Bilinear Programming

Transcription

1 Bilinear Programming Artyom G. Nahapetyan Center for Applied Optimization Industrial and Systems Engineering Department University of Florida Gainesville, Florida address: 1 Introduction A function f(x, y) is called bilinear if it reduces to a linear one by fixing the vector x or y to a particular value. In general, a bilinear function can be represented as follows: f(x, y) = a T x + x T Qy + b T y, where a, x R n, b, y R m, and Q is a matrix of dimension n m. It is easy to see that bilinear functions compose a subclass of quadratic functions. We refer to optimization problems with bilinear objective and/or constraints as bilinear problems, and they can be viewed as a subclass of quadratic programming. Bilinear programming has various applications in constrained bimatrix games, Markovian assignment and complementarity problems. Many 0-1 integer programs can be formulated as bilinear problems. An extensive discussion of different applications can be found in [3]. Concave piecewise linear and fixed charge network flow problems, which are very common in the supply chain management, can be also solved using bilinear formulations (see, e.g., [5] and [6]). 2 Formulation Despite a variety of different bilinear problems, most of the practical problems involve a bilinear objective function and linear constraints, and theoretical results are derived for those cases. In our discussion we consider the following bilinear problem, which we refer to as BP. min x X,y Y f(x, y) = at x + x T Qy + b T y, 1

2 where X and Y are nonempty polyhedra. The BP formulation is also known as a bilinear problem with a disjoint feasible region because the feasibility of x (y) is independent form the choice of the vector y (x). 2.1 Equivalence to Other Problems Below we discuss some theoretical results, which reveal the equivalence between bilinear problems and some of concave minimization problems. Let V (x) and V (y) denote the set of vertices of X and Y, respectively, and g(x) = min y Y f(x, y) = a T x + min y Y {x T Qy + b T y}. Note that min y Y f(x, y) is a linear programm. Because the solution of a linear problem attains on a vertex of the feasible region, g(x) = min y Y f(x, y) = min y V (Y ) f(x, y). Using those notations, the BP problem can be restated as min f(x, y) = min {min x X,y Y x X y Y f(x, y)} = min x X { min y V (Y ) f(x, y)} = min g(x). (1) x X Observe that the set of vertices of Y is finite, and for each y Y, f(x, y) is a linear function of x; therefore, function g(x) is a piecewise linear concave function of x. From the later it follows that BP is equivalent to a piecewise linear concave minimization problem with linear constraints. It also can be shown that any concave minimization problem with a piecewise linear separable objective function can be reduced to a bilinear problem. To establish this relationship consider the following optimization problem: min x X φ i (x i ), (2) i where X is an arbitrary nonempty set of feasible vectors, and φ i (x i ) is a concave piecewise linear function of only one component x i, i.e., c 1 i x i + s 1 i (= φ 1 i (x i )) x i [λ 0 i, λ 1 i ) c φ i (x i ) = 2 i x i + s 2 i (= φ 2 i (x i )) x i [λ 1 i, λ 2 i ), c n i i x i + s n i i (= φ n i i (x i )) x i [λ n i 1 i with c 1 i > c 2 i > > c n i i. Let K i = {1, 2,..., n i }. Because of the concavity of φ i (x i ), the function can be written in the following alternative form, λ n i φ i (x i ) = min k K i {φ k i (x i )} = min k K i {c k i x i + s k i }. (3) Construct the following bilinear problem: min f(x, y) = φ k i (x i )yi k = (c k i x i + s k i )yi k (4) x X,y Y i k K i i k K i where Y = [0, 1] i K i. The proof of the following theorem follows directly from equation (3), and for details we refer to the paper [5]. 2 i ]

3 Theorem 1 If (x, y ) is a solution of the problem (4) then x is a solution of the problem (2). Observe that X is not required to be a polytop. If X is a polytop then the structure of the problem (4) is similar to BP. Furthermore, it can be shown that any quadratic concave minimization problem can be reduced to a bilinear problem. Specifically, consider the following optimization problem: min φ(x) = x X 2aT x + x T Qx, (5) where Q is a symmetric negative semi-definite matrix. Construct the following bilinear problem min f(x, y) = x X,y Y at x + a T y + x T Qy, (6) where Y = X. Theorem 2 (see [2]) If x is a solution of the problem (5) then (x, x ) is a solution of the problem (6). If (ˆx, ŷ) is a solution of the problem (6) then ˆx and ŷ solve the problem (5). 2.2 Properties of a Solution In the previous section we have shown that BP is equivalent to a piecewise linear concave minimization problem. On the other hand it is well known that a concave minimization problem over a polytop attains its solution on a vertex (see, for instance, [1]). The following theorem follows from this observation. Theorem 3 (see [2] and [1]) If X and Y are bounded then there is an optimal solution of BP, (x, y ), such that x V (X) and y V (Y ). Let (x, y ) denote a solution of BP. By fixing the vector x to the value of the vector x, the BP problem reduces to a linear one, and y should be a solution of the resulting problem. From the symmetry of the problem, a similar result holds by fixing the vector y to the value of the vector y. The following theorem is a necessary optimality condition, and it is a direct consequence of the above discussion. Theorem 4 (see [2] and [1]) If (x, y ) is a solution of the BP problem, then min x X f(x, y ) = f(x, y ) = min y Y f(x, y) (7) However, (7) is not a sufficient condition. In fact it can only guarantee a local optimality of (x, y ) under some additional requirements. In particular, y has to be the unique solution of min y Y f(x, y) problem. From the later it follows that f(x, y ) < f(x, y), y V (Y ), y y. Because of the continuity of the function f(x, y), for any y V (y), y y, f(x, y ) < f(x, y) in a small neighborhood U y of the point x. Let U = y V (Y ),y y U y. 3

4 Then f(x, y ) < f(x, y), x U, y V (Y ), y y. At last observe that Y is a polytop, and any point of the set can be expressed through a convex combination of its vertices. From the later it follows that f(x, y ) f(x, y), x U, y Y, which completes the proof of the following theorem. Theorem 5 If (x, y ) satisfies the condition (7) and y is the unique solution of the problem min y Y f(x, y) then (x, y ) is a local optimum of BP. Recall that BP is equivalent to a piecewise concave minimization problem. Under the assumptions of the theorem, it is easy to show that x is a local minimum of the function g(x) as well (see [2]). 3 Methods In this section we discuss methods to find a solution of a bilinear problem. Because BP is equivalent to a piecewise linear concave minimization problem, any solution algorithm for the later can be used to solve the former. In particular, one can employ a cutting plain algorithms developed for those problems. However, the symmetric structure of the BP problem allows constructing more efficient cuts. In the paper [4], the author discusses an algorithm, which converges to a solution that satisfies condition (7), and then proposes a cutting plain algorithm to find the global minimum of the problem. Assume that X and Y are bounded. Algorithm 1, which is also known as the mountain climbing procedure, starts from an initial feasible vector y 0 and iteratively solves two linear problems. The first LP is obtained by fixing the vector y to the value of the vector y m 1. The solution of the problem is used to fix the value of the vector x and construct the second LP. If f(x m, y m 1 ) f(x m, y m ), then we continue solving the linear problems by fixing the vector y to the value of y m. If the stopping criteria is satisfied, then it is easy to show that the vector (x m, y m ) satisfies the condition (7). In addition, observe that V (X) and V (Y ) are finite. From the later and the fact that f(x m, y m 1 ) f(x m, y m ) it follows that the algorithm converges in a finite number of iterations. Let (x, y ) denote the solution obtained by the Algorithm 1. Assuming that the vertex x is not degenerate, denote by D the set of directions d j along the ages emanating from the point x. Recall that g(x) = min y Y f(x, y) is a concave function. To construct a valid cut, for each direction d j find the maximum value of θ j such that g(x + θ j d j ) f(x, y ) ε, i.e., θ j = argmax{θ j g(x + θ j d j ) f(x, y ) ε}, Algorithm 1 : Mountain Climbing Procedure Step 1: Let y 0 Y denote an initial feasible solution, and m 1. Step 2: Let x m = argmin x X {f(x, y m 1 )}, and y m = argmin y Y {f(x m, y)}. Step 3: If f(x m, y m 1 ) = f(x m, y m ) then stop. Otherwise, m m + 1 and go to Step 2. 4

5 Algorithm 2 : Cutting Plane Algorithm Step 1: Apply Algorithm 1 to find a vector (x, y ) that satisfies the relationship (7). Step 2: Based on the solution (x, y ), compute the appropriate cuts and construct the sets X 1 and Y 1. Step 3: If X 1 = or Y 1 =, then stop; (x, y ) is a global ε-optimal solution. Otherwise, X X 1, Y Y 1, and go to Step 1. where ε is a small positive number. Let C = (d 1,..., d n ), { ( 1 1 x = x,..., 1 ) T C 1 (x x ) 1}, θ 1 θ n and X 1 = X 1 x. If X 1 = then min f(x, y) x X,y Y f(x, y ) ε, and (x, y ) is a global ε-optimum of the problem. If X 1 then one can replace X by the set X 1, i.e, consider the optimization problem min f(x, y), x X 1,y Y and run Algorithm 1 to find a better solution. However, because of the symmetric structure of the problem, a similar procedure can be applied to construct a cut for the set Y. Let 1 y denote the corresponding half-space, and Y 1 = Y 1 y. By updating both sets, i.e., considering the optimization problem min f(x, y), x X 1,y Y 1 the cutting plane algorithm (see Algorithm 2) might find a global solution of the problem using less number of iterations. References [1] Horst R, Pardalos P, Thoai N. Introduction to global optimization, 2-nd Edition, Boston: Springer, [2] Horst R, Tuy H. Global Optimization, 3-rd Edition, New York: Springer, [3] Konno H. A Bilinear Programming: Part II. Applications of Bilinear Programming. Technical Report 71-10, Operations Research House, Stanford University, Stanford, CA. 5

6 [4] Konno H. A Cutting Plane Algorithm for Solving Bilinear Programs. Mathematical Programming 1976;11: [5] Nahapetyan A, Pardalos P. A Bilinear Relaxation Based Algorithm for Concave Piecewise Linear Network Flow Problems. Journal of Industrial and Management Optimization 3(1), pp , [6] Nahapetyan A, Pardalos P. Adaptive Dynamic Cost Updating Procedure for Solving Fixed Charge Network Flow Problems. Computational Optimization and Applications, submitted for publication. 6