.. _tutorial: ===== Usage ===== SciPy Compatible Interface ========================== For simple cases where you do not need the full power of sparse and structured Jacobians etc, ``cyipopt`` provides the function ``minimize_ipopt`` which has the same behaviour as ``scipy.optimize.minimize``, for example:: >>> from scipy.optimize import rosen, rosen_der >>> from cyipopt import minimize_ipopt >>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2] >>> res = minimize_ipopt(rosen, x0, jac=rosen_der) >>> print(res) fun: 2.1256746564022273e-18 info: {'x': array([1., 1., 1., 1., 1.]), 'g': array([], dtype=float64), 'obj_val': 2.1256746564022273e-18, 'mult_g': array([], dtype=float64), 'mult_x_L': array([0., 0., 0., 0., 0.]), 'mult_x_U': array([0., 0., 0., 0., 0.]), 'status': 0, 'status_msg': b'Algorithm terminated successfully at a locally optimal point, satisfying the convergence tolerances (can be specified by options).'} message: b'Algorithm terminated successfully at a locally optimal point, satisfying the convergence tolerances (can be specified by options).' nfev: 200 nit: 37 njev: 39 status: 0 success: True x: array([1., 1., 1., 1., 1.]) In order to demonstrate the usage of sparse jacobians, let's assume we want to minimize the well-known rosenbrock function .. math:: f(x) = \sum_{i=1}^{4} 100 (x_{i+1} - x_i^2)^2 + (1-x_i)^2 subject to some constraints, i.e. we want to solve the constraint optimization problem .. math:: \min_{x \in \mathbb{R}^5} f(x) \quad \text{s.t.} \quad 10 - x_2^2 - x_3 \geq 0, \quad 100 - x_5^2 \geq 0. We won't implement the rosenbrock function and its derivatives here, since all three can be imported from ``scipy.optimize``. The constraint function :math:`c` and the jacobian :math:`J_c` are given by .. math:: c(x) &= \begin{pmatrix} c_1(x) \\ c_2(x) \end{pmatrix} = \begin{pmatrix} 10 - x_1^2 + x^3 \\ 100 - x_5^2 \end{pmatrix} \geq 0 \\ J_c(x) &= \begin{pmatrix} 0 & -2x_2 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -2x_5 \end{pmatrix} and we can implement the constraint and the sparse jacobian by means of an ``scipy.sparse.coo_array`` like this:: from scipy.sparse import coo_array def con(x): return np.array([ 10 -x[1]**2 - x[2], 100.0 - x[4]**2 ]) def con_jac(x): # Dense Jacobian: # J = (0 -2*x[1] -1 0 0 ) # (0 0 0 0 -2*x[4] ) # Sparse Jacobian (COO) rows = np.array([0, 0, 1]) cols = np.array(([1, 2, 4])) data = np.array([-2*x[1], -1, -2*x[4]]) return coo_array((data, (rows, cols))) In addition, we would like to pass the hessian of the objective and the constraints. Note that Ipopt expects the hessian :math:`\nabla^2_x L` of the lagrangian function .. math:: L(x, \lambda) = f(x) + \lambda^\top c(x) = f(x) + \sum_{j=1}^{2} \lambda_j c_j(x), which is given by .. math:: \nabla^2_x L(x, \lambda) = \nabla^2 f(x) + \sum_{j=1}^2 \lambda_j \nabla^2 c_j(x). Hence, we need to pass the hessian-vector-product of the constraint hessians :math:`\nabla^2 c_1(x)` and :math:`\nabla^2 c_2(x)` and the lagrangian multipliers :math:`\lambda` (also known as dual variables). In code: :: def con_hess(x, _lambda): H1 = np.array([ [0, 0, 0, 0, 0], [0, -2, 0, 0, 0 ], [0, 0, 0, 0, 0 ], [0, 0, 0, 0, 0 ], [0, 0, 0, 0, 0 ] ]) H2 = np.array([ [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, -2] ]) return _lambda[0] * H1 + _lambda[1] * H2 Ipopt only uses the lower triangle of the hessian-vector-product under the hood, due to the symmetry of the hessians. Similar to sparse jacobians, it also supports sparse hessians, but this isn't supported by the scipy interface yet. However, you can use cyipopt's problem interface in case you need to pass sparse hessians. Finally, after defining the constraint and the initial guess, we can solve the problem:: from scipy.optimize import rosen, rosen_der, rosen_hess constr = {'type': 'ineq', 'fun': con, 'jac': con_jac, 'hess': con_hess} # initial guess x0 = np.array([1.1, 1.1, 1.1, 1.1, 1.1]) # solve the problem res = minimize_ipopt(rosen, jac=rosen_der, hess=rosen_hess, x0=x0, constraints=constr) Algorithmic Differentation -------------------------- Computing derivatives by hand can be quite error-prone. In case you don't provide the (exact) objective gradient or the jacobian of the constraint function, the scipy interface will approximate the missing derivatives by finite differences similar to ``scipy.optimize.minimize``. However, finite differences are prone to truncation errors due to floating point arithmetic and computationally expensive especially for evaluating jacobians. A more efficient and accurate way to evaluate derivatives is algorithmic differentation (AD). In this example we use AD by means of the `JAX`_ library to compute derivatives and we use cyipopt's scipy interface to solve an example problem, namely number 71 from the Hock-Schittkowsky test suite [1]_, .. math:: \min_{x \in \mathbb{R}^4}\ &x_1 x_4 (x_1 + x_2 + x_3 ) + x_3 \\ s.t.\ &x_1 x_2 x_3 x_4 \geq 25 \\ &x_1^2 + x_2^2 + x_3^2 + x_4^2 = 40 \\ &1 \leq x_1, x_2, x_3, x_4 \leq 5, \\ with the starting point, .. math:: x_0 = (1,\ 5,\ 5,\ 1), and the optimal solution, .. math:: x_* = (1.0,\ 4.743,\ 3.821,\ 1.379) We start by importing all required libraries:: from jax.config import config # Enable 64 bit floating point precision config.update("jax_enable_x64", True) # We use the CPU instead of GPU und mute all warnings if no GPU/TPU is found. config.update('jax_platform_name', 'cpu') import jax.numpy as np from jax import jit, grad, jacfwd, jacrev from cyipopt import minimize_ipopt Then we define the objective and constraint functions:: def objective(x): return x[0]*x[3]*np.sum(x[:3]) + x[2] def eq_constraints(x): return np.sum(x**2) - 40 def ineq_constrains(x): return np.prod(x) - 25 Next, we build the derivatives and just-in-time (jit) compile the functions (more details regarding ``jit``, ``grad`` and ``jacfwd`` can be found in the `JAX autodiff cookbook`_):: # jit the functions obj_jit = jit(objective) con_eq_jit = jit(eq_constraints) con_ineq_jit = jit(ineq_constrains) # build the derivatives and jit them obj_grad = jit(grad(obj_jit)) # objective gradient obj_hess = jit(jacrev(jacfwd(obj_jit))) # objective hessian con_eq_jac = jit(jacfwd(con_eq_jit)) # jacobian con_ineq_jac = jit(jacfwd(con_ineq_jit)) # jacobian con_eq_hess = jacrev(jacfwd(con_eq_jit)) # hessian con_eq_hessvp = jit(lambda x, v: con_eq_hess(x) * v[0]) # hessian vector-product con_ineq_hess = jacrev(jacfwd(con_ineq_jit)) # hessian con_ineq_hessvp = jit(lambda x, v: con_ineq_hess(x) * v[0]) # hessian vector-product Finally, we can call ``minimize_ipopt`` similar to ``scipy.optimize.minimize``:: # constraints cons = [ {'type': 'eq', 'fun': con_eq_jit, 'jac': con_eq_jac, 'hess': con_eq_hessvp}, {'type': 'ineq', 'fun': con_ineq_jit, 'jac': con_ineq_jac, 'hess': con_ineq_hessvp} ] # starting point x0 = np.array([1.0, 5.0, 5.0, 1.0]) # variable bounds: 1 <= x[i] <= 5 bnds = [(1, 5) for _ in range(x0.size)] # executing the solver res = minimize_ipopt(obj_jit, jac=obj_grad, hess=obj_hess, x0=x0, bounds=bnds, constraints=cons, options={'disp': 5}) Problem Interface ================= In this example we will use cyipopt problem class interface to solve the aforementioned test problem. Getting started --------------- Before you can use cyipopt, you have to import it:: import cyipopt This problem will also make use of NumPy:: import numpy as np Defining the problem -------------------- The first step is to define a class that computes the objective and its gradient, the constraints and its Jacobian, and the Hessian. The following methods can be defined on the class: - :func:`cyipopt.Problem.objective` - :func:`cyipopt.Problem.gradient` - :func:`cyipopt.Problem.constraints` - :func:`cyipopt.Problem.jacobian` - :func:`cyipopt.Problem.hessian` The :func:`cyipopt.Problem.jacobian` and :func:`cyipopt.Problem.hessian` methods should return the non-zero values of the respective matrices as flattened arrays. The hessian should return a flattened lower triangular matrix. The Jacobian and Hessian can be dense or sparse. If sparse, :func:`cyipopt.Problem.jacobian` and :func:`cyipopt.Problem.hessian` methods should return only the non-zero values of the respective matrices and you must also define: - :func:`cyipopt.Problem.jacobianstructure` - :func:`cyipopt.Problem.hessianstructure` which should return a tuple of indices (row indices, column indices) that indicate the location of the non-zero values of the Jacobian and Hessian matrices, respectively. If not defined then these matrices are assumed to be dense. The :func:`cyipopt.Problem.intermediate` method is called every Ipopt iteration algorithm and can be used to perform any needed computation at each iteration. Define the problem class:: class HS071(): def objective(self, x): """Returns the scalar value of the objective given x.""" return x[0] * x[3] * np.sum(x[0:3]) + x[2] def gradient(self, x): """Returns the gradient of the objective with respect to x.""" return np.array([ x[0]*x[3] + x[3]*np.sum(x[0:3]), x[0]*x[3], x[0]*x[3] + 1.0, x[0]*np.sum(x[0:3]) ]) def constraints(self, x): """Returns the constraints.""" return np.array((np.prod(x), np.dot(x, x))) def jacobian(self, x): """Returns the Jacobian of the constraints with respect to x.""" return np.concatenate((np.prod(x)/x, 2*x)) def hessianstructure(self): """Returns the row and column indices for non-zero vales of the Hessian.""" # NOTE: The default hessian structure is of a lower triangular matrix, # therefore this function is redundant. It is included as an example # for structure callback. return np.nonzero(np.tril(np.ones((4, 4)))) def hessian(self, x, lagrange, obj_factor): """Returns the non-zero values of the Hessian.""" H = obj_factor*np.array(( (2*x[3], 0, 0, 0), (x[3], 0, 0, 0), (x[3], 0, 0, 0), (2*x[0]+x[1]+x[2], x[0], x[0], 0))) H += lagrange[0]*np.array(( (0, 0, 0, 0), (x[2]*x[3], 0, 0, 0), (x[1]*x[3], x[0]*x[3], 0, 0), (x[1]*x[2], x[0]*x[2], x[0]*x[1], 0))) H += lagrange[1]*2*np.eye(4) row, col = self.hessianstructure() return H[row, col] def intermediate(self, alg_mod, iter_count, obj_value, inf_pr, inf_du, mu, d_norm, regularization_size, alpha_du, alpha_pr, ls_trials): """Prints information at every Ipopt iteration.""" msg = "Objective value at iteration #{:d} is - {:g}" print(msg.format(iter_count, obj_value)) Now define the lower and upper bounds of :math:`x` and the constraints:: lb = [1.0, 1.0, 1.0, 1.0] ub = [5.0, 5.0, 5.0, 5.0] cl = [25.0, 40.0] cu = [2.0e19, 40.0] Define an initial guess:: x0 = [1.0, 5.0, 5.0, 1.0] Define the full problem using the :class:`cyipopt.Problem` class:: nlp = cyipopt.Problem( n=len(x0), m=len(cl), problem_obj=HS071(), lb=lb, ub=ub, cl=cl, cu=cu, ) The constructor of the :class:`cyipopt.Problem` class requires: - ``n``: the number of variables in the problem, - ``m``: the number of constraints in the problem, - ``lb`` and ``ub``: lower and upper bounds on the variables, - ``cl`` and ``cu``: lower and upper bounds of the constraints. - ``problem_obj`` is an object whose methods implement ``objective``, ``gradient``, ``constraints``, ``jacobian``, and ``hessian`` of the problem. Setting optimization parameters ------------------------------- Setting optimization parameters is done by calling the :func:`cyipopt.Problem.add_option` method, e.g.:: nlp.add_option('mu_strategy', 'adaptive') nlp.add_option('tol', 1e-7) The different options and their possible values are described in the `ipopt documentation `_. Executing the solver -------------------- The optimization algorithm is run by calling the :func:`cyipopt.Problem.solve` method, which accepts the starting point for the optimization as its only parameter:: x, info = nlp.solve(x0) The method returns the optimal solution and an info dictionary that contains the status of the algorithm, the value of the constraints multipliers at the solution, and more. Accessing iterate and infeasibility vectors in an intermediate callback ======================================================================= When debugging an Ipopt solve that converges slowly or not at all, it can be very useful to track the primal/dual iterate and infeasibility vectors to get a sense for the variable and constraint coordinates that are causing a problem. This can be done with Ipopt's ``GetCurrentIterate`` and ``GetCurrentViolations`` functions, which were added to Ipopt's C interface in Ipopt version 3.14.0. These functions are accessed in CyIpopt via the ``get_current_iterate`` and ``get_current_violations`` methods of ``cyipopt.Problem``. These methods should only be called during an intermediate callback. To access them, we define our problem as a subclass of ``cyipopt.Problem`` and access the ``get_current_iterate`` and ``get_current_violations`` methods on ``self``. In contrast to the previous example, we now define the HS071 problem as a subclass of ``cyipopt.Problem``:: import cyipopt import numpy as np class HS071(cyipopt.Problem): def objective(self, x): """Returns the scalar value of the objective given x.""" return x[0] * x[3] * np.sum(x[0:3]) + x[2] def gradient(self, x): """Returns the gradient of the objective with respect to x.""" return np.array([ x[0]*x[3] + x[3]*np.sum(x[0:3]), x[0]*x[3], x[0]*x[3] + 1.0, x[0]*np.sum(x[0:3]) ]) def constraints(self, x): """Returns the constraints.""" return np.array((np.prod(x), np.dot(x, x))) def jacobian(self, x): """Returns the Jacobian of the constraints with respect to x.""" return np.concatenate((np.prod(x)/x, 2*x)) def hessianstructure(self): """Returns the row and column indices for non-zero vales of the Hessian.""" # NOTE: The default hessian structure is of a lower triangular matrix, # therefore this function is redundant. It is included as an example # for structure callback. return np.nonzero(np.tril(np.ones((4, 4)))) def hessian(self, x, lagrange, obj_factor): """Returns the non-zero values of the Hessian.""" H = obj_factor*np.array(( (2*x[3], 0, 0, 0), (x[3], 0, 0, 0), (x[3], 0, 0, 0), (2*x[0]+x[1]+x[2], x[0], x[0], 0))) H += lagrange[0]*np.array(( (0, 0, 0, 0), (x[2]*x[3], 0, 0, 0), (x[1]*x[3], x[0]*x[3], 0, 0), (x[1]*x[2], x[0]*x[2], x[0]*x[1], 0))) H += lagrange[1]*2*np.eye(4) row, col = self.hessianstructure() return H[row, col] def intermediate(self, alg_mod, iter_count, obj_value, inf_pr, inf_du, mu, d_norm, regularization_size, alpha_du, alpha_pr, ls_trials): """Prints information at every Ipopt iteration.""" iterate = self.get_current_iterate() infeas = self.get_current_violations() primal = iterate["x"] jac = self.jacobian(primal) print("Iteration:", iter_count) print("Primal iterate:", primal) print("Flattened Jacobian:", jac) print("Dual infeasibility:", infeas["grad_lag_x"]) Now, in the ``intermediate`` method of ``HS071``, we call ``self.get_current_iterate`` and ``self.get_current_violations``. These are implemented on ``cyipopt.Problem``. These methods return dicts that contain each component of the Ipopt iterate and infeasibility vectors. The primal iterate and constraint dual iterate can be accessed with ``iterate["x"]`` and ``iterate["mult_g"]``, while the primal and dual infeasibilities can be accessed with ``infeas["g_violation"]`` and ``infeas["grad_lag_x"]``. A full list of keys present in these dictionaries can be found in the ``cyipopt.Problem`` documentation. We can now set up and solve the optimization problem. Note that now we instantiate the ``HS071`` class and provide it the arguments that are required by ``cyipopt.Problem``. When we solve, we will see the primal iterate and dual infeasibility vectors printed every iteration:: lb = [1.0, 1.0, 1.0, 1.0] ub = [5.0, 5.0, 5.0, 5.0] cl = [25.0, 40.0] cu = [2.0e19, 40.0] x0 = [1.0, 5.0, 5.0, 1.0] nlp = HS071( n=len(x0), m=len(cl), lb=lb, ub=ub, cl=cl, cu=cu, ) x, info = nlp.solve(x0) While here we have implemented a very basic callback, much more sophisticated analysis is possible. For example, we could compute the condition number or rank of the constraint Jacobian to identify when constraint qualifications are close to being violated. Where to go from here ===================== Once you feel sufficiently familiar with the basics, feel free to dig into the :ref:`reference `. For more examples, check the :file:`examples/` subdirectory of the distribution. .. [1] W. Hock and K. Schittkowski. Test examples for nonlinear programming codes. Lecture Notes in Economics and Mathematical Systems, 187, 1981. .. _JAX: https://jax.readthedocs.io/en/latest/notebooks/autodiff_cookbook.html .. _JAX autodiff cookbook: https://jax.readthedocs.io/en/latest/notebooks/autodiff_cookbook.html