(ph-SrSSKrSSKJrJr SSKJrJrJ r SSK J r SSK J r JrJrJrJrJrJrJrJrJrJrJr SrS rS rS rg) a 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. References ---------- .. [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. .. [NumOpt] J. Nocedal and S. J. Wright, "Numerical optimization, 2nd edition". N)lstsqnorm)LinearOperatoraslinearoperatorlsmr)OptimizeResult) step_size_to_bound in_boundsupdate_tr_radiusevaluate_quadraticbuild_quadratic_1dminimize_quadratic_1d compute_gradcompute_jac_scalecheck_terminationscale_for_robust_loss_functionprint_header_nonlinearprint_iteration_nonlinearcd^^^TRup4UUU4SjnUUU4Sjn[X44XV[S9$)zCompute LinearOperator to use in LSMR by dogbox algorithm. `active_set` mask is used to excluded active variables from computations of matrix-vector products. cr>UR5R5nSUT'TRUT-5$Nr)ravelcopymatvec)xx_freeJop active_setds M/var/www/html/venv/lib/python3.13/site-packages/scipy/optimize/_lsq/dogbox.pyrlsmr_operator..matvec@s2!zzz!a%  c:>TTRU5-nSUT'U$r)rmatvec)rrrrr s r!r%lsmr_operator..rmatvecEs#  A * r#)rr%dtype)shaperfloat)rr rmnrr%s``` r! lsmr_operatorr-8s- 99DA!  1& NNr#c(X - nX0- n[R"XA*5n[R"XQ5n[R"Xd5n[R"Xu5n [R"Xa*5n [R"Xq5n XgXX4$)aFind intersection of trust-region bounds and initial bounds. Returns ------- 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. )npmaximumminimumequal) r tr_boundslbub lb_centered ub_centeredlb_totalub_totalorig_lorig_utr_ltr_us r!find_intersectionr>Msw&K&Kzz+z2Hzz+1H XXh ,F XXh ,F 88Hj )D 88H (D vt 99r#c[XXg5uppp[R"U[S9n[ XU 5(aXS4$[ [R"U5U*X5unn[ X4SU5S*U-nUU- n[ UUX5unnSUUS:U -'SUUS:U -'[R"US:U -US:U --5nUUU--UU4$)aFind dogleg step in a rectangular region. Returns ------- 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. r(Frr )r>r/ zeros_likeintr r rany)r newton_stepgabr3r4r5r8r9r:r;r<r= bound_hits to_bounds_ cauchy_step step_diff step_sizehitstr_hits r! dogleg_steprQjs 6G b62Hq,J11--%bmmA&6HOLIq)q)DAq & "1%C$C ;G9j9!DhAdGq=DhAdGAVVXFDAq A AID(#A&5aC@1Q"A#4Q #B yI Q C F!  $F64d;M OOr#)__doc__numpyr/ numpy.linalgrrscipy.sparse.linalgrrrscipy.optimizercommonr r r r rrrrrrrrr-r>rQrr#r!rsI)T$FF)7777O*::(CVvOr#