Operations research: linear programming - fundamentals and matrix formulation · Introduction

Introduction

What this guide covers

An exhaustive tour of the topology of linear convex optimization: from algebraic synthesis and polyhedral geometry to the algebraic analysis of the Simplex Primal-Dual method.

Overview

Topics covered

Introduction
Fundamentals, Mathematical Structure and Matrix Formulation
Introduction and History of Linear Programming
Historical Evolution and Relevance
Structural Assumptions of the Linear Programming Model
Mathematical Formulation of a Linear Program

Related concepts

Main connections

Compatible notation

Mathematical sample in LaTeX

The expression is rendered accessibly, with its LaTeX source preserved below.

\begin{aligned} \text{Optimize} \quad & z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n \\ \text{sujeto a:} \quad & a_{i1} x_1 + a_{i2} x_2 + \cdots + a_{in} x_n \leq b_i, \quad i \in I_1 \\ & a_{j1} x_1 + a_{j2} x_2 + \cdots + a_{jn} x_n \geq b_j, \quad j \in I_2 \\ & a_{k1} x_1 + a_{k2} x_2 + \cdots + a_{kn} x_n = b_k, \quad k \in I_3 \\ & x_r \geq 0 \quad (r \in J_1), \quad x_s \leq 0 \quad (s \in J_2), \quad x_t \text{ libre} \quad (t \in J_3) \end{aligned}

View LaTeX source

\begin{aligned}
      \text{Optimize} \quad & z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n \\
      \text{sujeto a:} \quad & a_{i1} x_1 + a_{i2} x_2 + \cdots + a_{in} x_n \leq b_i, \quad i \in I_1 \\
                             & a_{j1} x_1 + a_{j2} x_2 + \cdots + a_{jn} x_n \geq b_j, \quad j \in I_2 \\
                             & a_{k1} x_1 + a_{k2} x_2 + \cdots + a_{kn} x_n = b_k, \quad k \in I_3 \\
                             & x_r \geq 0 \quad (r \in J_1), \quad x_s \leq 0 \quad (s \in J_2), \quad x_t \text{ libre} \quad (t \in J_3)
    \end{aligned}