dogleg algorithm with rectangular trust regions for least-squares minimization.
The description of the algorithm can be found in [Voglis]. The algorithm does trust-region iterations, but the shape of trust regions is rectangular as opposed to conventional elliptical. The intersection of a trust region and an initial feasible region is again some rectangle. Thus on each iteration a bound-constrained quadratic optimization problem is solved.
A quadratic problem is solved by well-known dogleg approach, where the function is minimized along piecewise-linear “dogleg” path [NumOpt], Chapter 4. If Jacobian is not rank-deficient then the function is decreasing along this path, and optimization amounts to simply following along this path as long as a point stays within the bounds. A constrained Cauchy step (along the anti-gradient) is considered for safety in rank deficient cases, in this situations the convergence might be slow.
If during iterations some variable hit the initial bound and the component of anti-gradient points outside the feasible region, then a next dogleg step won’t make any progress. At this state such variables satisfy first-order optimality conditions and they are excluded before computing a next dogleg step.
Gauss-Newton step can be computed exactly by numpy.linalg.lstsq (for dense Jacobian matrices) or by iterative procedure scipy.sparse.linalg.lsmr (for dense and sparse matrices, or Jacobian being LinearOperator). The second option allows to solve very large problems (up to couple of millions of residuals on a regular PC), provided the Jacobian matrix is sufficiently sparse. But note that dogbox is not very good for solving problems with large number of constraints, because of variables exclusion-inclusion on each iteration (a required number of function evaluations might be high or accuracy of a solution will be poor), thus its large-scale usage is probably limited to unconstrained problems.
|[Voglis]||C. Voglis and I. E. Lagaris, “A Rectangular Trust Region Dogleg Approach for Unconstrained and Bound Constrained Nonlinear Optimization”, WSEAS International Conference on Applied Mathematics, Corfu, Greece, 2004.|
||Compute LinearOperator to use in LSMR by dogbox algorithm.|
||Find intersection of trust-region bounds and initial bounds.|
||Find dogleg step in a rectangular region.|
lsmr_operator(Jop, d, active_set)¶
Compute LinearOperator to use in LSMR by dogbox algorithm.
active_set mask is used to excluded active variables from computations of matrix-vector products.
find_intersection(x, tr_bounds, lb, ub)¶
Find intersection of trust-region bounds and initial bounds.
- lb_total, ub_total : ndarray with shape of x
- Lower and upper bounds of the intersection region.
- orig_l, orig_u : ndarray of bool with shape of x
- True means that an original bound is taken as a corresponding bound in the intersection region.
- tr_l, tr_u : ndarray of bool with shape of x
- True means that a trust-region bound is taken as a corresponding bound in the intersection region.
dogleg_step(x, newton_step, g, a, b, tr_bounds, lb, ub)¶
Find dogleg step in a rectangular region.
- step : ndarray, shape (n,)
- Computed dogleg step.
- bound_hits : ndarray of int, shape (n,)
Each component shows whether a corresponding variable hits the initial bound after the step is taken:
- 0 - a variable doesn’t hit the bound.
- -1 - lower bound is hit.
- 1 - upper bound is hit.
- tr_hit : bool
- Whether the step hit the boundary of the trust-region.
dogbox(fun, jac, x0, f0, J0, lb, ub, ftol, xtol, gtol, max_nfev, x_scale, loss_function, tr_solver, tr_options, verbose)¶