MathProg (AMPL) -由另一个变量调整大小的变量数组

Question

我正在编写我的第一个GNU MathProg (AMPL)程序来查找给定基数、主机数量和二等分带宽的HyperX拓扑(图)的最小交换机(顶点)计数实例。这是一个简单的第一个程序，因为所有的方程都在下面的论文中描述过：http://cal.snu.ac.kr/files/2009.sc.hyperx.pdf

我已经阅读了规范和示例程序，但我被一个非常简单的语法错误卡住了。我需要有以下两个变量: L，网络中的维数，以及一个长度为L的数组S，其中S的每个元素都是每个维度中的交换机数量。在我的MathProg程序中，我将其表示为：

var L >= 1, integer;
var S{1 .. L} >= 2, integer;

但是，当我运行$ glpsol --check --math hyperx.mod时，我得到以下错误：

hyperx.mod:28: operand following .. has invalid type
Context: ...isec ; param radix ; var L >= 1 , integer ; var S { 1 .. L }

如果有人能帮助我解释我应该如何恰当地表达这种关系，我将不胜感激。此外，我还包括了我写的整个程序，以供参考和额外的帮助。我预计我的程序中会有许多语法错误，但在修复第一个错误之前，我无法找到其余的错误。

/* 
 * A MathProg linear program to find an optimal HyperX topology of a
 *  given network size, switch radix, and bisection bandwidth.  Optimal
 *  is simplistically defined as minimum switch count network. 
 *
 * A HyperX topology is a multi-dimensional network (graph) where, in
 *  each dimension, the switches are fully connected.  Every switch
 *  (vertex) is a point in an L-dimensional integer lattic.  Each switch
 *  is identified by a multi-index I = (I_1, ..., I_L) where 0 <= I_k <
 *  S_k for each k = 1..L, where S_k is the number of switches in each
 *  dimension.  A switch connects to all others whose multi-index is the
 *  same in all but one coordinate.
 */

/* Network size in number of hosts. */
param hosts;

/* Desired bisection bandwidth. */
param bisec;

/* Maximum switch radix. */
param radix;

/* The number of dimensions in the HyperX. */
var L >= 1, integer;

/* The number of switches in each dimension. */
var S{1 .. L} >= 2, integer;

/* 
 * Relative bandwidth of the dimension, i.e., the number of links in a
 * given dimension. 
 */
var K{1 .. L} >= 1, integer;

/* The number Terminals (hosts) per switch */
var T >= 1, integer;

/* Minimize the total number of switches. */
minimize cost: prod{i in 1..L} S[i];

/* The total number of links must be less than the switch radix. */
s.t. Radix: T + sum{i in 1..L} K[i] * (S[i] - 1) <= radix;

/* There must be enough hosts in the network. */
s.t. Hosts: T * prod{i in 1..L} S[i] >= hosts;

/* There must be enough bandwidth. */
s.t. Bandwidth: min{K[i]*S[i]} / (2 * T) >= bisec;

/* The order of the dimensions doesn't matter, so constrain them */
s.t. SwitchDimen: forall{i in 1..(L-1)} S[i] <= S[i+1];

/* 
 * Bisection bandwidth depends on the smallest S_i * K_i, so we know
 * that the smallest switch count dimension needs the most links.
 */
s.t. LinkDimen: forall{i in 1..(L-1)} K[i] >= K[i+1];

# TODO: I would like to constrain the search such that the number of
# terminals, T, is bounded to T >= (hosts / O), where O is the switch
# count of the smallest switch count topology discovered so far, but I
# don't know how to do this.

/* Data section */
data;

param hosts := 32

param bisec := 0.5

param radix := 64

end;

Answer 1

问题中变量的固定数量是求解器和代数建模语言(包括AMPL/MathProg )中的一个常见假设。因此，在索引表达式中只能使用常量表达式，尤其是参数，而不能使用变量。一种可能的解决方案是将L作为参数，解决L的不同值的问题，并选择给出最佳目标值的值。这可以通过一个简单的AMPL脚本来完成。