Solving Integer Programs - Discrete Modeling and Optimization - Lecture Slides, Slides of Discrete Structures and Graph Theory

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Solving Integer Programs

Solution idea #1:

Explicit enumeration : Try all possible solutions and pick the best one.

• Example: max 3x 1 + 2x2 + 4x s. t. x1 + x2 + x3 ≥ 1 x1 + 2x2 + x3 ≤ 2 x 1 + x2 - 2x 3 ≥ 0 x 1 , x2 , x 3 binary
• Checking for all possible solutions: x 1 1 1 1 1 0 0 0 0 x2 1 1 0 0 1 1 0 0 x3 1 0 1 0 1 0 1 0 feasible no no no yes no yes no no value 3 2
• Select the best solution among the feasible ones:

x 1 =1 x2 = 0 x3 = 0 with value 3

Solution idea #2: Solve the corresponding LP and round the solution.

• For every IP (or MIP) there is a corresponding linear program which is obtained by dropping the integrality constraints on the variables. That linear program is called LP-relaxation of the original IP (MIP).
• E.g., for the following IP:

max x 2 (IP) s.t. -2x 1 +2x2 ≤ 1 2x1 +2x2 ≤ 7 x1 , x2 ≥ 0 integer the LP-relaxation is: max x2 (LP) s.t. -2x 1 +2x2 ≤ 1 2x1 +2x2 ≤ 7 x1 , x2 ≥ 0 Docsity.com

• There are efficient algorithms (e.g., Simplex method) to solve LPs.

So a natural algorithm for solving an IP is (i) solve the LP-relaxation; (ii) round its solution to get an integral solution for the original IP.

• Let’s see how it works on our example. max x (^2) s.t. -2x 1 +2x2 ≤ 1 2x1 +2x2 ≤ 7 x1 , x2 ≥ 0
• Optimal solution is (1.5, 2)
• Rounded solutions are (1, 2) and (2, 2) ; but both are infeasible

2x 1 +2x 2 = 7 -2x 1 +2x 2 = 1

Z=

 Summarizing,

Solution idea #2 has the following two pitfalls:

1. The rounded solution might be infeasible.

2. The rounded solution might be too far from optimal solution.

 Solution idea #2 works well for a class of problems

which have the following property: All CPF solutions of LP-relaxation are integral.

For this class of problems,

the optimal solution of LP-relaxation is also optimal solution for the original IP.

 Generally, we need more sophisticated algorithms to solve IPs

(still using the idea of LP-relaxation).

LP-relaxation-based solution methods for Integer Programs

 Branch-and-Bound Technique

(next handout)

 Cutting Plane Algorithms