An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.
Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
If some decision variables are not discrete, the problem is known as a mixed-integer programming problem.
In integer linear programming, the canonical form is distinct from the standard form. An integer linear program in canonical form is expressed thus (note that it is the vector which is to be decided):
and an ILP in standard form is expressed as
where are vectors and is a matrix. As with linear programs, ILPs not in standard form can be converted to standard form by eliminating inequalities, introducing slack variables () and replacing variables that are not sign-constrained with the difference of two sign-constrained variables.
The plot on the right shows the following problem.
The feasible integer points are shown in red, and the red dashed lines indicate their convex hull, which is the smallest convex polyhedron that contains all of these points. The blue lines together with the coordinate axes define the polyhedron of the LP relaxation, which is given by the inequalities without the integrality constraint. The goal of the optimization is to move the black dashed line as far upward while still touching the polyhedron. The optimal solutions of the integer problem are the points and which both have an objective value of 2. The unique optimum of the relaxation is with objective value of 2.8. If the solution of the relaxation is rounded to the nearest integers, it is not feasible for the ILP.
The following is a reduction from minimum vertex cover to integer programming that will serve as the proof of NP-hardness.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This advanced undergraduate course treats basic principles on linear programming like the simplex algorithm, its complexity, and duality. Furthermore it gives an introduction on discrete optimization
Explores linear programming basics, including basic solutions, feasible solutions, optimal solutions, and challenges in solving integer programming problems.
This course is an introduction to linear and discrete optimization.Warning: This is a mathematics course! While much of the course will be algorithmic in nature, you will still need to be able to p
This course introduces the theory and applications of optimization. We develop tools and concepts of optimization and decision analysis that enable managers in manufacturing, service operations, marke
A first graduate course in algorithms, this course assumes minimal background, but moves rapidly. The objective is to learn the main techniques of algorithm analysis and design, while building a reper
In computational complexity theory, a problem is NP-complete when: It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) solution. The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions.
In mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory.
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems.
Future low-carbon societies will need to store vast amounts of electricity to stabilize electricity grids and to power electric vehicles. Vehicle-to-grid allows vehicle owners and grid operators to sh
Adjustable robust minimization problems where the objective or constraints depend in a convex way on the adjustable variables are generally difficult to solve. In this paper, we reformulate the origin
INFORMS2022
, ,
We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an