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## MLinExpr

Gurobi linear matrix expression object. A linear matrix expression results from an arithmetic operation with an MVar object. A common example is a matrix-vector product, where the matrix is a NumPy ndarray or a SciPy sparse matrix and the vector is a Gurobi MVar object. Linear matrix expressions are used to build linear objectives and constraints. They are temporary objects that typically have short lifespans.

You generally build linear matrix expressions using overloaded operators, typically by multiplying a 2-D matrix (dense or sparse) by a 1-D MVar object using the Python matrix multiply (@) operator (e.g., expr = A @ x). You can also promote an MVar object to an MLinExpr using arithmetic expressions (e.g., expr = x + 1). Most arithmetic operations are supported on MLinExpr objects, including addition and subtraction (e.g., expr = A @ x - B @ y), multiplication by a constant (e.g. expr = 2 * A @ x), and point-wise multiplication with an ndarray or a sparse matrix. An MLinExpr object containing empty expressions can be created using the zeros method.

An MLinExpr object has a shape representing its dimensions, a size that counts the total number of elements, and an ndim that gives the number of dimensions. These properties lean on their counterparts in NumPy's ndarray class.

When working with MLinExpr objects, you need to make sure that the operands' shapes are compatible. For matrix multiplication, we follow the rules of Python's matrix multiplication operator: both operands need to have at least one dimension, and their inner dimensions must agree. For more information we refer you to Python's documentation. Other binary operations such as addition and multiplication are straightforward to understand if both operands have the same shape: the operation is applied point wise on the matching indices. For operands that have different shapes, the arithmetic follows NumPy's broadcasting rules. We refer you to the NumPy documentation for more information.

The full list of overloaded operators on MLinExpr objects is as follows: +, +=, -, -=, *, *=, and @. In Python parlance, we've defined the following MLinExpr functions: __add__, __radd__, __iadd__, __sub__, __rsub__, __isub__, __neg__, __mul__, __rmul__, __imul__, __matmul__, and __rmatmul__.

We've also overloaded the comparison operators (==, <=, and >=), to make it easier to build constraints from linear matrix expressions.

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