Lineárisrelaxáció

Lineárisrelaxáció, often abbreviated as LS, is a computational method used to solve certain types of optimization problems, particularly those that can be formulated as linear programs. It is a relaxation technique, meaning it approximates a more complex problem with a simpler, related problem that is easier to solve. The solutions obtained from the linear relaxation can provide bounds or insights into the original problem.

In the context of integer programming, where variables are restricted to integer values, linear relaxation involves dropping the integer constraints and treating the variables as continuous. This transforms the integer program into a linear program, which can then be solved efficiently using standard algorithms like the simplex method or interior-point methods. The optimal value of the linear relaxation provides a lower bound for the optimal value of the minimization problem or an upper bound for the optimal value of the maximization problem. This bound is crucial in branch and bound algorithms, a common technique for solving integer programs.

The effectiveness of linear relaxation depends on the specific problem structure. For some problems, the optimal solution to the linear relaxation may coincidentally be integer, directly yielding the optimal solution to the original integer program. However, this is not generally the case. The gap between the optimal value of the relaxed problem and the optimal value of the original problem can indicate the difficulty of finding an integer solution.

Linear relaxation is also employed in other areas of optimization, such as combinatorial optimization and convex optimization. It is a fundamental tool for understanding the structure of optimization problems and developing efficient solution algorithms. The concept of duality in linear programming is closely related to linear relaxation and provides further theoretical underpinnings for its use.