Let any BLP model be represented by
Any minimization BLP can be converted into maximization form and vice versa. There are several strategies for solving mixed 0 - 1 integer problems that are presented in Adams and Sherali [
The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex quadratic programming problem is then solved by interior point algorithms. This settles one of the open problems of whether P = NP or not. The worst case complexity of interior point algorithms for the convex quadratic problem is polynomial. It can also be shown that every liner integer problem can be converted into binary linear problem.
The binary linear programming (BLP) model is NP-complete and up to now we have not been aware of any polynomial algorithm for this model. See for example Fortnow [
Let any BLP model be represented by
Any minimization BLP can be converted into maximization form and vice versa. There are several strategies for solving mixed 0 - 1 integer problems that are presented in Adams and Sherali [
Let a quadratic programming problem be represented by (2).
We assume that:
1) matrix Q is symmetric and positive definite,
2) function
3) since constraints are linear then the solution space is convex,
4) any maximization quadratic problem can be changed into a minimization and vice versa.
When the function
Our problem is to transform problem (1) into (2) and once that is done then (2) can be solved in polynomial time implying P = NP. Interior point algorithms can solve the convex/concave QP problem in polynomial time.
Binary variables have certain special features that we can capitalize on when solving.
Given any binary variable
Proof
Case 1: When
Case 2: When
For any binary variable
The proof is the same as the one given in 4.1.1. Note that it is only binary variables that can satisfy (4). Even though none binary values such as
The main weakness of the objective function given in (1) is that it does not force variables to assume binary values. In this paper we alleviate this challenge by adding a nonlinear extension to the objective function as given in (5).
where
Proof
Since from Rule 2,
In other words
A function
Proof
In this case
This has continuous second order partial derivatives and the 2n by 2n Hessian matrix is given by
Since all principal minors of
The function
where matrix
Thus matrix
The main reason for converting a BLP into a convex quadratic programming model is to take advantage of the availability of interior point algorithms which can solve convex QPs in polynomial time [
The proof is easily shown by reducing the convex quadratic objective function to the original linear form given in (1). The proposed objective function of the convex QP is reduced as follows:
Since
In other words
where
In this case the solution of the convex OP will not be integer. The objective,
forces variables to binary or integral values if an integer point exists in the solution space. If an integer point does not exists in the solution space the large constant
In some BLP problems that occur in real life, a fraction of some of the variables may not be restricted to integer values. In this case the enforcer
Any maximization BLP problem can be converted into a minimization BLP and vice versa. This can be done by the substitution given in (18).
where
Suppose the primal-dual pair of the convex QP is given by (19) and (20).
Primal:
Dual:
where
The first order optimality conditions for (19) and (20) are given by (21)
where e is a vector of ones. The primal-dual central path method can be used to solve the convex QP. Detailed information on this interior point algorithm and other variants can be obtained in Gondzio [
From the two versions of the same problem
The following numerical illustration shows how a BLP problem is transformed into convex quadratic programming model and then solved.
where
Transforming into a convex quadratic programming problem becomes (24)
where
The solution to the convex quadratic problem is given in (25).
In the case of a mixed binary linear programming problem, only the binary integer variables occupy the enforcer. In other words, if only the r binary variables
Suppose in 5.1, the variables
The transformation becomes as shown in (27).
The solution to the convex quadratic problem is:
The problems that occur in real life do not have binary variables only. These practical problems occur as general mixed integer problem (MIP) where variables assume integer values greater than 1. There are methods that can be used to solve these problems but we are not aware of any method that can solve these mixed integer problems in polynomial time up now. The obvious strategy is to expand the general mixed integer variable into binary ones.
Any MIP variable
where
Convert the following MIP into a BLP.
where
The following substitutions change the problem into a BLP.
where
The general BLP problem has been given so much attention by researchers all over the world for over half a century without a breakthrough. A difficult category of BLP models includes the traveling salesman, generalized assignment, quadratic assignment and set covering problems. The paper presented a technique to solve BLP problems by first transforming them into convex QPs and then applying interior point algorithms to solve them in polynomial time. We also showed that the proposed technique worked for both pure and mixed BLPs and also for the general linear integer model where variables were expanded into BLPs. We hope the proposed approach will give more clues to researchers in the hunt for efficient solutions to the general difficult integer programming problem.
The author is thankful to the referees for their helpful and constructive comments.
EliasMunapo, (2016) Solving the Binary Linear Programming Model in Polynomial Time. American Journal of Operations Research,06,1-7. doi: 10.4236/ajor.2016.61001