For linear objective functions, the simplex method is most commonly used. Although its worst-case behavior is not satisfactory, it works very well for most real-world problems. The mathematical formulation of the problem is to maximize wTv, subject to Av < b and v > 0.
The last inequation means that all entries of vector v are positive, A is a matrix with as many rows as the dimension of vector b and as many columns as dimension of v. T is the transpose operator; hence, the objective function is the scalar product of a fixed vector w and the objective v.
The idea is to first construct a solution at a vertex of the set of feasible solutions, which is shaped like a polyhedron. Then, the algorithm goes to edges with increasing values, orienting itself by using the gradient technique described above.
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