This document summarizes a lecture on combinatorial optimization and the simplex algorithm. It defines local and global optima, convex sets, convex functions, and convex programming problems. It then explains that the simplex algorithm, developed by George Dantzig in 1947, is an iterative method for solving linear programs by generating a sequence of basic feasible solutions until an optimal solution is found. Finally, it provides the standard form of a linear programming problem and defines the basis and properties of the submatrix in the simplex method.

1. Combinatorial Optimization
CS-724
Lec-5:
Date: 02-03-2021
Dr. Parikshit Saikia
Assistant Professor
Department of Computer Science and Engineering
NIT Hamirpur (HP)
India

2.  Local and global optima
 Convex Set
 Convex Functions
 Convex Programming Problem
 Forms of Linear Programming Problem

3. Simplex Algorithm
 [1947]. George B. Dantzig developed a technique to solve linear
programs----known as Simplex Algorithm
 The simplex method is an iterative method that generates a sequence
of basic feasible solutions (corresponding to different bases) and
eventually stops when it has found an optimal basic feasible solution.

4.  The given LP is in the Standard form.
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 (𝑜𝑟 𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒) 𝑐′𝑥
𝐴𝑥 = 𝑏
𝑥 ≥ 0
 𝑏 ≥ 0
 There exists a collection 𝑩 of m variables called a basis such that
 the submatrix 𝑨𝑩 of 𝑨 consisting of the columns of 𝑨 corresponding to the variables in
𝑩 is the 𝑚 × 𝑚 identity matrix and
 the cost coefficients corresponding to the variables in 𝑩 are all equal to 𝟎.