Global optimization of robust truss topology via mixed integer semidefinite programming

Yonekura, Kazuo; Kanno, Yoshihiro

doi:10.1007/s11081-010-9107-1

Cited by 37 publications

(20 citation statements)

References 19 publications

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“…It can be found that, among totally 51 binary discrete variables, the optimal values of 18 0-1 variables (x i , i ∈ I 1 = {5, 9,10,14,15,17,19,22,24,27,28,32,33,38,40,43,45, 50}) can be determined directly in the first-round sensitivity bounding analysis. With these obtained values being fixed another 15 values of x i , i ∈ I 2 = {1, 4,12,13,18,20,23,26,29,34,35,37, 39, 48, 51} can be found from monotonic analysis in the second-round sensitivity bounding process. After that, the 0-1 programming with 18 remaining variables is solved by the optimizer CPLEX with global optimality.…”

Section: A 51-bar Truss Arch Bridgementioning

confidence: 99%

Exact response bound analysis of truss structures via linear mixed 0‐1 programming and sensitivity bounding technique

Wei

et al. 2018

Numerical Meth Engineering

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In the present paper, finding the exact bound of structural response for truss structures is considered under bounded interval type uncertainty. This problem is challenging since seeking the exact bound corresponds to locating the global optima of a multivariate function (generally nonconvex). Traditional treatment of this problem involves the solution of a linear mixed 0-1 programming problem, which is a highly computationally demanding task especially when large-scale structures are taken into consideration. In order to alleviate the computational effort, a sensitivity bounding technique is developed in this work using the tools from convex analysis to disclose the monotonicity of concerned structural response function with respect to 0-1 variables. It is shown that this technique can not only reduce the number of 0-1 variables substantially but also change the computational complexity of the considered problem from nondeterministic polynomial-hard to nondeterministic polynomial-hard in some cases. The proposed approach provides the possibility of finding the exact bound of structural response for large-scale truss structures within a reasonable time, and its effectiveness is demonstrated through several numerical examples. KEYWORDS bounds of structural response, global optimization, interval uncertainty, linear mixed 0-1 programming, sensitivity bounding Int J Numer Methods Eng. 2018;116:21-42. wileyonlinelibrary.com/journal/nmewhere x, f (x), and g i (x; p) are the design variable vector, objective function, and ith constraint, respectively. p denotes the vector of uncertain parameters and U p is the corresponding uncertainty set. It is clear that the objective of a WCDO problem is essentially to optimize the performance of a design while ensuring the its safety under the worst combination of uncertain parameters. Therefore, Equation (1) is actually a nested bilevel program. The upper level program aims at finding the best design by selection of optimal design variables, while the lower level program identifies the worst-case structural responses and determines the feasibility of a given design. To predict the extreme structural response with respect to the uncertain parameter efficiently, interval analysis technique or convex modeling methods cooperating with sensitivity analysis and gradient-based algorithms are commonly adopted in literatures. [8][9][10][11][12] It should be pointed out that the lower level program in Equation (1) must be solved with global optimality; otherwise, the feasibility of an "optimal solution" cannot be fully guaranteed. 13 This issue, however, cannot been guaranteed by the general gradient-based algorithms used in most related literature studies, especially for WCDO problems involving nonconvex objective/constraint functions, which are ubiquitous in practical engineering applications. Under this circumstance, the obtained solution may be totally meaningless since the worst-case structural response may be severely underestimated. Actually, it is very difficult to find the global ...

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Section: A 51-bar Truss Arch Bridgementioning

confidence: 99%

Exact response bound analysis of truss structures via linear mixed 0‐1 programming and sensitivity bounding technique

Wei

et al. 2018

Numerical Meth Engineering

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“…From an optimization perspective, mixed-integer semi-definite optimization (MI-SDP) has received a lot of attention in recent years, for they naturally appear in robust optimization problems with ellipsoidal uncertainty sets [4] or as reformulations of combinatorial problems [58]. Problem-specific MI-SDP strategies have been developed for problems such as binary quadratic programming [33], robust truss topology [63] or the max-cut problem [51]. More recently, rounding and Gomory cuts [12,1], branch-and-bound [29] and outer-approximation schemes [43] have also been developed, in an attempt to provide the same level of general-purpose solvers for MI-SDP as there are for mixed-integer linear optimization.…”

Section: Current Approachesmentioning

confidence: 99%

Certifiably optimal sparse inverse covariance estimation

Bertsimas¹,

Lamperski²,

Pauphilet³

2019

Math. Program.

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We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity.We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approach uses techniques from mixed-integer optimization and convex optimization, and provides a high-quality solution with a guarantee on its suboptimality, even if the algorithm is terminated early. Using a variety of synthetic and real datasets, we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1, 000s. We also demonstrate that our approach produces significantly sparser solutions than Glasso and other popular learning procedures, makes less false discoveries, while still maintaining state-of-the-art accuracy.

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“…Guna-wa和Azarm [25~27] 提出一种新的区间参数灵敏度评估方法, 并将该方法推广到单目标和多目标的鲁棒性优化问题中. 为了获得带有区间参数鲁棒性优化的全局最优解, Yonekura和Kanno [28] 采用分支定界方法处理桁架结构鲁棒性拓扑优化问题. Siddiqui等人 [29] 通过在原有的Bender's分解方法中增加额外的鲁棒性判断, 将Bender's分解方法拓展到区间非线性鲁棒性优化问题中, 提高了鲁棒性优化问题的求解效率.…”

Section: 因此在不确定性领域发展了一种非概率不确定unclassified