Solving dual problems using a coevolutionary optimization algorithm

In solving certain optimization problems, the corresponding Lagrangian dual problem is sometimes solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that for all optimization problems the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for coevolving two populations – one involving Lagrange multipliers and other involving decision variables – to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested algorithms and a couple of previously suggested coevolutionary algorithms. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed algorithm compared to usual nested and other coevolutionary approaches remain as the hallmark of the current study.