A new simplicial algorithm for box constrained nonlinear optimization

The paper proposes a new simplex-based direct search algorithm for box-constrained nonlinear programming (NLP) problems. As is typical of such methods, the optimal solution is obtained by iteratively transforming a multi-dimensional simplex and no gradient information is calculated. In contrast to unconstrained optimization problems, where the decision about way of transforming an iterate simplex is based solely on the consideration of optimality, in the proposed algorithm, both optimality and constraint satisfaction affect the iterations. First novel feature of the proposed algorithm is a fast way of checking if a planned iteration of current simplex violates any of the bounds and, if yes, a systematic way of choosing an alternative without compromising on the optimality considerations. The proposed procedure, in fact, guarantees that each iterate simplex lies entirely within the feasible region. Second novel feature of the algorithm is the use of so-called free and bounded variables that allow for different dimensions of simplices in different iterations. This feature helps convergence of the algorithm when one or more of the decision variables lie on bounds. The performance of the proposed algorithm is demonstrated on some well known test problems.