Algorithms

Numerous methods have been developed for solving the binary quadratic problem or the associated combinatorial problems. The exact methods are categorized into five groups: branch-and-bound methods, cutting plane methods, branch-and-cut meth- ods, algebraic methods and continuous methods.

Branch and bound

The branch-and-bound method is based on the idea of implicitly and intelligently enumerating all the feasible solutions. Given n being the dimension of the binary quadratic problem, there are 2n _{feasible solutions. It is hopeless to examine each}

solution to find the global solution even when n is moderate. The branch-and- bound method breaks the problem into a series of smaller problems that can be easily tackled, and then puts the information together again to obtain an optimal solution for the original problem. The construction of smaller problems is based on

successive partitioning of the solution set. The branch in branch-and-bound refers to this partitioning process; the bound refers to lower and upper bounds that are used to provide a proof of optimality.

There are many strategies in how a branch-and-bound algorithm is implemented. At each branching step of determining which node to branch from, the usual al- ternatives are least-lower-bound-next, depth-first and breadth-first. At each node, the upper bound for the corresponding subproblem can be provided by any feasible solutions, and the lower bound is normally obtained by relaxation, duality or some other methods, which will be surveyed below.

A variety of branch-and-bound methods have been introduced in literatures for the binary quadratic problem. They are equipped with di↵erent strategies of branch- ing and lower bounding. The bounding techniques used in branch-and-bound meth- ods are convex quadratic program relaxations [22, 16], linearization techniques [18, 63, 3], linear programming relaxations [9, 10], semidefinite programming relaxations [68, 67, 31, 60, 101], Second order cone programming relaxations [76, 91], roof dual- ity [66, 18, 19], DC programming [125], one row relaxations [126, 127] and geometric property [84].

Cutting plane

For mixed-integer linear programming, the cutting plane methods work by first solv- ing the linear relaxation, which is obtained by replacing the integer constraints with box constraints. The theory of the linear programming guarantees that under mild assumptions, one can always find an extreme point or a corner point that is an op- timal solution. Then the obtained optimal solution is examined: if it satisfies the integer constraints, an optimal solution for the original problem is already found; if it not, a linear inequality can be constructed that separates the optimal solution from the convex hull of the feasible region of the original problem. Such an inequality is called a cut, which can then be added to the relaxation problem to cut o↵ the non- integer optimal solution and tighten the feasible region. This process is repeated until an optimal integer solution is found. The two most prominent cuts for the mixed-integer linear programming are Gomory cut [6] and lift-and-project cut [5].

In [9, 132], cutting plane method are used to solve the binary quadratic problem, which is firstly transformed into a mixed-integer linear programming problem by linearization techniques (see [63] and references there). The most used linearization technique is to replace xixj with a new variable zij. Here, the diagonal entries of

matrix Q are supposed to be zeros. Then the following problem is obtained:

min x,z X i<j qijzij n X i=1 fixi (5.3) s.t. zij  xi, zij  xj, xi+ xj 1 zij zij 0, 1 i < j  n x_{2 {0, 1}}n

In fact, zij can also be stated as 0-1 integer variables, since under other constraints

they can only be zero or one. Thus, the problem (5.3) becomes a constrained 0-1 linear programming problem.

The convex hull of the feasible region of the problem (5.3) is called the Boolean quadric polytope [92], and three families of facets for this polytope is identified: the clique-inequality, the cut-inequality and the generalized cut inequality, which can be used to construct sufficient cuts.

Branch and cut

A combination of cutting planes and branch-and-bound search generates the so called branch-and-cut methods. At each node on the search tree, cutting planes are added to tighten the linear relaxations, and thus the lower bound is improved. Work in [93, 6, 126, 127] gives more details on the implementation of branch-and-cut methods.

Algebraic methods

In [30, 19], the binary quadratic problem is transformed into pseudo-Boolean opti- mization problems, where pseudo-Boolean functions are minimized over binary con- straints x 2 {0, 1}n_{. A pseudo-boolean function is a real-valued function of 0-1}

variables. Any pseudo-Boolean function can be written uniquely as a multi-linear polynomial: g(x) = a +X i aixi+ X i<j aijxixj+ X i<j<k aijkxixjxk+· · · .

Because of the property x2

i = xi, the function ⇧(x) can be written as a pseudo-

Boolean function. The basic algorithm for pseudo-Boolean optimization determines the minimum of the pseudo-Boolean function by recursively eliminating variables, following the dynamic programming principle. However, computationally, the pro- cedure could be very expensive. Thus, a branch-and-bound scheme is proposed to the variable elimination [30].

Continuous methods

In [94], a continuous approach is described for solving the problem (_Pbqp). Rather

than using relaxations and bounding information in a tree search scheme, the authors employ Fischer-Burmeister nonlinear complementarity function to reformulate the problem as a continuous problem with equilibrium constraints. The binary constraint xi 2 { 1, 1} is always equivalent to conditions:

1_{ x}i  1, (1 + xi)(1 xi) = 0,

where, by employing the Fischer-Burmeister function, the latter complementarity condition is equivalent to the following equality

F B(1 + xi, 1 xi) =

q

2 + 2x2

Then, the quadratic penalty function and logarithmic barrier function are used to remove the equality and inequality constraints, and a global smoothing function is constructed, which promises convexity in a large subset of its domain.