One-Center Location With Block and Euclidean Distance

A geometrical analysis is made of the dual simplex algorithm applied to a linear programming formulation of the one-center location problem in IR2 using block distance. A geometric rule is given, and shown to be equivalent to the minimum ratio rule of the simplex algorithm, for updating the dual basis. The geometric analysis is applied to the Euclidean distance one-center problem and yields an alternative updating procedure for the dual algorithm.

linear programming formulations based on characterizations of block distance in terms of fundamental directions and polar directions.
This paper considers the two linear programming formulations of the one-center problem with block distances, as presented by Ward and Wendell. The equivalence of these two formulations follows from the equivalence of the block distance representations. The dual simplex algorithm is applied to the linear programming formulation based on polar directions of the block distance, and a geometric interpretation is presented. This interpretation is applied to the Euclidean distance onecenter problem and provides an alternative update procedure for the dual algorithm.
This paper actually considers a generalization of P1, the weighted one-center problem, in which there is a positive weight w i associated with each point p i , i = 1, ..., m. The problem is denoted by P2 and the constrained version is stated as follows: P2: min z s.t. z ≥ w i d(p i , x) i = 1, ..., m.

Block Distance
Block distance is a special case of general norms and were introduced to location problems by Witzgall [9] and by Ward and Wendell [7,8]. Block distance is defined in the plane with respect to a symmetric polytope as its unit ball, denoted by B. The polytope B is assumed to have 2p distinct extreme points, for some integer p ≥ 2. The vectors corresponding to the extreme points are called fundamental directions, and are denoted by b 1 , b 2 , ..., b 2p where b p+k = -b k for k = 1, ..., p. Assume that the fundamental directions are ordered counter clockwise, and for notational convenience, let b 2p+k = b k for k = 1, ..., p. Figure 1 shows an example with p = 4 fundamental directions and the corresponding unit ball.
The block distance between the points x o and x d with respect to a given set of 2p fundamental directions b 1 , ..., b 2p , is denoted d p (x o , x d ) and is defined to be the objective function value of the following linear programming problem, denoted by LPD: For any two points x o and x d ∈ IR 2 , the vector x o -x d must be in some cone generated by two adjacent fundamental directions, that is, for some nonnegative scalars α k and α k+1 . The vector x d -x o might also be in one or more cones generated by pairs of nonadjacent fundamental vectors. However, the following Property shows that an optimal basis to the linear program LPD must correspond to adjacent fundamental directions.
Suppose that x o -x d ∈ Γ(b k , b k+1 ) for some k. Then the 2 by 2 matrix [ b k b k+1 ] is a feasible basis for the linear program with and where e T = (1, 1).
Block distance may also be characterized in terms of the polar set B 0 of the polytope B. The polar set B 0 is also a symmetric polytope defined by B 0 = {v : b T k v ≤ 1, k = 1, ..., 2p}. In general, the facets of B are in one-toone correspondence with the extreme points of B 0 . In IR 2 , B 0 has the same number of extreme points as B, which correspond to polar directions and are denoted by b 0 k for k = 1, ..., 2p. It may be shown that the polar directions are given by Consider the dual of LPD, stated below as DLPD: The constraint set of DLPD is the polar set B 0 which, by the Representation Theorem [5], may be written as B 0 = Substituting into the dual objective function gives the equivalent characterization of block distance in terms of polar directions: Block distances are used to model travel distance in which the directions of travel are restricted to the fundamental directions. The l 1 distance is an example of a block distance with p = 2. Its fundamental directions are given by b 1 = ε ε 1 , b 2 = ε ε 2 , b 3 = -ε ε 1 , and b 4 = -ε ε 2 , where ε ε i is the ith unit vector in IR 2 . The polar directions for the l 1 distance are given by b 1 0T = (1, 1), b 2 0T = (-1, 1), b 0 3 = -b 0 1 , and b 0 4 = -b 0 2 , which are also the fundamental directions of the l ∞ distance, a block distance with p = 2.

Linear Programming Formulations
Ward and Wendell [8] presented two linear programming formulations of the one facility minmax location problem (with all w i = 1) using block distance: one in terms of fundamental directions and one in terms of polar directions. These two formulations are given below for problem P2, and denoted as LP1 and LP2. The expression of block distance in terms of polar directions is substituted into problem P2 to obtain the following: which is restated below as a linear program: .., m, and k = 1, ..., 2p.
Since the dual constraints are of rank three, a dual basis has the form: and the dual basic variables are denoted by π i 1 ,k 1 , π i 2 ,k 2 , π i 3 ,k 3 , where i j ∈ {1, ..., m} and k j ∈ {1, ..., 2p} for j = 1, 2, 3. The three weighted polar directions that determine a dual feasible basis are called basic weighted polar directions.
Given a dual feasible basis, the dual simplex algorithm proceeds as follows. The basic weighted polar directions in the dual feasible basis correspond to active (equality) constraints in the primal, so that the variables z*, x* are determined by a solution to the following system of linear equations.
If z*, x* are primal feasible, that is, if -x*), for all i = 1, ..., m, and k = 1, ..., 2p, then x* and z* are optimal. Otherwise, for some point p q and some direction b τ which implies that the point p q is outside the ball centered at x* with radius z*/w q . Choosing the most violated constraint corresponds to choosing a point of greatest weighted distance from x*.
The direction b τ 0 and the point p q determine the column that enters the dual basis. The leaving column may be determined by using the simplex rules, that is by using the following equations to compute the components d1, d2 and d3 of the direction vector corresponding to the basic columns: Then the step size α and the leaving basic column are computed using the minimum ratio test: and the column leaves the basis. The weighted polar directions in the new dual feasible basis are . The algorithm continues until primal feasibility is achieved. Problem LP1 is bounded and feasible, so that an optimal solution exists.

Geometric Interpretation of the Dual Basis Update
The update of the dual basis in the simplex algorithm applied to problem LP2 is analyzed in terms of the geometry associated with the basic weighted polar Consider a dual feasible basis with basic dual variables π i 1 , k 1 , π i 2 , k 2 , π i 3 , k 3 , for i j ∈ {1, ..., m}, k j ∈ {1, ..., 2p} and j = 1, 2, 3. Assume the weighted polar directions in the dual feasible basis are ordered counterclockwise with respect to j = 1, 2, 3. We adopt the notation that if j = 1, then j -1 = 3, and if j = 3, then j + 1 = 1.
Geometrically, a dual feasible basis implies that the vector 0 may be expressed as a convex combination of basic weighted polar directions, that is, If the dual feasible basis is non-degenerate, then π i j , k j > 0 for j = 1, 2, 3 and 0 is a strict convex combination of the basic weighted polar directions. In this case each basic weighted polar direction is contained in the cone generated by the negative of the other two basic weighted polar directions, that is, Also, for any weighted polar direction w q b 0 τ , there is some j = 1, 2, 3 so that ). Figure 2 illustrates the non-degenerate case with solid arrows corresponding to the basic weighted polar directions w i j b 0 k j and dashed arrows corresponding to -w i j b 0 k j , for j = 1, 2, 3. For the non-degenerate case the basic weighted polar directions form a simplex in IR 2 . Figure  2 ).
In the degenerate case, π i j , k j = 0 for exactly one j = 1, 2, or 3, and π i j-1 ,k j-1 > 0, π i j+1 , k j+1 > 0. Figure 3 illustrates the degenerate case. Note that if π i j , k j = 0, then . Also, if π i j , k j = 0, then for any weighted polar direction )}. For the non-degenerate case, the basis update rule is given as follows: ), for some j, and since w i j- )), then the new basic weighted polar directions are non-degenerate.
, then there is a tie for the replaced weighted polar direction and the new basis is degener- ) and either w i j-1 b 0 k j-1 or w i j b 0 k j may be replaced. For the degenerate case suppose π i j , k j = 0. If The new basic weighted polar directions are w q b 0 τ , w i j b 0 k j and )), then the new basic weighted polar directions are non-degenerate. If ) and either w i j-1 b 0 k j-1 or w i j+1 b 0 k j+1 may be replaced and the new basis is degenerate. However, if w q b 0 τ coincides with - ) only, so that only ), is analogous to the degenerate case in the preceding paragraph with j + 1 interchanged with j -1 throughout.
Finally, consider the degenerate case with π i j , k j = 0 but and remains degenerate.
The following Property shows that the new weighted polar directions determined by the geometric procedures above are basic feasible.   The next property shows that the geometric replacement rule corresponds to the minimum ratio rule of the simplex algorithm.

Property 3: Suppose the weighted polar directions
, are basic feasible, and suppose w q b 0 τ is the entering weighted polar direction and that ). The geometric rule that w q b 0 τ replaces w i j-1 b 0 k j-1 is equivalent to the minimum ratio rule of the simplex algorithm. Proof: The proof is given for the non-degenerate case. The proof of the degenerate case is similar. First we show that if ), then min j=1,2,3 { : d j < 0} = . Dual feasibility implies the following equations: The components of the direction vector determined by the vector w q b 0 τ are given by the following equations: The assumption that where β i j , k j ≥ 0 and β i j+1 , k j+1 ≥ 0. Equations (3) and (5) combine to give and equation (4) implies that Dividing through the last equation by the right hand side, and comparing the resulting two equations to (1) and (2), shows that so that d i j-1 < 0 since π i j-1 , k j-1 > 0. Equation (1) implies that Substitute this expression into equation (3), multiply through by π i j-1 , k j-1 and divide through by -d j-1 to get If d j ≥ 0, the first coefficient is positive. If d j < 0, the first coefficient is non-negative if and only if A similar argument holds for the second coefficient. The third coefficient is positive. Thus, the new coefficients are non-negative if and only if α = is the minimum ratio.
To prove the converse, suppose that min j=1,2,3 { : d j < 0} = , and show that w q b 0 τ ∈ Γ(-w i j b 0 k j , -w i j+1 b 0 k j+1 ). By the minimum ratio assumption, d j-1 < 0. If d j < 0, then by the minimum ratio, and the first coefficient in equation (6) is non-negative. If d j ≥ 0, the first coefficient in equation (6) is non-negative. A similar argument holds for d j+1 and the second coefficient in equation (6). Thus w q b 0 τ ∈ Γ(-w i j b 0 k j , -w i j+1 b 0 k j+1 ).
Properties 2 and 3 provide a geometric rule that could be used to determine the leaving column of a dual basis in the basis update step. However, the equivalent minimum ratio rule of the simplex algorithm is more efficient.
In the next section, the equivalence is used to show that the minimum ratio rule may be used to update the dual algorithm applied to the Euclidean distance onecenter problem, which is an improvement over existing geometrical update rules.