A Continuous Approach to Binary Quadratic Problems

This paper presents a continuous method for solving binary quadratic programming problems. First, the original problem is converted into an equivalent continuous optimization problem by using NCP (Nonlinear Complementarity Problem) function, which can be further carry on the smoothing processing by aggregate function. Therefore, the original combinatorial optimization problem could be transformed into a general differential nonlinear programming problem, which can be solved by mature optimization technique. Through some numerical experiments, the applicability, robustness, and solution quality of the approach are proved, which could be applied to large scale problems.


Introduction
Binary quadratic programming (BQP) problem is a kind of typical combinatorial optimization problem, and has a variety of applications in computer aided design, traffic management, cellular mobile communication frequency allocation, operations research and engineering.And many combinatorial optimization problems with constraints can be transformed into BQP problems form by certain transformation, so these problems can be on behalf of kinds of important problems in combinatorial optimization.Hammer [1] pointed out that any integer programming problems where the objective function is quadratic or linear and the constraint condition is linear can be described as BQP problems.Glover et al. [2] have successfully used this transformation method to transform the quadratic knapsack problem into BQP problem for solving.Considering the fea-Z.Liu where Q is a real symmetric n n × matrix, and the objective function can be followed by another linear term, but it can be omitted because there is no substantial influence on the method introduced later.Of course, in the following discussion, we will mainly focus on the so-called 0 -1 quadratic programming: However, the continuous method proposed later in this paper is also applicable to the general binary quadratic programming (1), which can be easily accomplished by simple variable transformation.It is well known that 0 -1 programming problem will increase exponentially with the increase of the scale of the problem, and within limited computer resources and time the traditional solution method will be difficult to realize.Because of the wide application prospect and difficult characteristic of 0 -1 programming, how to solve this kind of combinatorial optimization problem effectively has been the focus of many scholars.The classical methods for solving 0 -

Continuous Formulation of BQP
The simplest and most intuitive way to solve 0 -1 planning problem is to adopt the round integration technology, that is, to consider 0 -1 variable as a continuous variable, and to replace the variable constraint in the original problem with the following interval constraint: Then a relaxation optimization problem is obtained, and the components of the relaxation solution are rounded to the nearest discrete value.Although this method is easy to implement, but the theory of this technology is not strict.It cannot guarantee that the optimal solution obtained is the global optimal solution, or even a feasible solution.
The other method is the penalty function method, which is easy to see that is always equivalent to a complementarity condition: ( ) Therefore, the punishment function can be constructed as follows: where 0 α > is a large enough penalty parameter.Adding this penalty function to the object function can obtain an equivalent continuous optimization problem: ( ) ( ) Due to the strong concavity of ( ) T 1 x x α − , the object function in (6) is concave.The equivalence of (2) and ( 6) is based on the fact that the concavity function obtains the minimum value at a vertex, as well as ( ) T 1 0 x x − = has included 0 x = and 1 x = the sum is included.Although the vertex of the feasible domain is not necessarily the vertex of the unit hypercube, if α is sufficiently large, the global minimum can only be obtained while ( ) However, the specific value of α cannot be determined.If the value is too small, the penalty term will not play its role.If the value is too large, due to the strong concavity of the function, this penalty function method cannot effectively obtain the optimal solution of the original problem, so it is not an effective method.It is pointed out in literature [14], but only a new viewpoint is provided, and no more effective method is given to solve the original problem.Therefore, it is not ideal to solve the original problem directly with (6).Great-Ming Ng later constructed a logarithmic barrier term: Z.This barrier term can be used to replace the boxed constraint conditions as well as to avoid falling into the local minimum during the iteration, so as to find the global optimal solution for the original problem.
At this point, the following unconstrained optimization problems can be obtained: where 0 α > is a penalty parameter and 0 µ > is a barrier parameter.In a sense, ( 8) is an improvement on ( 6), but in essence they both replace constraints with penalty terms or logarithmic barrier terms.
For the BQP problem, another continuous method is proposed in this paper.
Consider the following two sets of constraints: , .
Obviously, these two sets of constraint conditions and binary constraint conditions are equivalent.In fact, these two constraints skillfully reform complementary constraints: ( ) The advantage of constructing constraint conditions in this way is that: the two constraint conditions of (9) work at the same time, and to each variable there is a unique complementary constraint condition ( ) And more importantly, complementary constraints (10) and the general constraint conditions are different: for general complementary constraints, each , , , n x x x x =  , and for (10), the value of every ( ) ( ) only has relationship with the values of i x , nothing to do with the other variables.Namely, each of the complementary constraint of ( 10) is independent of each other.
Through the above, the problem (2) can be equivalently transformed into the following mathematical programming problems with complementary constraints: ( ) ( ) For the above complementary constraints, we can use NCP function [15] to replace them further.To better describe NCP functions, we first introduce the ( ) ( ) Thus, the global optimal solution of ( 12) is the exact solution of problem (2).

Smoothing Method for BQP
According to the definition of NCP function given above, functions satisfying conditions can take various forms.Several commonly used NCP functions can be given: In the following section, we mainly choose the minimum value function to study.
It is easy to know − − , and for the non-differentiable maximum function: we often use aggregate function to carry out smooth approximation to it [16] [17].The so-called aggregate function is a function in the following form: ( ) ( ) where the smooth parameter µ is large enough, and ( ),  [18], which can be found from the following properties: Lemma 3.1 The function which is defined by (15) satisfies the following conditions: , and there is at least one indicator that makes ( ) ( ) 2) When µ → ∞ , the above conclusion can be proved by formula (16).
Z. Liu et al.
Based on the above introduction, we can handle the minimum function as follows [19]: where the smooth parameter µ is large enough, as shown by lemma 3.1, when µ → ∞ , For ease of expression, let's call: At this point, the problem ( 12) can be further transformed into an equivalent continuous and smooth nonlinear constraint optimization problem: By this continuous method, the global optimal solution of the original problem (2) can be solved by solving the corresponding nonlinear constrained optimization problem (19).Many mature optimization algorithms have been developed for solving nonlinear equality constraint optimization problems.This paper mainly uses the multiplier penalty function method to solve the above problems (19).Multiplier method is an optimization algorithm independently For convenience's sake, let's first: The Lagrange function of problem ( 19) is [20]: where is the Lagrangian multiplier vector, and ( ) x λ is set as KT pairs of problem (19), then it can be known from the optimality condition: In addition, it is not difficult to find any x in the feasible domain is satisfied: The above equation shows that if the multiplier vector * λ is known, the Then the external penalty function method is considered to solve the problem (24).The augmented objective function is In this way, we can fix λ λ = and find a minimum of ( ) , , L x µ λ α , then change the value of λ appropriately to find a new value x , until we get the * x and * λ that we want.Specifically, when solving the minimum ( ) λ α in the k-th iteration of the unconstrained subproblem, the necessary conditions for taking the extreme values are known: And the KT-point of ( ) x λ the original problem satisfies: In order for after comparing the above two expressions, the updating formula of the multiplier sequence , It can be seen from Equation (28) that the sufficient and necessary condition for And then we proof that x λ being a KT-point to the (19) has a necessary and sufficient condition of Proof The necessity is obvious.The following proof is sufficient, since ( ) x is a minimum of (29) and therefore, for any feasible point: is also a minimum of (19).On the other hand, it is noted that ( ) k x is also the stable point of (29), therefore The above formula indicates that ( ) k λ is the Lagrangian multiplier vector with respect to ( ) k x , that is to say x λ is also the KT-point of (19).
Based on the above discussion, before giving the detailed steps to solve the Proof According to the properties of aggregate function: Proof Easy to obtain by lemma 3.2 that: From the above equation we know: ( Proof According to the definition of ( ) − , it can be seen that: ( ,1 ln exp 1 ,1 ln exp 1 ( ) e e e , e e e .

Algorithm
According to lemma 3.5, we can use the augmented Lagrange penalty function to solve the problem (19).For a strictly monotonic increasing sequence { } k µ and { } k α , the solution of unconstrained optimization Based on the previous analysis, the basic algorithm for solving the problem ( 19) is as follows [19]: Algorithm 1 Step 1 Given parameters ( ) Step 3 If , , Step 6; else go to Step 4; Step 4 If Step 5; else update the parameter , and go to Step 1; Step 5 Set ( ) , modify ( ) Step 6 Finish.
The optimal solution sequence  1 that in the process of solving these problems, the optimal solution obtained by our algorithm is basically close to the best known solution.Through the comparison in the table above, it can be seen that the proposed continuous algorithm is basically close to the known best solution for solving the general unconstrained 0 -1 quadratic programming problem.

Conclusion
In this paper, we have reformulated an unconstrained BQP problem as a MPEC problem by the equivalent complementarity conditions of a binary vector.To seek a global minimizer of the resulting continuous optimization problem, we construct a global smoothing function and develop a global continuation algorithm via a sequence of unconstrained minimization.And the numerical results indicate that the continuous approach proposed is extremely promising, especially for those large problems, in terms of the quality of the optimal values generated and the computational work involved.In addition, this new approach can be extended to general nonlinear binary optimization problems, and we can Z.Liu et al.

=
concept of them: Definition (NCP function): If a function is set , then the function of φ is called an NCP function.Therefore, we can replace the complementary constraint in (11) with a series Z. Liu et al. of equation equations.Finally, we get the continuous optimization model as follows: proposed by Powell and Hestenes in 1969 to solve equality constraint optimization problems.Later, it was extended to solve inequality constraint optimization problems by Rockfellar in 1973.The basic idea is to start from the Lagrangian function of the original problem and add an appropriate penalty function, so as to transform the original problem into a series of unconstrained optimization subproblems.
point of the unconstrained optimization problem: are large enough, as long as the region where x satisfies the following equation: defined by Equation (25) is convex in this region.
the corresponding Lagrange multiplier is { } k λ , and is modified according to (28) in the iteration.

2 1σ
> .Select a starting point ( ) ( ) et al.Due to the wide application background of this problem and the difficulties caused by NP (Non-Deterministic Polynomial) hard properties, it is of great academic value to study effective algorithms for solving such problems.
DOI: 10.4236/jamp.2018.681471721 Journal of Applied Mathematics and Physics sibility of this transformation, BQP problem has more extensive practical application.

Table 1 .
Comparison of numerical results.
Φconverge to 0 at a speed of at least 0.1.If this requirement is not met in an iteration, the penalty factor and smooth parameter should be increased automatically.Z.Liu et al.It's not hard to see from the Table