A convergent simplicial algorithm with ω-subdivision and ω-bisection strategies

A modified simplicial algorithm for convex maximization based on an extension of $$\omega $$-subdivision

“…This guarantees the convergence of the algorithm to an optimal solution of (1) if the successor S i+1 to S i is chosen in best-bound-first order, i.e., S i+1 is a simplex with the largest upper bound among all active descendants of S 1 . In addition to this simple bisection, there are several rules for subdividing S i which guarantee the convergence of the algorithm [9,10,11,12]. Among others, the most poplar is the ω-subdivision rule, where u is placed at ω ω ω i for each i.…”

“…Empirically, it is known that the ω-subdivision rule runs the algorithm much more efficiently than bisection [9]. Whichever rule is adopted, in order to make the algorithm converge to an optimal solution of (1), the successor S i+1 to S i has to be chosen in best-bound-first order.…”

“…If there is a large variability in the edge lengths of S i , we cannot obtain stable solutions to the systems (6) and (9). A simple way to avoid this is to occasionally apply the usual bisection rule in (12) and shorten a longest edge of S i by half.…”

“…In this paper, we show that the computational efficiency of the simplicial algorithm is considerably improved by dropping the constraint on the simplex S from the linearized subproblem. As a consequence of this modification, the optimal solution ω ω ω of the linearized subproblem does not always lie in S. In that case, we cannot apply the ω-subdivision rule in the branching process, despite its empirical efficiency [9,10], since the rule is configured to partition S radially around ω ω ω assumed to be in S. To cope with such a case, we develop a new simplicial subdivision rule, named extended ω-subdivision, which uses ω ω ω to partition S even if ω ω ω is not a point in S, and show that the simplicial algorithm works properly under this subdivision rule. In Section 2, we give a formal definition of the convex maximization problem, and illustrate how the simplicial algorithm behaves on the problem.…”

A generalization of $$\omega $$-subdivision ensuring convergence of the simplicial algorithm

“…Kuno-Ishihama [8] showed that a more moderate condition holds always and guarantees the convergence for the conical algorithm. In a similar way, Kuno-Buckland [7] proved the convergence for the simplicial algorithm, but instead allowed an error in the feasibility of the algorithm output within a specified tolerance. They also provided another subdivision rule, called ω-bisection, combining ω-subdivision and bisection, and reported in [7,8] that it improves the computational efficiency of both algorithms.…”