egészértékoptimálásban

Integer optimization, or egészértékoptimálás in Hungarian, is a subfield of mathematical optimization where the objective is to find the optimal solution to a problem with the constraint that one or more variables must be integers. This is in contrast to continuous optimization, where variables can take any real value. Integer optimization problems are generally more difficult to solve than their continuous counterparts because the feasible region, which consists of discrete points, is not convex.

The general form of an integer optimization problem is to minimize or maximize an objective function f(x) subject to constraints g_i(x) <= b_i for all i, where x is a vector of variables, and at least one variable in x must be an integer. If all variables must be integers, it is called a pure integer programming problem. If some variables are integers and others are continuous, it is called a mixed-integer programming problem. If the variables can only take values of 0 or 1, it is a binary integer programming problem, often used for decision-making tasks.

Various algorithms exist for solving integer optimization problems, including branch and bound, cutting plane methods, and heuristics. These methods often involve relaxing the integer constraints to obtain a continuous problem, solving it, and then using the solution to guide the search for an integer solution. Despite the computational challenges, integer optimization is a powerful tool with wide-ranging applications in fields such as operations research, logistics, finance, scheduling, and artificial intelligence.