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The following keys are recognized." } ] }, { "__type": "DefList", "__tag": 4033, "children": [ { "__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": "node_limit" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "node_limit" } ] } ] }, { "__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": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "time_limit" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "time_limit" } ] } ] }, { "__type": "DefListItem", "__tag": 4037, "dt": { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "mip_rel_gap" } ] }, "dd": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "mip_rel_gap" } ] } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Solves problems of the following form:" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\min_x \\ & c^T x \\\\\n\\mbox{such that} \\ & b_l \\leq A x \\leq b_u,\\\\\n& l \\leq x \\leq u, \\\\\n& x_i \\in \\mathbb{Z}, i \\in X_i" }, { "__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_l" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineMath", "__tag": 4057, "value": "b_u" }, { "__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; " }, { "__type": "InlineMath", "__tag": 4057, "value": "A" }, { "__type": "Text", "__tag": 4046, "value": " is a matrix, and " }, { "__type": "InlineMath", "__tag": 4057, "value": "X_i" }, { "__type": "Text", "__tag": 4046, "value": " is the set of indices of decision variables that must be integral. (In this context, a variable that can assume only integer values is said to be \"integral\"; it has an \"integrality\" constraint.)" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Alternatively, that's:" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "minimize " } ] }, { "__type": "Code", "__tag": 4050, "value": "c @ x", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "such that " } ] }, { "__type": "Code", "__tag": 4050, "value": "b_l <= A @ x <= b_u\nl <= x <= u\nSpecified elements of x must be integers", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "By default, " }, { "__type": "InlineCode", "__tag": 4051, "value": "l = 0" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "u = np.inf" }, { "__type": "Text", "__tag": 4046, "value": " unless 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/_milp.py", "item_line": 154, "item_type": "function", "aliases": [ "scipy.optimize.milp" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Consider the problem at\nhttps://en.wikipedia.org/wiki/Integer_programming#Example, which is\nexpressed as a maximization problem of two variables. Since `milp` requires\nthat the problem be expressed as a minimization problem, the objective\nfunction coefficients on the decision variables are:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nc = -np.array([0, 1])\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nNote the negative sign: we maximize the original objective function\nby minimizing the negative of the objective function.\n\nWe collect the coefficients of the constraints into arrays like:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "A = np.array([[-1, 1], [3, 2], [2, 3]])\nb_u = np.array([1, 12, 12])\nb_l = np.full_like(b_u, -np.inf, dtype=float)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nBecause there is no lower limit on these constraints, we have defined a\nvariable ``b_l`` full of values representing negative infinity. This may\nbe unfamiliar to users of `scipy.optimize.linprog`, which only accepts\n\"less than\" (or \"upper bound\") inequality constraints of the form\n``A_ub @ x <= b_u``. By accepting both ``b_l`` and ``b_u`` of constraints\n``b_l <= A_ub @ x <= b_u``, `milp` makes it easy to specify \"greater than\"\ninequality constraints, \"less than\" inequality constraints, and equality\nconstraints concisely.\n\nThese arrays are collected into a single `LinearConstraint` object like:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.optimize import LinearConstraint\nconstraints = LinearConstraint(A, b_l, b_u)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nThe non-negativity bounds on the decision variables are enforced by\ndefault, so we do not need to provide an argument for `bounds`.\n\nFinally, the problem states that both decision variables must be integers:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "integrality = np.ones_like(c)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nWe solve the problem like:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.optimize import milp\nres = milp(c=c, constraints=constraints, integrality=integrality)\nres.x\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nNote that had we solved the relaxed problem (without integrality\nconstraints):\n\n" }, { "__type": "Code", "__tag": 4050, "value": "res = milp(c=c, constraints=constraints) # OR:\nres.x\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nwe would not have obtained the correct solution by rounding to the nearest\nintegers.\n\nOther examples are given :ref:`in the tutorial `." } ], "title": [], "level": 0, "target": null }, "see_also": [], "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": "integrality", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "bounds", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "constraints", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "options", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "milp" }, "references": [ ".. [1] Huangfu, Q., Galabova, I., Feldmeier, M., and Hall, J. A. J.", " \"HiGHS - high performance software for linear optimization.\"", " https://highs.dev/", ".. [2] 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._milp:milp", "arbitrary": [], "local_refs": [ "bounds", "c", "constraints", "integrality", "options", "res" ] }