Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators

<p style='text-indent:20px;'>In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.</p>

Uncertain Optimization of Discrete Supply Networks with Order Delivery Disruption and Risk Preference in the Postepidemic Era

Epidemic blockade leads to increased uncertainty and dynamic supply network disruption. This study considers an uncertain optimization of dynamic supply networks with risk preference and order delivery disruption. Taking the subjective utility of downstream enterprises as a reference point for the utility measurement of order delivery disruption and risk preference, this study constructs a biobjective optimization model with the goal of maximizing the downstream firm’s subjective utility and minimizing the manufacturer’s cost. The influence of each parameter in the downstream firm’s subjective utility function on the integrated optimization was analysed. The research found that the uncertain optimization model with the risk preference of downstream firms for order delivery disruption better controls the actual manufacturer’s order allocation and distribution problems when considering the downstream firms’ behaviour preference characteristics under bounded rationality. When allocating orders, manufacturers should consider that changes in order delivery disruption will cause changes in the subjective utility of downstream enterprises. In the process of multiperiod cooperation between manufacturers and downstream firms, they can obtain downstream firm risk preferences through repeated investigations.

Modeling, Simulation and Uncertain Optimization of the Gun Engraving System

The system designed to accomplish the engraving process of a rotating band projectile is called the gun engraving system. To obtain higher performance, the optimal design of the size parameters of the gun engraving system was carried out. First, a fluid–solid coupling computational model of the gun engraving system was built and validated by the gun launch experiment. Subsequently, three mathematic variable values, like performance evaluation indexes, were obtained. Second, a sensitivity analysis was performed, and four high-influence size parameters were selected as design variables. Finally, an optimization model based on the affine arithmetic was set up and solved, and then the optimized intervals of performance evaluation indexes were obtained. After the optimal design, the percent decrease of the maximum engraving resistance force ranged from 6.34% to 18.24%; the percent decrease of the maximum propellant gas temperature ranged from 1.91% to 7.45%; the percent increase of minimum pressure wave of the propellant gas ranged from 0.12% to 0.36%.

Robust strong duality for nonconvex optimization problem under data uncertainty in constraint

<abstract><p>This paper deals with the robust strong duality for nonconvex optimization problem with the data uncertainty in constraint. A new weak conjugate function which is abstract convex, is introduced and three kinds of robust dual problems are constructed to the primal optimization problem by employing this weak conjugate function: the robust augmented Lagrange dual, the robust weak Fenchel dual and the robust weak Fenchel-Lagrange dual problem. Characterizations of inequality (1.1) according to robust abstract perturbation weak conjugate duality are established by using the abstract convexity. The results are used to obtain robust strong duality between noncovex uncertain optimization problem and its robust dual problems mentioned above, the optimality conditions for this noncovex uncertain optimization problem are also investigated.</p></abstract>