4[gٲ dZddlZddlZddlmZddlm Z ddl m Z ddl m Z mZmZddlmZdZdZ ddlZdd lmZdd lmZ ddlZ dd Z dd ZdZdZ dZ!dZ"dZ#dZ$ddZ%dZ& ddZ'y#e$rd ZY=wxYw#e$rd ZYEwxYw)aInterior-point method for linear programming The *interior-point* method uses the primal-dual path following algorithm outlined in [1]_. This algorithm supports sparse constraint matrices and is typically faster than the simplex methods, especially for large, sparse problems. Note, however, that the solution returned may be slightly less accurate than those of the simplex methods and will not, in general, correspond with a vertex of the polytope defined by the constraints. .. versionadded:: 1.0.0 References ---------- .. [1] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. N)warn) LinAlgError)OptimizeWarningOptimizeResult_check_unknown_options) _postsolveT)cholesky)analyzeFc` |r|rdfd }|S|r2 tjjtj}|St r#|r!t jj}|St jj|j} |S|rfd}|S|r&tjjfd}|S|ffd } |S#t$r6t t_tjjYwxYw#t$rt$rYywxYw)ae Given solver options, return a handle to the appropriate linear system solver. Parameters ---------- M : 2-D array As defined in [4] Equation 8.31 sparse : bool (default = False) True if the system to be solved is sparse. This is typically set True when the original ``A_ub`` and ``A_eq`` arrays are sparse. lstsq : bool (default = False) True if the system is ill-conditioned and/or (nearly) singular and thus a more robust least-squares solver is desired. This is sometimes needed as the solution is approached. sym_pos : bool (default = True) True if the system matrix is symmetric positive definite Sometimes this needs to be set false as the solution is approached, even when the system should be symmetric positive definite, due to numerical difficulties. cholesky : bool (default = True) True if the system is to be solved by Cholesky, rather than LU, decomposition. This is typically faster unless the problem is very small or prone to numerical difficulties. permc_spec : str (default = 'MMD_AT_PLUS_A') Sparsity preservation strategy used by SuperLU. Acceptable values are: - ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering. See SuperLU documentation. Returns ------- solve : function Handle to the appropriate solver function cJtjj|dSNr)spslinalglsqrrsym_posMs U/var/www/html/bid-api/venv/lib/python3.12/site-packages/scipy/optimize/_linprog_ip.pysolvez_get_solver..solveWs::??1a033) permc_speccJtjj|dSr)sprlstsq)rrs rrz_get_solver..solvejs99??1a033rcDtjj|SN)rr cho_solve)rLs rrz_get_solver..solveos99..q!44rc|r"tjj|dStjj|S)Npos)assume_a)rrrrs rrz_get_solver..solvets4!yyq!eDD!yyq!44rNF) _get_solvercholmod_factorcholesky_inplace Exceptioncholmod_analyze has_umfpackrr factorizedsplurr cho_factorKeyboardInterrupt)rsparserrr rrr s` @rr%r%*s4T, 4T LQC ..??B$22@ L=7JJ11!4E: L7 JJOOA*OEKKE6 L14. L+II((+5$ L&-5 LG!C1@1CK...??BC>  sV DDCD'D-+DD$&D D.r_ps3wq!!rcL|zjj|z |z Sr)r=r<)rMrNrOrIrKs rr_dz_indicators..r_ds#3w#a''rcP|j|zj|z Srr)rLrMrPrJrKs rr_gz_indicators..r_gs#quuQx!%%(**rcp|j|tj||zt|dzz SNr)r<rBr;)rLrOrNrPs rrYz_indicators..mus-a266#u--#a&1*==rc@tjj|Sr)rBrnorm)as rrz_indicators..normsyy~~a  rr)rr:r<r=max)rIrJrKc0rLrMrNrOrPrrrrrrrrrYobjrr_p0r_d0r_g0mu_0rho_Arho_prho_drho_grho_mus``` r _indicatorsrsW 08BBf"(+> %%C.2 C! r4=D r2t D r2v D b$F #D accggaj( )S4 3C-C DE Q AtDz 2 2E Q3 3q$t*#5 5E Q5! "SDJ%7 7E 35 !D (F %vs 22rc |rtddddddd}t|jt|t|t|t|tr|n t|t|t|y) a\ Print indicators of optimization status to the console. Parameters ---------- rho_p : float The (normalized) primal feasibility, see [4] 4.5 rho_d : float The (normalized) dual feasibility, see [4] 4.5 rho_g : float The (normalized) duality gap, see [4] 4.5 alpha : float The step size, see [4] 4.3 rho_mu : float The (normalized) path parameter, see [4] 4.5 obj : float The objective function value of the current iterate header : bool True if a header is to be printed References ---------- .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. zPrimal Feasibility zDual Feasibility zDuality Gap zStep zPath Parameter zObjective z<{0:<20.13}{1:<20.13}{2:<20.13}{3:<17.13}{4:<20.13}{5:<20.13}N)printr2float isinstancestr)rrrr]rrheaderfmts r _display_iterrsq: ### ##  % IC #** e  e  e E3'U5\ f  c  rctd}t|j\}}}}}| r|nd}t||||||||| \}}}}}}||kDxs ||kDxs||kD}|rt|||d||d|2t ||z |\}} }!}"t || |!|"|dddddd }#||#d}$d }%| rt j|}|r|dz }|rd}&d }'n$| rdn|tj||zz}&|&fd }' t|||||||||&|'| | | | | ||\}(})}*}+},|rHd }-t||||||(|)|*|+|,|- \}}}}}d||dk<d||dk<td|}td|}d}n.t||(||*||+||,| }-t||||||(|)|*|+|,|- \}}}}}t||||||||| \}}}}}}||kDxs ||kDxs||kD}|rt||||-|||2t ||z |\}} }!}"t || |!|"|dddddd }#||#||kxr ||kxr||kxr||td|zk}.||kxr||t'd|zk}/|.s|/r3|j)j+||kDrd}$nd}$t%|$}%n||k\rd}$t%|$}%n|r||z }0|0|$|%|fS#ttt t"f$rd }$t%|$}%Y3wxYw)a Solve a linear programming problem in standard form: Minimize:: c @ x Subject to:: A @ x == b x >= 0 using the interior point method of [4]. Parameters ---------- A : 2-D array 2-D array such that ``A @ x``, gives the values of the equality constraints at ``x``. b : 1-D array 1-D array of values representing the RHS of each equality constraint (row) in ``A`` (for standard form problem). c : 1-D array Coefficients of the linear objective function to be minimized (for standard form problem). c0 : float Constant term in objective function due to fixed (and eliminated) variables. (Purely for display.) alpha0 : float The maximal step size for Mehrota's predictor-corrector search direction; see :math:`\beta_3`of [4] Table 8.1 beta : float The desired reduction of the path parameter :math:`\mu` (see [6]_) maxiter : int The maximum number of iterations of the algorithm. disp : bool Set to ``True`` if indicators of optimization status are to be printed to the console each iteration. tol : float Termination tolerance; see [4]_ Section 4.5. sparse : bool Set to ``True`` if the problem is to be treated as sparse. However, the inputs ``A_eq`` and ``A_ub`` should nonetheless be provided as (dense) arrays rather than sparse matrices. lstsq : bool Set to ``True`` if the problem is expected to be very poorly conditioned. This should always be left as ``False`` unless severe numerical difficulties are frequently encountered, and a better option would be to improve the formulation of the problem. sym_pos : bool Leave ``True`` if the problem is expected to yield a well conditioned symmetric positive definite normal equation matrix (almost always). cholesky : bool Set to ``True`` if the normal equations are to be solved by explicit Cholesky decomposition followed by explicit forward/backward substitution. This is typically faster for moderate, dense problems that are numerically well-behaved. pc : bool Leave ``True`` if the predictor-corrector method of Mehrota is to be used. This is almost always (if not always) beneficial. ip : bool Set to ``True`` if the improved initial point suggestion due to [4]_ Section 4.3 is desired. It's unclear whether this is beneficial. permc_spec : str (default = 'MMD_AT_PLUS_A') (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = True``.) A matrix is factorized in each iteration of the algorithm. This option specifies how to permute the columns of the matrix for sparsity preservation. Acceptable values are: - ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering. This option can impact the convergence of the interior point algorithm; test different values to determine which performs best for your problem. For more information, refer to ``scipy.sparse.linalg.splu``. callback : callable, optional If a callback function is provided, it will be called within each iteration of the algorithm. The callback function must accept a single `scipy.optimize.OptimizeResult` consisting of the following fields: x : 1-D array Current solution vector fun : float Current value of the objective function success : bool True only when an algorithm has completed successfully, so this is always False as the callback function is called only while the algorithm is still iterating. slack : 1-D array The values of the slack variables. Each slack variable corresponds to an inequality constraint. If the slack is zero, the corresponding constraint is active. con : 1-D array The (nominally zero) residuals of the equality constraints, that is, ``b - A_eq @ x`` phase : int The phase of the algorithm being executed. This is always 1 for the interior-point method because it has only one phase. status : int For revised simplex, this is always 0 because if a different status is detected, the algorithm terminates. nit : int The number of iterations performed. message : str A string descriptor of the exit status of the optimization. postsolve_args : tuple Data needed by _postsolve to convert the solution to the standard-form problem into the solution to the original problem. Returns ------- x_hat : float Solution vector (for standard form problem). status : int An integer representing the exit status of the optimization:: 0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded 4 : Serious numerical difficulties encountered message : str A string descriptor of the exit status of the optimization. iteration : int The number of iterations taken to solve the problem References ---------- .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear Programming based on Newton's Method." Unpublished Course Notes, March 2004. Available 2/25/2017 at: https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf rF-T)rr) rLfunslackconnitphasecompleter~messagesuccessr|cyrr}gs rrRz_ip_hsd..etasrc d|z Srr}rs rrRz_ip_hsd..etas 1u rg?r4)rr:rrr rr csc_matrixrBmeanrorrrGrFloatingPointErrorrEZeroDivisionErrorrrHr>r<)1rIrJrKrrtbetamaxiterdisptolr/rrr rSrTrcallbackpostsolve_args iterationrLrMrNrOrPrrrrrrgox_orrrresr~rrQrRr^rmr_r`rar]inf1inf2x_hats1 r_ip_hsdr*sbI+1773Aq!S%uB/: 1aQ1c5/*+E5%  2 2us{B eUE3DI)!C%@S%3sU%(Q*/1)+?@  F5G NN1  Q E  AA!6E " ,61aAq#ueSw"b*-F )Cc5'&.q!S%c'0#1ae!a%!a%!SkAu "!S!S#"'A'/q!S%c3w'O#1ae3> q!RAq#u3./ueUFC S[ 6ECK 653;  %ufc B  #-ae^#D CeS s3),Y.3q-/E"CDC SM   cAu GA|9cC5M&9 9 4{{}  #c)"6*G  ' !F"6*G m p GE &'9 ,,W/-/  F"6*G  s)BJ&J76J7c t||s|| rts|rtdtdd}| r| rtdtd| r|rtdtdd}|j |vr#td t |zd ztdd }| s |r t d |xs |duxr| xr| }t||||| | |||| | | ||||||\}}}}||||fS) a], Minimize a linear objective function subject to linear equality and non-negativity constraints using the interior point method of [4]_. Linear programming is intended to solve problems of the following form: Minimize:: c @ x Subject to:: A @ x == b x >= 0 User-facing documentation is in _linprog_doc.py. Parameters ---------- c : 1-D array Coefficients of the linear objective function to be minimized. c0 : float Constant term in objective function due to fixed (and eliminated) variables. (Purely for display.) A : 2-D array 2-D array such that ``A @ x``, gives the values of the equality constraints at ``x``. b : 1-D array 1-D array of values representing the right hand side of each equality constraint (row) in ``A``. callback : callable, optional Callback function to be executed once per iteration. postsolve_args : tuple Data needed by _postsolve to convert the solution to the standard-form problem into the solution to the original problem. Options ------- maxiter : int (default = 1000) The maximum number of iterations of the algorithm. tol : float (default = 1e-8) Termination tolerance to be used for all termination criteria; see [4]_ Section 4.5. disp : bool (default = False) Set to ``True`` if indicators of optimization status are to be printed to the console each iteration. alpha0 : float (default = 0.99995) The maximal step size for Mehrota's predictor-corrector search direction; see :math:`\beta_{3}` of [4]_ Table 8.1. beta : float (default = 0.1) The desired reduction of the path parameter :math:`\mu` (see [6]_) when Mehrota's predictor-corrector is not in use (uncommon). sparse : bool (default = False) Set to ``True`` if the problem is to be treated as sparse after presolve. If either ``A_eq`` or ``A_ub`` is a sparse matrix, this option will automatically be set ``True``, and the problem will be treated as sparse even during presolve. If your constraint matrices contain mostly zeros and the problem is not very small (less than about 100 constraints or variables), consider setting ``True`` or providing ``A_eq`` and ``A_ub`` as sparse matrices. lstsq : bool (default = False) Set to ``True`` if the problem is expected to be very poorly conditioned. This should always be left ``False`` unless severe numerical difficulties are encountered. Leave this at the default unless you receive a warning message suggesting otherwise. sym_pos : bool (default = True) Leave ``True`` if the problem is expected to yield a well conditioned symmetric positive definite normal equation matrix (almost always). Leave this at the default unless you receive a warning message suggesting otherwise. cholesky : bool (default = True) Set to ``True`` if the normal equations are to be solved by explicit Cholesky decomposition followed by explicit forward/backward substitution. This is typically faster for problems that are numerically well-behaved. pc : bool (default = True) Leave ``True`` if the predictor-corrector method of Mehrota is to be used. This is almost always (if not always) beneficial. ip : bool (default = False) Set to ``True`` if the improved initial point suggestion due to [4]_ Section 4.3 is desired. Whether this is beneficial or not depends on the problem. permc_spec : str (default = 'MMD_AT_PLUS_A') (Has effect only with ``sparse = True``, ``lstsq = False``, ``sym_pos = True``, and no SuiteSparse.) A matrix is factorized in each iteration of the algorithm. This option specifies how to permute the columns of the matrix for sparsity preservation. Acceptable values are: - ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering. This option can impact the convergence of the interior point algorithm; test different values to determine which performs best for your problem. For more information, refer to ``scipy.sparse.linalg.splu``. unknown_options : dict Optional arguments not used by this particular solver. If `unknown_options` is non-empty a warning is issued listing all unused options. Returns ------- x : 1-D array Solution vector. status : int An integer representing the exit status of the optimization:: 0 : Optimization terminated successfully 1 : Iteration limit reached 2 : Problem appears to be infeasible 3 : Problem appears to be unbounded 4 : Serious numerical difficulties encountered message : str A string descriptor of the exit status of the optimization. iteration : int The number of iterations taken to solve the problem. Notes ----- This method implements the algorithm outlined in [4]_ with ideas from [8]_ and a structure inspired by the simpler methods of [6]_. The primal-dual path following method begins with initial 'guesses' of the primal and dual variables of the standard form problem and iteratively attempts to solve the (nonlinear) Karush-Kuhn-Tucker conditions for the problem with a gradually reduced logarithmic barrier term added to the objective. This particular implementation uses a homogeneous self-dual formulation, which provides certificates of infeasibility or unboundedness where applicable. The default initial point for the primal and dual variables is that defined in [4]_ Section 4.4 Equation 8.22. Optionally (by setting initial point option ``ip=True``), an alternate (potentially improved) starting point can be calculated according to the additional recommendations of [4]_ Section 4.4. A search direction is calculated using the predictor-corrector method (single correction) proposed by Mehrota and detailed in [4]_ Section 4.1. (A potential improvement would be to implement the method of multiple corrections described in [4]_ Section 4.2.) In practice, this is accomplished by solving the normal equations, [4]_ Section 5.1 Equations 8.31 and 8.32, derived from the Newton equations [4]_ Section 5 Equations 8.25 (compare to [4]_ Section 4 Equations 8.6-8.8). The advantage of solving the normal equations rather than 8.25 directly is that the matrices involved are symmetric positive definite, so Cholesky decomposition can be used rather than the more expensive LU factorization. With default options, the solver used to perform the factorization depends on third-party software availability and the conditioning of the problem. For dense problems, solvers are tried in the following order: 1. ``scipy.linalg.cho_factor`` 2. ``scipy.linalg.solve`` with option ``sym_pos=True`` 3. ``scipy.linalg.solve`` with option ``sym_pos=False`` 4. ``scipy.linalg.lstsq`` For sparse problems: 1. ``sksparse.cholmod.cholesky`` (if scikit-sparse and SuiteSparse are installed) 2. ``scipy.sparse.linalg.factorized`` (if scikit-umfpack and SuiteSparse are installed) 3. ``scipy.sparse.linalg.splu`` (which uses SuperLU distributed with SciPy) 4. ``scipy.sparse.linalg.lsqr`` If the solver fails for any reason, successively more robust (but slower) solvers are attempted in the order indicated. Attempting, failing, and re-starting factorization can be time consuming, so if the problem is numerically challenging, options can be set to bypass solvers that are failing. Setting ``cholesky=False`` skips to solver 2, ``sym_pos=False`` skips to solver 3, and ``lstsq=True`` skips to solver 4 for both sparse and dense problems. Potential improvements for combatting issues associated with dense columns in otherwise sparse problems are outlined in [4]_ Section 5.3 and [10]_ Section 4.1-4.2; the latter also discusses the alleviation of accuracy issues associated with the substitution approach to free variables. After calculating the search direction, the maximum possible step size that does not activate the non-negativity constraints is calculated, and the smaller of this step size and unity is applied (as in [4]_ Section 4.1.) [4]_ Section 4.3 suggests improvements for choosing the step size. The new point is tested according to the termination conditions of [4]_ Section 4.5. The same tolerance, which can be set using the ``tol`` option, is used for all checks. (A potential improvement would be to expose the different tolerances to be set independently.) If optimality, unboundedness, or infeasibility is detected, the solve procedure terminates; otherwise it repeats. The expected problem formulation differs between the top level ``linprog`` module and the method specific solvers. The method specific solvers expect a problem in standard form: Minimize:: c @ x Subject to:: A @ x == b x >= 0 Whereas the top level ``linprog`` module expects a problem of form: Minimize:: c @ x Subject to:: A_ub @ x <= b_ub A_eq @ x == b_eq lb <= x <= ub where ``lb = 0`` and ``ub = None`` unless set in ``bounds``. The original problem contains equality, upper-bound and variable constraints whereas the method specific solver requires equality constraints and variable non-negativity. ``linprog`` module converts the original problem to standard form by converting the simple bounds to upper bound constraints, introducing non-negative slack variables for inequality constraints, and expressing unbounded variables as the difference between two non-negative variables. References ---------- .. [4] Andersen, Erling D., and Knud D. Andersen. "The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm." High performance optimization. Springer US, 2000. 197-232. .. [6] Freund, Robert M. "Primal-Dual Interior-Point Methods for Linear Programming based on Newton's Method." Unpublished Course Notes, March 2004. Available 2/25/2017 at https://ocw.mit.edu/courses/sloan-school-of-management/15-084j-nonlinear-programming-spring-2004/lecture-notes/lec14_int_pt_mthd.pdf .. [8] Andersen, Erling D., and Knud D. Andersen. "Presolving in linear programming." Mathematical Programming 71.2 (1995): 221-245. .. [9] Bertsimas, Dimitris, and J. Tsitsiklis. "Introduction to linear programming." Athena Scientific 1 (1997): 997. .. [10] Andersen, Erling D., et al. Implementation of interior point methods for large scale linear programming. HEC/Universite de Geneve, 1996. NzPSparse cholesky is only available with scikit-sparse. Setting `cholesky = False`rr6FzEOption combination 'sparse':True and 'lstsq':True is not recommended.zvInvalid option combination 'lstsq':True and 'cholesky':True; option 'cholesky' has no effect when 'lstsq' is set True.)NATURALMMD_ATA MMD_AT_PLUS_ACOLAMDzInvalid permc_spec option: 'zc'. Acceptable values are 'NATURAL', 'MMD_ATA', 'MMD_AT_PLUS_A', and 'COLAMD'. Reverting to default.rzInvalid option combination 'sym_pos':False and 'cholesky':True: Cholesky decomposition is only possible for symmetric positive definite matrices.)r has_cholmodrrupperrrEr)rKrrIrJrrrrrrtrr/rrr rSrTrunknown_optionsvalid_permc_specrLr~rrs r _linprog_ipr5s3J?+ H$&  . Q 0 % #  ,  $ , I!11 +c*o =A33 ,%  x 89 9 GH,FFYH$+Aq!R,3T3,17H,.J,: %<!Avw fgy ((r)FFTTr)FFTTTFrr$) ig:0yE>FgSt$?r9FFTNTFr)(__doc__numpyrBscipyr scipy.sparser/rwarningsr scipy.linalgr _optimizerrr _linprog_utilr r*rsksparsesksparse.cholmodr cholmodr r) ImportErrorscikits.umfpackscikitsr%rorArGrrrrrrrr}rrrs($NN%  4; 7;*9WtAFEJ)F)R(4"J($*.3b-`H-VJNIN9>*q)gKKs#A6B6B?BB  B