A Study On Some Algorithms For Confidence Structural Robust Optimization And Their Extensions

Uncertainties are unavoidable in practical engineering. In order to deal with the structural design problems in a reliable way, robust structural optimization method has attracted great attentions recently. In the last few years, a new confidence structural robust optimization formulation was proposed, which can guarantee the strict feasibility of robust optimal solution theoretically. Under this framework, in the present thesis, several algorithms are proposed and applied to discrete robust structural optimization and fail-safe design. The content of this dissertation includes the following four parts:1. Based on the sensitivity bounding technique, an accurate truss structure extreme response analysis method is developed for the structure extreme response analysis problem which is concerned with the worst case of structural design and optimization problems. The original method completely depends on integer programming, which has non-polynomial time computational complexity. In this dissertation, sensitivity bounding and monotonic analysis method has been employed simultaneously to determine the critical values of some uncertain parameters in order to reduce the computational efforts of the problem。Then the mixed 0-1 linear programming has been applied to the reduced problem to obtain the accurate structural response. Since sensitivity bounding technique only exhibits polynomial-time computational complexity, the overall computational complexity can be reduced substantially.2. Confidence non-linear semi-definite programming (NLSDP)-based single-level formulations are proposed for structural robust optimization under non-probabilistic stiffness uncertainties. Both static and dynamic steady-state conditions are considered by using some the tools of S-procedure and quadratic embedding in convex analysis. Compared with the traditional Bi-level approach, this method can improve computational efficiency and convergence. Moreover, the robustness of the optimal solution can be assured theoretically. Several numerical examples illustrate the effectiveness of the proposed approach.3. Based on the idea of robust optimization, a robust rounding method is proposed for discrete optimization. A robust optimization formulation equivalent to the original discrete optimization problem is also proposed. These formulations can then be solved by NLSDP method with strict feasibility. This method is more reliable than traditional method. A NLSDP-based formulation for robust discrete optimization problems are also proposed based on the same idea. Numerical examples show the effectiveness of the proposed approaches.4. Traditional fail-safe optimal design is usually performed based on presumed failure mode. However, the assumed failure mode may not necessarily be the most dangerous one. If this happens, the reliability of the obtained optimal solution cannot be guaranteed. To overcome this problem, in this dissertation, fail-safe optimal design is developed in a robust optimization framework. The problem is formulated in a Bi-level form. The upper level program aims at finding the optimal design variables while the lower level program is used to find the worst case of the failure modes. In order to find the global optimal solution of the lower level program, a mixed 0-1 programming is proposed and solved by the branch-and-bound algorithm. The effectiveness of the proposed approach for fail-safe design and optimization is illustrated by some numerical examples.