New Algorithm for the Conical Combination Representation Problem of a Vector

An important problem in constrained optimization is to determine whether or not a vector can be represented as the conical combination (i.e., linear nonnegative combination) of some given vectors. This problem can be transformed into a special linear programming problem (SLP). A new approach, the variable-dimension boundary-point algorithm (VDBPA), based on the projection of a vector into a variable-dimension subspace, is proposed to solve this problem. When a conical combination representation (CCR) of a vector exists, the VDBPA can compute its CCR coefficients; otherwise, the algorithm certificates the nonexistence of such representation. In order to assure convergence, the VDBPA may call the lexicographically ordered method (LOM) for linear programming at the final stage. In fact, the VDBPA terminates often by solving SLP for most instances before calling the LOM. Numerical results indicate that the VDBPA works more efficiently than the LOM for problems that have more variables or inequality constraints. Also, we have found instances of the SLP, when the number of inequality constraints is about twice the number of variables, which are much more difficult to solve than other instances. In addition, the convergence of the VDBPA without calling the LOM is established under certain conditions.