None) if ieqcons[i]: # Wrap ieqcons and ieqcons_prime ceval, ieqcons[i] = wrap_function(ieqcons[i],args) geval, ieqcons_prime[i] = wrap_function(approx_fprime, (ieqcons[i],epsilon)) # Transform x0 into an array. x = asfarray(x0).flatten() # Set the parameters that SLSQP will need # meq = The number of equality constraints if f_eqcons: meq = len(f_eqcons(x)) else: meq = len(eqcons) if f_ieqcons: mieq = len(f_ieqcons(x)) else: mieq = len(ieqcons) # m = The total number of constraints m = meq + mieq # la = The number of constraints, or 1 if there are no constraints la = array([1,m]).max() # n = The number of independent variables n = len(x) # Define the workspaces for SLSQP n1 = n+1 mineq = m - meq + n1 + n1 len_w = (3*n1+m)*(n1+1)+(n1-meq+1)*(mineq+2) + 2*mineq+(n1+mineq)*(n1-meq) \ + 2*meq + n1 +(n+1)*n/2 + 2*m + 3*n + 3*n1 + 1 len_jw = mineq w = zeros(len_w) jw = zeros(len_jw) # Decompose bounds into xl and xu if len(bounds) == 0: bounds = [(-1.0E12, 1.0E12) for i in range(n)] elif len(bounds) != n: raise IndexError('SLSQP Error: If bounds is specified, ' 'len(bounds) == len(x0)') else: for i in range(len(bounds)): if bounds[i][0] > bounds[i][1]: raise ValueError('SLSQP Error: lb > ub in bounds[' + str(i) + '] ' + str(bounds[4])) xl = array( [ b[0] for b in bounds ] ) xu = array( [ b[1] for b in bounds ] ) # Initialize the iteration counter and the mode value mode = array(0,int) acc = array(acc,float) majiter = array(iter,int) majiter_prev = 0 # Print the header if iprint >= 2 if iprint >= 2: print "%5s %5s %16s %16s" % ("NIT","FC","OBJFUN","GNORM") while 1: if mode == 0 or mode == 1: # objective and constraint evaluation requird # Compute objective function fx = func(x) # Compute the constraints if f_eqcons: c_eq = f_eqcons(x) else: c_eq = array([ eqcons[i](x) for i in range(meq) ]) if f_ieqcons: c_ieq = f_ieqcons(x) else: c_ieq = array([ ieqcons[i](x) for i in range(len(ieqcons)) ]) # Now combine c_eq and c_ieq into a single matrix if m == 0: # no constraints c = zeros([la]) else: # constraints exist if meq > 0 and mieq == 0: # only equality constraints c = c_eq if meq == 0 and mieq > 0: # only inequality constraints c = c_ieq if meq > 0 and mieq > 0: # both equality and inequality constraints exist c = append(c_eq, c_ieq) if mode == 0 or mode == -1: # gradient evaluation required # Compute the derivatives of the objective function # For some reason SLSQP wants g dimensioned to n+1 g = append(fprime(x),0.0) # Compute the normals of the constraints if fprime_eqcons: a_eq = fprime_eqcons(x) else: a_eq = zeros([meq,n]) for i in range(meq): a_eq[i] = eqcons_prime[i](x) if fprime_ieqcons: a_ieq = fprime_ieqcons(x) else: a_ieq = zeros([mieq,n]) for i in range(mieq): a_ieq[i] = ieqcons_prime[i](x) # Now combine a_eq and a_ieq into a single a matrix if m == 0: # no constraints a = zeros([la,n]) elif meq > 0 and mieq == 0: # only equality constraints a = a_eq elif meq == 0 and mieq > 0: # only inequality constraints a = a_ieq elif meq > 0 and mieq > 0: # both equality and inequality constraints exist a = vstack((a_eq,a_ieq)) a = concatenate((a,zeros([la,1])),1) # Call SLSQP slsqp(m, meq, x, xl, xu, fx, c, g, a, acc, majiter, mode, w, jw) # Print the status of the current iterate if iprint > 2 and the # major iteration has incremented if iprint >= 2 and majiter > majiter_prev: print "%5i %5i % 16.6E % 16.6E" % (majiter,feval[0], fx,linalg.norm(g)) # If exit mode is not -1 or 1, slsqp has completed if abs(mode) != 1: break majiter_prev = int(majiter) # Optimization loop complete. Print status if requested if iprint >= 1: print exit_modes[int(mode)] + " (Exit mode " + str(mode) + ')' print " Current function value:", fx print " Iterations:", majiter print " Function evaluations:", feval[0] print " Gradient evaluations:", geval[0] if not full_output: return x else: return [list(x), float(fx), int(majiter), int(mode), exit_modes[int(mode)] ]