This document summarizes a lecture on combinatorial optimization and linear programming. It defines key concepts like local and global optima, convex sets and functions. It presents an example linear programming problem to maximize consumption of grains subject to nutritional constraints. It also describes how to convert a linear programming problem from general to canonical and standard form, including introducing slack and surplus variables and changing inequality constraints to equality constraints. An example problem is converted step-by-step from its original form to the standard form.

1. Combinatorial Optimization
CS-724
Lec-5:
Date: 01-03-2021
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

3. Starch Proteins Vitamins Cost/Kg
G1 5 4 2 6
G2 7 2 1 35
x1: number of units of grain G1 to be consumed per day, •
x2: number of units of grain G2 to be consumed per day.
The requirement per day of starch, proteins and vitamins
is 8, 15 and 3 respectively.

4.  General LP
 Canonical LP
 Standard LP

8. The following one is an equality constraint in general
form
σ𝒋=𝟏
𝒏
𝒂i,j xj = bj
The equivalent canonical form is
σ𝒋=𝟏
𝒏
𝒂i,j xj ≥ bj
and
σ𝒋=𝟏
𝒏
(−𝒂i,j ) xj ≥ (− bj )

9.  Given an unconstrained (free) variable xj in the general form
xj ≶ 𝟎
In the corresponding canonical form we create two variables xj
+ and xj
–
and we write
xj = xj
+ − xj
−
where xj
+ ≥ 0 and xj
−
≥ 0

10.  The following is in Canonical LP
σ𝒋=𝟏
𝒏
𝒂i,j xj ≥ bj
In the standard form we introduce a variable si (called surplus variable) and write
σ𝒋=𝟏
𝒏
𝒂i,j xj − si = bj , si ≥ 0
 If the inequality is in the following form
σ𝒋=𝟏
𝒏
𝒂i,j xj ≤ bj
Then the corresponding standard form is (here si is called slack variable)
σ𝒋=𝟏
𝒏
𝒂i,j xj + si = bj , si ≥ 0

11.  Example:
𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 3𝑥1 + 5𝑥2 − 𝑥3
subject to 𝑥1+2𝑥2 + 4𝑥3 ≤ −4 … … … … … … … . 1
−5𝑥1 − 3𝑥2 + 𝑥3 ≥ 15 … … … … … … … . (2)
𝑥1 ≶ 0
𝑥2 ≤ 0
𝑥3 ≥ 0

12.  Convert the maximization problem to minimization one by multiplying the objective
function by −1 .
 Next, we introduce a slack variable 𝑥4 in the first constraint to convert constraint 1
into
𝑥1 + 2𝑥2 + 4𝑥3 + 𝑥4 = −4, 𝑥4 ≥ 0
And then multiply it by −1 to make the RHS positive.
−𝑥1 − 2𝑥2 − 4𝑥3 − 𝑥4 = 4, 𝑥4 ≥ 0
For the constrained 2 a surplus variable 𝑥5 is introduced
−5𝑥1 − 3𝑥2 + 𝑥3 − 𝑥5 = 15, 𝑥5 ≥ 0

13.  Next substitute 𝑥1 = 𝑥6 − 𝑥7 in every expression
 Finally substitute 𝑥2 with −𝑥2
′
 The standard LP form is as follows.
Minimize 5𝑥2
′
+ 𝑥3 − 3𝑥6 + 3𝑥7
Subject to 2𝑥2
′
− 4𝑥3 − 𝑥4 + 0𝑥5 − 𝑥6 + 𝑥7 = 4
3𝑥2
′
+ 𝑥3 + 0𝑥4 − 𝑥5 − 5𝑥6 + 5𝑥7 = 15
𝑥2
′
, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑥7 ≥ 0