A set-valued variational model for risk-based decision optimization in civil engineering systems

Abstract

The paper introduces a minimax variational optimization framework that is applicable to uncertain civil engineering systems that require risk-based decision making. Most optimization methods are usually based on deterministic parameters and single-valued operations, which make such methods inefficient for modeling several uncertain possibilities. In order to resolve the problem, the new method combines the concepts of minimax theory, set-valued analysis, and variational inequalities to establish an efficient mathematical optimization model. The uncertain decision-making situation is modeled using set-valued mapping, and a worst-case minimax formulation is then formulated as a set-valued variational inequality problem. The new framework formulates theoretical properties that include the existence, uniqueness, stability under perturbations of uncertainty, and convergence of the computational process. The paper proposes a projection-based iterative algorithm to compute the optimal solution of the equilibrium while taking into account feasible constraints. The proposed model will be tested using numerical experiments based on risk minimization and convergence characteristics. These results clearly show that the risk value obtained by the proposed set-valued minimax variational model is 0.12, which is relatively smaller when compared to classical minimax, robust optimization, and stochastic variational inequality models with risk values 0.31, 0.25, and 0.21, respectively. In addition, the convergence achieved by the proposed method requires only 75 iterations, with a convergence tolerance of 0.0019, thereby proving the superiority of the method from the aspect of optimization efficiency and convergence. The applicability of the methodology is validated with respect to civil engineering problems under uncertain conditions.