9i/SrSSKrSSKJrJr SSKJrJrJ r SSK J r SSK J r SSKJrJrJrJrJrJrJrJrJrJrJrJr S rS rS rS S jrg)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)_call_callback_maybe_halt) 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 Z/var/www/html/land-doc-ocr/venv/lib/python3.13/site-packages/scipy/optimize/_lsq/dogbox.pyrlsmr_operator..matvecBs2!zzz!a%  c:>TTRU5-nSUT'U$r)rmatvec)rrrr r!s r"r&lsmr_operator..rmatvecGs#  A * r$)rr&dtype)shaperfloat)rr!r mnrr&s``` r" lsmr_operatorr.:s- 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?Osw&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?r0 zeros_likeintr r rany)r newton_stepgabr4r5r6r9r:r;r<r=r> bound_hits to_bounds_ cauchy_step step_diff step_sizehitstr_hits r" dogleg_steprRls 6G b62Hq,J11--%bmmA&6HOLIq)q)W;U>S'[AUU>5(aSn OGMU cSn [?UUUUU'U(UUUU S9 $)Nr g?rjac)ordg?rArBdTexact)rcondrggg?) cost_only)rfunnitnfevcost) rr^r[rTgrad optimality active_maskr]njevstatus)!rr0sumrdotr isinstancestrrrinfrCrDr3 empty_likesizerrrrrr.rrRfillrclipallisfiniter rrr )?r[rTx0f0J0r5r6ftolxtolgtolmax_nfevx_scale loss_function tr_solver tr_optionsverbosecallbackff_truer]Jrcrhor^rG jac_scalescale scale_invDeltaon_boundrsteptermination_status iteration step_normactual_reductionr free_setg_freeg_fullg_normrlb_freeub_free scale_freeJ_freerFrHrIrlsmr_opr4 step_free on_bound_freerQpredicted_reductionx_newf_new step_h_normcost_newratiomaskintermediate_results? r"dogboxrs_ A VVXF D A D ARVVCF^#-aC81RVVAq\!QA7C(=W-=I,Q/y"AKy iRVV ,E z}}Rs+H!#HRXXb !"HRXXb  A == D77S=II!| \A% ;8* aRVV$ D=!"  a< %it=M&/ 9  )TX-= 8X,X,8_   q({^F"5a8K&ffvg>DAq & "1%C$C ;G9j9!DhAdGq=DhAdGAVVXFDAA AID(#A&5aC@1Q"A#4Q #B yI Q   "0 #6 *2  '(-&("[ ^!  $F64d;M OOr$)N)__doc__numpyr0 numpy.linalgrrscipy.sparse.linalgrrrscipy.optimizerscipy._lib._utilr commonr r r rrrrrrrrrr.r?rRrr$r"rsS)T$FF)67777O*::(CXDHBOr$