{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This section describes the available solvers that can be selected by the 'method' parameter." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'highs-ds' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ", and " }, { "__type": "InlineRole", "__tag": 4003, "value": "'highs-ipm' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " are interfaces to the HiGHS simplex and interior-point method solvers " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "13" }, { "__type": "Text", "__tag": 4046, "value": ", respectively. " }, { "__type": "InlineRole", "__tag": 4003, "value": "'highs' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " (default) chooses between the two automatically. These are the fastest linear programming solvers in SciPy, especially for large, sparse problems; which of these two is faster is problem-dependent. The other solvers are legacy methods and will be removed when " }, { "__type": "ParamRef", "__tag": 4071, "name": "callback" }, { "__type": "Text", "__tag": 4046, "value": " is supported by the HiGHS methods." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Method " }, { "__type": "InlineRole", "__tag": 4003, "value": "'highs-ds' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ", is a wrapper of the C++ high performance dual revised simplex implementation (HSOL) " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "13" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "14" }, { "__type": "Text", "__tag": 4046, "value": ". Method " }, { "__type": "InlineRole", "__tag": 4003, "value": "'highs-ipm' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " is a wrapper of a C++ implementation of an " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "i" } ] }, { "__type": "Text", "__tag": 4046, "value": "terior-\\ " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "p" } ] }, { "__type": "Text", "__tag": 4046, "value": "oint " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "m" } ] }, { "__type": "Text", "__tag": 4046, "value": "ethod " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "13" }, { "__type": "Text", "__tag": 4046, "value": "; it features a crossover routine, so it is as accurate as a simplex solver. Method " }, { "__type": "InlineRole", "__tag": 4003, "value": "'highs' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " chooses between the two automatically. For new code involving " }, { "__type": "CrossRef", "__tag": 4002, "value": "linprog", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._linprog:linprog" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", we recommend explicitly choosing one of these three method values." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.6.0" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Method " }, { "__type": "InlineRole", "__tag": 4003, "value": "'interior-point' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " uses the primal-dual path following algorithm as outlined in " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "4" }, { "__type": "Text", "__tag": 4046, "value": ". 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." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.0.0" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Method " }, { "__type": "InlineRole", "__tag": 4003, "value": "'revised simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " uses the revised simplex method as described in " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "9" }, { "__type": "Text", "__tag": 4046, "value": ", except that a factorization " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "11" }, { "__type": "Text", "__tag": 4046, "value": " of the basis matrix, rather than its inverse, is efficiently maintained and used to solve the linear systems at each iteration of the algorithm." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.3.0" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Method " }, { "__type": "InlineRole", "__tag": 4003, "value": "'simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " uses a traditional, full-tableau implementation of Dantzig's simplex algorithm " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": " (" }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "not" } ] }, { "__type": "Text", "__tag": 4046, "value": " the Nelder-Mead simplex). This algorithm is included for backwards compatibility and educational purposes." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 0.15.0" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Before applying " }, { "__type": "InlineRole", "__tag": 4003, "value": "'interior-point' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineRole", "__tag": 4003, "value": "'revised simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ", or " }, { "__type": "InlineRole", "__tag": 4003, "value": "'simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": ", a presolve procedure based on " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "8" }, { "__type": "Text", "__tag": 4046, "value": " attempts to identify trivial infeasibilities, trivial unboundedness, and potential problem simplifications. Specifically, it checks for:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rows of zeros in " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_ub" }, { "__type": "Text", "__tag": 4046, "value": ", representing trivial constraints;" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "columns of zeros in " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq" }, { "__type": "Text", "__tag": 4046, "value": " " }, { "__type": "InlineRole", "__tag": 4003, "value": "and", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_ub" }, { "__type": "Text", "__tag": 4046, "value": ", representing unconstrained variables;" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "column singletons in " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq" }, { "__type": "Text", "__tag": 4046, "value": ", representing fixed variables; and" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "column singletons in " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_ub" }, { "__type": "Text", "__tag": 4046, "value": ", representing simple bounds." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If presolve reveals that the problem is unbounded (e.g. an unconstrained and unbounded variable has negative cost) or infeasible (e.g., a row of zeros in " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq" }, { "__type": "Text", "__tag": 4046, "value": " corresponds with a nonzero in " }, { "__type": "InlineCode", "__tag": 4051, "value": "b_eq" }, { "__type": "Text", "__tag": 4046, "value": "), the solver terminates with the appropriate status code. Note that presolve terminates as soon as any sign of unboundedness is detected; consequently, a problem may be reported as unbounded when in reality the problem is infeasible (but infeasibility has not been detected yet). Therefore, if it is important to know whether the problem is actually infeasible, solve the problem again with option " }, { "__type": "InlineCode", "__tag": 4051, "value": "presolve=False" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If neither infeasibility nor unboundedness are detected in a single pass of the presolve, bounds are tightened where possible and fixed variables are removed from the problem. Then, linearly dependent rows of the " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq" }, { "__type": "Text", "__tag": 4046, "value": " matrix are removed, (unless they represent an infeasibility) to avoid numerical difficulties in the primary solve routine. Note that rows that are nearly linearly dependent (within a prescribed tolerance) may also be removed, which can change the optimal solution in rare cases. If this is a concern, eliminate redundancy from your problem formulation and run with option " }, { "__type": "InlineCode", "__tag": 4051, "value": "rr=False" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "presolve=False" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Several potential improvements can be made here: additional presolve checks outlined in " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "8" }, { "__type": "Text", "__tag": 4046, "value": " should be implemented, the presolve routine should be run multiple times (until no further simplifications can be made), and more of the efficiency improvements from " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "5" }, { "__type": "Text", "__tag": 4046, "value": " should be implemented in the redundancy removal routines." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "After presolve, the problem is transformed to standard form by converting the (tightened) 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. Optionally, the problem is automatically scaled via equilibration " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "12" }, { "__type": "Text", "__tag": 4046, "value": ". The selected algorithm solves the standard form problem, and a postprocessing routine converts the result to a solution to the original problem." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "res", "annotation": "OptimizeResult", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.optimize.OptimizeResult", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.optimize._optimize:OptimizeResult" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " consisting of the fields below. Note that the return types of the fields may depend on whether the optimization was successful, therefore it is recommended to check " }, { "__type": "InlineRole", "__tag": 4003, "value": "OptimizeResult.status", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " before relying on the other fields:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "x" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "x" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "fun" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "fun" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "slack" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "slack" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "con" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "con" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "success" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "success" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "status" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "status" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "nit" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "nit" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "message" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "message" } ] } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Linear programming: minimize a linear objective function subject to linear equality and inequality constraints." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "c", "annotation": "1-D array", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The coefficients of the linear objective function to be minimized." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "A_ub", "annotation": "2-D array, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The inequality constraint matrix. Each row of " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_ub" }, { "__type": "Text", "__tag": 4046, "value": " specifies the coefficients of a linear inequality constraint on " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "b_ub", "annotation": "1-D array, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The inequality constraint vector. Each element represents an upper bound on the corresponding value of " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_ub @ x" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "A_eq", "annotation": "2-D array, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The equality constraint matrix. Each row of " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq" }, { "__type": "Text", "__tag": 4046, "value": " specifies the coefficients of a linear equality constraint on " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "b_eq", "annotation": "1-D array, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The equality constraint vector. Each element of " }, { "__type": "InlineCode", "__tag": 4051, "value": "A_eq @ x" }, { "__type": "Text", "__tag": 4046, "value": " must equal the corresponding element of " }, { "__type": "InlineCode", "__tag": 4051, "value": "b_eq" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "bounds", "annotation": "sequence, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A sequence of " }, { "__type": "InlineCode", "__tag": 4051, "value": "(min, max)" }, { "__type": "Text", "__tag": 4046, "value": " pairs for each element in " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": ", defining the minimum and maximum values of that decision variable. If a single tuple " }, { "__type": "InlineCode", "__tag": 4051, "value": "(min, max)" }, { "__type": "Text", "__tag": 4046, "value": " is provided, then " }, { "__type": "InlineCode", "__tag": 4051, "value": "min" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "max" }, { "__type": "Text", "__tag": 4046, "value": " will serve as bounds for all decision variables. Use " }, { "__type": "InlineCode", "__tag": 4051, "value": "None" }, { "__type": "Text", "__tag": 4046, "value": " to indicate that there is no bound. For instance, the default bound " }, { "__type": "InlineCode", "__tag": 4051, "value": "(0, None)" }, { "__type": "Text", "__tag": 4046, "value": " means that all decision variables are non-negative, and the pair " }, { "__type": "InlineCode", "__tag": 4051, "value": "(None, None)" }, { "__type": "Text", "__tag": 4046, "value": " means no bounds at all, i.e. all variables are allowed to be any real." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "method", "annotation": "str, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The algorithm used to solve the standard form problem. The following are supported." } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'highs' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " (default)" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'highs-ds' ", "domain": null, "role": "ref", "inventory": null } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'highs-ipm' ", "domain": null, "role": "ref", "inventory": null } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'interior-point' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " (legacy)" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'revised simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " (legacy)" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "'simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " (legacy)" } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The legacy methods are deprecated and will be removed in SciPy 1.11.0." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "callback", "annotation": "callable, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If a callback function is provided, it will be called at least once per iteration of the algorithm. The callback function must accept a single " }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.optimize.OptimizeResult", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "*", "kind": "api", "path": "scipy.optimize._optimize:OptimizeResult" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " consisting of the following fields:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "x" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "x" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "fun" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "fun" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "success" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "success" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "slack" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "slack" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "con" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "con" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "phase" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "phase" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "status" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "status" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "nit" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "nit" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "message" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "message" } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Callback functions are not currently supported by the HiGHS methods." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "options", "annotation": "dict, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A dictionary of solver options. All methods accept the following options:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maxiter" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "maxiter" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "disp" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "disp" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "presolve" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "presolve" } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "All methods except the HiGHS solvers also accept:" } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "tol" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "tol" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "autoscale" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "autoscale" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rr" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rr" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rr_method" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "rr_method" } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For method-specific options, see " }, { "__type": "CrossRef", "__tag": 4002, "value": "show_options('linprog')", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._optimize:show_options" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "x0", "annotation": "1-D array, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Guess values of the decision variables, which will be refined by the optimization algorithm. This argument is currently used only by the " }, { "__type": "InlineRole", "__tag": 4003, "value": "'revised simplex' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " method, and can only be used if " }, { "__type": "ParamRef", "__tag": 4071, "name": "x0" }, { "__type": "Text", "__tag": 4046, "value": " represents a basic feasible solution." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "integrality", "annotation": "1-D array or int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Indicates the type of integrality constraint on each decision variable." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "0" }, { "__type": "Text", "__tag": 4046, "value": "Continuous variable; no integrality constraint." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "1" }, { "__type": "Text", "__tag": 4046, "value": "Integer variable; decision variable must be an integer within " }, { "__type": "ParamRef", "__tag": 4071, "name": "bounds" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "2" }, { "__type": "Text", "__tag": 4046, "value": "Semi-continuous variable; decision variable must be within " }, { "__type": "ParamRef", "__tag": 4071, "name": "bounds" }, { "__type": "Text", "__tag": 4046, "value": " or take value " }, { "__type": "InlineCode", "__tag": 4051, "value": "0" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "3" }, { "__type": "Text", "__tag": 4046, "value": "Semi-integer variable; decision variable must be an integer within " }, { "__type": "ParamRef", "__tag": 4071, "name": "bounds" }, { "__type": "Text", "__tag": 4046, "value": " or take value " }, { "__type": "InlineCode", "__tag": 4051, "value": "0" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "By default, all variables are continuous." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For mixed integrality constraints, supply an array of shape " }, { "__type": "InlineCode", "__tag": 4051, "value": "c.shape" }, { "__type": "Text", "__tag": 4046, "value": ". To infer a constraint on each decision variable from shorter inputs, the argument will be broadcast to " }, { "__type": "InlineCode", "__tag": 4051, "value": "c.shape" }, { "__type": "Text", "__tag": 4046, "value": " using " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.broadcast_to", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy:broadcast_to" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This argument is currently used only by the " }, { "__type": "InlineRole", "__tag": 4003, "value": "'highs' ", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " method and is ignored otherwise." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Linear programming solves problems of the following form:" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\min_x \\ & c^T x \\\\\n\\mbox{such that} \\ & A_{ub} x \\leq b_{ub},\\\\\n& A_{eq} x = b_{eq},\\\\\n& l \\leq x \\leq u ," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": " is a vector of decision variables; " }, { "__type": "InlineMath", "__tag": 4057, "value": "c" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "b_{ub}" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "b_{eq}" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "l" }, { "__type": "Text", "__tag": 4046, "value": ", and " }, { "__type": "InlineMath", "__tag": 4057, "value": "u" }, { "__type": "Text", "__tag": 4046, "value": " are vectors; and " }, { "__type": "InlineMath", "__tag": 4057, "value": "A_{ub}" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "A_{eq}" }, { "__type": "Text", "__tag": 4046, "value": " are matrices." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Alternatively, that's:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "minimize " } ] }, { "__type": "Code", "__tag": 4050, "value": "c @ x", "execution_status": null } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "such that " } ] }, { "__type": "Code", "__tag": 4050, "value": "A_ub @ x <= b_ub\nA_eq @ x == b_eq\nlb <= x <= ub", "execution_status": null } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that by default " }, { "__type": "InlineCode", "__tag": 4051, "value": "lb = 0" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "ub = None" }, { "__type": "Text", "__tag": 4046, "value": ". Other bounds can be specified with " }, { "__type": "InlineCode", "__tag": 4051, "value": "bounds" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/optimize/_linprog.py", "item_line": 178, "item_type": "function", "aliases": [ "scipy.optimize.linprog" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Consider the following problem:\n\n.. math::\n\n \\min_{x_0, x_1} \\ -x_0 + 4x_1 & \\\\\n \\mbox{such that} \\ -3x_0 + x_1 & \\leq 6,\\\\\n -x_0 - 2x_1 & \\geq -4,\\\\\n x_1 & \\geq -3.\n\nThe problem is not presented in the form accepted by `linprog`. This is\neasily remedied by converting the \"greater than\" inequality\nconstraint to a \"less than\" inequality constraint by\nmultiplying both sides by a factor of :math:`-1`. Note also that the last\nconstraint is really the simple bound :math:`-3 \\leq x_1 \\leq \\infty`.\nFinally, since there are no bounds on :math:`x_0`, we must explicitly\nspecify the bounds :math:`-\\infty \\leq x_0 \\leq \\infty`, as the\ndefault is for variables to be non-negative. After collecting coeffecients\ninto arrays and tuples, the input for this problem is:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.optimize import linprog\nc = [-1, 4]\nA = [[-3, 1], [1, 2]]\nb = [6, 4]\nx0_bounds = (None, None)\nx1_bounds = (-3, None)\nres = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds])\nres.fun\nres.x\nres.message\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThe marginals (AKA dual values / shadow prices / Lagrange multipliers)\nand residuals (slacks) are also available.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res.ineqlin\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nFor example, because the marginal associated with the second inequality\nconstraint is -1, we expect the optimal value of the objective function\nto decrease by ``eps`` if we add a small amount ``eps`` to the right hand\nside of the second inequality constraint:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "eps = 0.05\nb[1] += eps\nlinprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nAlso, because the residual on the first inequality constraint is 39, we\ncan decrease the right hand side of the first constraint by 39 without\naffecting the optimal solution.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "b = [6, 4] # reset to original values\nb[0] -= 39\nlinprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds]).fun\n", "execution_status": "success" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "show_options", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.optimize._optimize:show_options" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Additional options accepted by the solvers." } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "c", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "A_ub", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "b_ub", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "A_eq", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "b_eq", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "bounds", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "(0, None)" }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "highs" }, { "__type": "SigParam", "__tag": 4030, "name": "callback", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "options", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "x0", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "integrality", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "linprog" }, "references": [ ".. [1] Dantzig, George B., Linear programming and extensions. Rand", " Corporation Research Study Princeton Univ. Press, Princeton, NJ,", " 1963", ".. [2] Hillier, S.H. and Lieberman, G.J. (1995), \"Introduction to", " Mathematical Programming\", McGraw-Hill, Chapter 4.", ".. [3] Bland, Robert G. New finite pivoting rules for the simplex method.", " Mathematics of Operations Research (2), 1977: pp. 103-107.", ".. [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.", ".. [5] Andersen, Erling D. \"Finding all linearly dependent rows in", " large-scale linear programming.\" Optimization Methods and Software", " 6.3 (1995): 219-227.", ".. [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", ".. [7] Fourer, Robert. \"Solving Linear Programs by Interior-Point Methods.\"", " Unpublished Course Notes, August 26, 2005. Available 2/25/2017 at", " http://www.4er.org/CourseNotes/Book%20B/B-III.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.", ".. [11] Bartels, Richard H. \"A stabilization of the simplex method.\"", " Journal in Numerische Mathematik 16.5 (1971): 414-434.", ".. [12] Tomlin, J. A. \"On scaling linear programming problems.\"", " Mathematical Programming Study 4 (1975): 146-166.", ".. [13] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J.", " \"HiGHS - high performance software for linear optimization.\"", " https://highs.dev/", ".. [14] Huangfu, Q. and Hall, J. A. J. \"Parallelizing the dual revised", " simplex method.\" Mathematical Programming Computation, 10 (1),", " 119-142, 2018. DOI: 10.1007/s12532-017-0130-5" ], "qa": "scipy.optimize._linprog:linprog", "arbitrary": [], "local_refs": [ "A_eq", "A_ub", "b_eq", "b_ub", "bounds", "c", "callback", "integrality", "method", "options", "res", "x0" ] }