Logistics Network Design includes network configuration decisions having long-standing influences on other tactical and operational decisions. Recently, regarding environmental issues and customer awareness, Closed-loop logistics network (CLLN) design is taken into consideration. This paper presents a nonlinear mixed integer programming for a dynamic multi-period multi-products forward/reverse logistics network with seven layers: suppliers, Plants, distribution centers (DCs), Customer zones, and inspection and disposal centers. The possibility of opening and closing of each facility at each period is one of the primary characteristics of the proposed model. Moreover, operational decisions such as inventory cost and capacity development in each period as well as the optimum assignment flow between consecutive layers of the proposed network are considered in CLLN modeling. To cope with such an NP-hard problem an advanced continuous Genetic Algorithm (cGA) has been applied. The primary focus of cGA lies in its innovative representation method, which allows it to avoid the time-consuming repair mechanisms due to fixing the infeasible solutions. Furthermore, through the implementation of this representation, additional operators can be seamlessly integrated into the algorithm. Numerical results show the efficiency of the proposed method in large scale problems.

In the context of mixed-integer nonlinear problems (MINLPs), it is well-known that strong duality does not hold in general if the standard Lagrangian dual is used. Hence, we consider the augmented Lagrangian dual (ALD), which adds a nonlinear penalty function to the classic Lagrangian function. For this setup, we study conditions under which the ALD leads to a zero duality gap for general MINLPs. In particular, under mild assumptions and for a large class of penalty functions, we show that the ALD yields zero duality gaps if the penalty parameter goes to infinity. If the penalty function is a norm, we also show that the ALD leads to zero duality gaps for a finite penalty parameter. Moreover, we show that such a finite penalty parameter can be computed in polynomial time in the mixed-integer linear case. This generalizes the recent results on linearly constrained mixed-integer problems by Bhardwaj et al. (2024), Boland and Eberhard (2014), Feizollahi et al. (2016), and Gu et al. (2020).

In this talk we present a sequential mixed-integer quadratic programming (MIQP) algorithm for solving mixed-integer nonlinear problems (MINLPs). The algorithm employs a three-step method in each iterate: First, the algorithm linearizes the MINLP at a given iterate. Second, an MIQP with its feasible set restricted to a specific region around the current linearization point is formulated and solved. Third, the integer variables obtained from the MIQP solution are fixed, and only an NLP in the continuous variables is solved. The outcome of the third step is compared to previous iterates, and the best iterate so far is used as a linearization point in the next iterate. The polyhedral region in the second step is computed based on objectives and derivatives from all previous iterates, and build on concepts from generalized Benders' decomposition. Although the presented MINLP algorithm is a heuristic method without any global optimality guarantee, it converges to the exact integer solution when applied to convex MINLP with a linear outer structure. The conducted numerical experiments on existing benchmarks demonstrate that the proposed algorithm is competitive with other open-source solvers for MINLP.