MIP heuristic solution: Local Search)

* This article mainly records and shares the learned knowledge, which is not original.

* For references, see the link.

This article describes the heuristic algorithms used to solve the MIP problem.

The objective of heuristic algorithms is to obtain optimal solutions in a short time.

I personally think that local search includes almost all of the core frameworks of the heuristic algorithm for solving MIP, from simple Hill-climbing) to complex tabu search, from a hill-climbing algorithm starting from an initial solution to a group of genetic algorithms (Genetic Algorithm ), its core framework is local search.

Therefore, to learn the heuristic algorithm for solving MIP, you must master the idea and algorithm process of local search.

Directory

Introduction

Local search process

Elements of Local Search

Introduction

One of the difficulties in solving a MIP problem is that it takes a lot of time to obtain the optimal solution when the solution space is large. However, what we can do is to obtain the optimal solution from a small solution space in a short time.

Just like a standard linear programming problem, we don't want to judge the size of all solutions in the solution space one by one to determine the optimal solution, but want to have a more efficient method. Because we know a standard linear programming problem, its solution space can be expressed as a polyhedron, and the optimal solution only occurs at the vertex of a polygon, therefore, we can only determine the solution size on the vertex to obtain the optimal solution.

How can we find the solution "most likely to be the optimal solution?

We know that a global optimal solution must be a local optimal solution. If a solution is not a local optimal solution, it cannot be a global optimal solution. Therefore, we think that the local optimal solution is the most likely to become the global optimal solution.

So can we find all the local optimal solutions?
The global optimal solution can be determined only by determining the size of all local optimal solutions. However, because the solution space is large, it takes time to find all the local optimal solutions. From the perspective of time, we can only obtain part of the local optimal solution and find a better solution to the problem.

So how can we find the local optimal solution to achieve the iteration of the current solution?

The idea and process of local search to find the local optimal solution is similar to the gradient descent method.

Local search process

(1) generate an initial solution: the algorithm starts from an initial solution or several initial solutions;

Starting from an initial solution, such as tabu search,

Starting from several initial solutions, such as genetic algorithm

(2) define the neighborhood and obtain the neighborhood of the solution: Define the neighborhood of the solution and generate several neighborhood solutions;

Different algorithms consider different neighbor structures. The definition of the neighborhood structure has a great impact on the algorithm performance.

(3) determining the New Territories: determine a new solution from the solution;

The simplest choice strategy is to use the optimal solution in the solution as the new solution. However, different algorithms may consider different selection strategies.

(4) iteration: Repeat the search process until the conditions for termination are met. During this period, parameters may be adjusted.

The termination conditions may be time and number of iterations.

Elements of Local Search

(1) Target function: used to determine the merits of the solution;

(2) method for generating the initial solution;

The quality of the initial solution may have a great impact on the results, which is related to specific algorithms.

(3) neighborhood definition;

(4) determining rules for new solutions;

(5) Conditions for termination.

* Local search includes a heuristic algorithm with local search as the core framework. Generally, two aspects of the algorithm must be considered during Algorithm Design:

(1) intensification: Deep Search. Find the local optimal solution as much as possible.

(2) diversification: extensive search. Search for more space as much as possible, and avoid falling into the local optimal solution.

 

References

Http://www.cnblogs.com/JiePro/p/Metaheuristics_0.html

Https://mp.weixin.qq.com? _ Biz = mzu0nzgymjgwng ==& mid = 2247484621 & idx = 1 & Sn = f2e92f44c2306b58034cf158647bc737 & scene = 19 # wechat_redirect

 

MIP heuristic solution: Local Search)