This paper presents two procedures for solving mixed-variables programming problems, which involve maximizing a linear function subject to linear and nonlinear constraints. The problems are of the form: \[ \max \{c^T x + f(y) \mid Ax + F(y) \leq b, \quad x \in R_p, y \in S\}, \] where \(x\) and \(y\) are vectors in \(R_p\) and \(R_q\) respectively, and \(S\) is a subset of \(R_q\). The paper discusses various applications, such as mixed-integer programming and problems with both linear and nonlinear variables. The core idea is to partition the variables into two disjoint subsets, leading to the solution of a linear programming problem and a sub-problem defined on \(S\). Two multi-step procedures are introduced to efficiently determine the optimal solution without calculating a complete set of constraints. These procedures are based on earlier work by the author and are detailed in this paper. The paper assumes familiarity with convex polyhedral sets and the simplex method for linear programming.This paper presents two procedures for solving mixed-variables programming problems, which involve maximizing a linear function subject to linear and nonlinear constraints. The problems are of the form: \[ \max \{c^T x + f(y) \mid Ax + F(y) \leq b, \quad x \in R_p, y \in S\}, \] where \(x\) and \(y\) are vectors in \(R_p\) and \(R_q\) respectively, and \(S\) is a subset of \(R_q\). The paper discusses various applications, such as mixed-integer programming and problems with both linear and nonlinear variables. The core idea is to partition the variables into two disjoint subsets, leading to the solution of a linear programming problem and a sub-problem defined on \(S\). Two multi-step procedures are introduced to efficiently determine the optimal solution without calculating a complete set of constraints. These procedures are based on earlier work by the author and are detailed in this paper. The paper assumes familiarity with convex polyhedral sets and the simplex method for linear programming.