%% Design of Collateralised Debt Obligations Squared (CDO^2) Transactions %% %% Problem described in: %% Pierre Flener, Justin Pearson, Luis G. Reyna, Olof Sivertsson: %% Design of Financial CDO Squared Transactions Using Constraint Programming. %% Constraints 12(2):179-205, 2007 %% http://dx.doi.org/10.1007/s10601-006-9014-4 %% %% Also solved with: %% LS with VNS: http://dx.doi.org/10.1016/j.endm.2014.11.017 %% CBLS with set variables: http://dx.doi.org/10.1007/11564751_7 %% %% Authors: Pierre Flener and Jean-Noel Monette %% License: CC-BY-4.0 %% %% Model loosely based on Ralph Becket's BIBD model at %% https://github.com/MiniZinc/minizinc-benchmarks/blob/master/bibd/bibd.mzn % An OPD (v, b, r) problem is to find a binary matrix of v rows % and b columns such that each row sums to r, and % the dot product beween any pair of distinct rows is minimal. %Requires MiniZinc >= 2.0.2 for the symmetry_breaking_constraint predicate include "lex_greatereq.mzn"; %instance data int: v; int: b; int: r; set of int: rows = 1..v; set of int: cols = 1..b; %computing a lower bound for lambda int: rv = r*v; int: rvmodb = rv mod b; int: floorrv = rv div b; int: ceilrv = rv div b + bool2int(rv mod b != 0); int: num = (ceilrv*ceilrv*rvmodb+floorrv*floorrv*(b-rvmodb)-rv); int: denom = v*(v-1); int: lb_lambda = num div denom + bool2int( num mod denom !=0); % This line is there for debugging purposes, it can be safely removed int: tmp = trace("Computed lower bound for lambda: "++show(lb_lambda)++"\n",lb_lambda); % lambda is called objective for the MiniZinc Challenge var lb_lambda..b: objective; array [rows, cols] of var 0..1: m; % Every row must sum to r. constraint forall (i in rows) (sum (j in cols) (m[i, j]) = r); % The dot product of every pair of distinct rows must be at most lambda for an OPD and a PD, and equal to lambda for a BIBD constraint forall (i_a, i_b in rows where i_a < i_b) ( sum (j in cols) (m[i_a, j] * m[i_b, j]) <= objective ); % Break row symmetry in the incidence matrix. constraint symmetry_breaking_constraint( forall(i in rows diff {max(rows)})( lex_greatereq([m[i, j] | j in cols], [m[i+1, j] | j in cols]) ) ); % Break column symmetry in the incidence matrix. constraint symmetry_breaking_constraint( forall(j in cols diff {max(cols)})( lex_greatereq([m[i, j] | i in rows], [m[i, j+1] | i in rows]) ) ); solve :: seq_search([int_search([m[i, j] | i in rows, j in cols], input_order, indomain_max, complete), int_search([objective], input_order,indomain_min,complete)]) minimize objective; % Disabled the full solution. Printing only the objective value, lower bound, and parameters. output ["opd: (v = ", show(v), ", b = ", show(b), ", r = ", show(r), "). Found lambda = ", show(objective),"\tlb: ", show(lb_lambda)] % ++["\n\n"] ++ % [ ( if j > b then "\n" else % if fix(m[i,j])=1 then "*" else " " endif % endif ) % | i in rows, j in 1..(b + 1) ] ;