A Gradient-Based Interior-Point Method to Solve the Many-to-Many Assignment Problems

Algorithm 1

A Gradient based interior point method.

Input:   and potential matrix (i.e., )

Output:   (i.e., the final solution)

 begin

 Initialization: ;

Cardinality constraint detection is performed using Equation (7);

Expansion of the potential matrix (i.e., ) is executed using Equation (8);

Conversion of the objective function into a minimization problem

 (i.e., Equation (10)) is conducted using Equation (9);

Hypercube embedding is performed & objective function is defined by Equation (16);

 while () do/ Start the LBF-based gradient projection method /

  repeat

   for 

     ;

     / by Equation (21) and Equation (33)/

   end

   return (i.e., value when );

   evaluate by Equation (35)/

   Compute for the new value of by Equation (16);

   case: then

      update, ;

   case: then

      update, ;

   end case

  until the gradient based interior point method converges

  ;

 end

return ;

 end