Diagonal Quadratic Approximation for Parallelization of Analytical Target Cascading

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Yanjing Li ◽  
Zhaosong Lu ◽  
Jeremy J. Michalek
Keyword(s):  
Optimization Problems ◽  
Computational Cost ◽  
Descent Method ◽  
Quadratic Approximation ◽  
Test Problems ◽  
Analytical Target Cascading ◽  
Decomposition Approach ◽  
Diagonal Quadratic Approximation ◽  
Target Cascading ◽  
Nested Loop

Analytical target cascading (ATC) is an effective decomposition approach used for engineering design optimization problems that have hierarchical structures. With ATC, the overall system is split into subsystems, which are solved separately and coordinated via target/response consistency constraints. As parallel computing becomes more common, it is desirable to have separable subproblems in ATC so that each subproblem can be solved concurrently to increase computational throughput. In this paper, we first examine existing ATC methods, providing an alternative to existing nested coordination schemes by using the block coordinate descent method (BCD). Then we apply diagonal quadratic approximation (DQA) by linearizing the cross term of the augmented Lagrangian function to create separable subproblems. Local and global convergence proofs are described for this method. To further reduce overall computational cost, we introduce the truncated DQA (TDQA) method, which limits the number of inner loop iterations of DQA. These two new methods are empirically compared to existing methods using test problems from the literature. Results show that computational cost of nested loop methods is reduced by using BCD, and generally the computational cost of the truncated methods is superior to the nested loop methods with lower overall computational cost than the best previously reported results.

Diagonal Quadratic Approximation for Parallelization of Analytical Target Cascading

2007 ◽  
Author(s):  
Yanjing Li ◽  
Zhaosong Lu ◽  
Jeremy J. Michalek
Keyword(s):  
Optimization Problems ◽  
Computational Cost ◽  
Descent Method ◽  
Quadratic Approximation ◽  
Test Problems ◽  
Analytical Target Cascading ◽  
Decomposition Approach ◽  
Diagonal Quadratic Approximation ◽  
Target Cascading ◽  
Nested Loop

Analytical Target Cascading (ATC) is an effective decomposition approach used for engineering design optimization problems that have hierarchical structures. With ATC, the overall system is split into subsystems, which are solved separately and coordinated via target/response consistency constraints. As parallel computing becomes more common, it is desirable to have separable subproblems in ATC so that each subproblem can be solved concurrently to increase computational throughput. In this paper, we first examine existing ATC methods, providing an alternative to existing nested coordination schemes by using the block coordinate descent method (BCD). Then we apply diagonal quadratic approximation (DQA) by linearizing the cross term of the augmented Lagrangian function to create separable subproblems. Local and global convergence proofs are described for this method. To further reduce overall computational cost, we introduce the truncated DQA (TDQA) method that limits the number of inner loop iterations of DQA. These two new methods are empirically compared to existing methods using test problems from the literature. Results show that computational cost of nested loop methods is reduced by using BCD and generally the computational cost of the truncated methods, TDQA and ALAD, are superior to other nested loop methods with lower overall computational cost than the best previously reported results.

Cutting Plane Methods for Analytical Target Cascading With Augmented Lagrangian Coordination

2013 ◽  
Vol 135 (10) ◽  
Author(s):  
Wenshan Wang ◽  
Vincent Y. Blouin ◽  
Melissa K. Gardenghi ◽  
Georges M. Fadel ◽  
Margaret M. Wiecek ◽  
...  
Keyword(s):  
Large Scale ◽  
Augmented Lagrangian ◽  
Optimization Problems ◽  
Cutting Plane ◽  
Analytical Target Cascading ◽  
Cutting Plane Methods ◽  
Subgradient Optimization ◽  
Decomposition Approach ◽  
Target Cascading ◽  
Augmented Lagrangian Coordination

Analytical target cascading (ATC), a hierarchical, multilevel, multidisciplinary coordination method, has proven to be an effective decomposition approach for large-scale engineering optimization problems. In recent years, augmented Lagrangian relaxation methods have received renewed interest as dual update methods for solving ATC decomposed problems. These problems can be solved using the subgradient optimization algorithm, the application of which includes three schemes for updating dual variables. To address the convergence efficiency disadvantages of the existing dual update schemes, this paper investigates two new schemes, the linear and the proximal cutting plane methods, which are implemented in conjunction with augmented Lagrangian coordination for ATC-decomposed problems. Three nonconvex nonlinear example problems are used to show that these two cutting plane methods can significantly reduce the number of iterations and the number of function evaluations when compared to the traditional subgradient update methods. In addition, these methods are also compared to the method of multipliers and its variants, showing similar performance.

A Sequential Linear Programming Coordination Algorithm for Analytical Target Cascading

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
Jeongwoo Han ◽  
Panos Y. Papalambros
Keyword(s):  
Design Optimization ◽  
Optimization Problems ◽  
Trust Region Method ◽  
Trust Region ◽  
Test Problems ◽  
Analytical Target Cascading ◽  
Step Size ◽  
Coordination Strategy ◽  
Target Cascading ◽  
Solution Accuracy

Decomposition-based strategies, such as analytical target cascading (ATC), are often employed in design optimization of complex systems. Achieving convergence and computational efficiency in the coordination strategy that solves the partitioned problem is a key challenge. A new convergent strategy is proposed for ATC that coordinates interactions among subproblems using sequential linearizations. The linearity of subproblems is maintained using infinity norms to measure deviations between targets and responses. A subproblem suspension strategy is used to suspend temporarily inclusion of subproblems that do not need significant redesign, based on trust region and target value step size. An individual subproblem trust region method is introduced for faster convergence. The proposed strategy is intended for use in design optimization problems where sequential linearizations are typically effective, such as problems with extensive monotonicities, a large number of constraints relative to variables, and propagation of probabilities with normal distributions. Experiments with test problems show that, relative to standard ATC coordination, the number of subproblem evaluations is reduced considerably while the solution accuracy depends on the degree of monotonicity and nonlinearity.

A Sequential Linear Programming Coordination Algorithm for Analytical Target Cascading

2007 ◽  
Author(s):  
Jeongwoo Han ◽  
Panos Papalambros
Keyword(s):  
Linear Programming ◽  
Optimization Problems ◽  
Trust Region ◽  
Test Problems ◽  
Sequential Linear Programming ◽  
Analytical Target Cascading ◽  
Step Size ◽  
Normal Distributions ◽  
Coordination Strategy ◽  
Target Cascading

Decomposition-based strategies, such as analytical target cascading (ATC), are often employed in design optimization of complex systems. Achieving convergence and computational efficiency in the coordination strategy that solves the partitioned problem is a key challenge. A new convergent strategy is proposed for ATC, which coordinates the interactions among subproblems using sequential lineralizations. Linearity of subproblems is maintained using L∞ norms to measure deviations between targets and responses. A subproblem suspension strategy is used to temporarily suspend inclusion of subproblems that do not need significant redesign, based on trust region and target value step size. The proposed strategy is intended for use in optimization problems where sequential linearizations are typically effective, such as problems with extensive monotonicities, large number of constraints relative to variables, and propagation of probabilities with normal distributions. Experiments with test problems show that, relative to standard ATC coordination, the number of subproblem evaluations is reduced considerably while maintaining accuracy.

Feasibility Robust Optimization via Scenario Generation and Reduction

2018 ◽  
Author(s):  
Eliot Rudnick-Cohen ◽  
Jeffrey W. Herrmann ◽  
Shapour Azarm
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Optimization Techniques ◽  
Scenario Generation ◽  
Test Problems ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Worst Case ◽  
Robust Solution

Feasibility robust optimization techniques solve optimization problems with uncertain parameters that appear only in their constraint functions. Solving such problems requires finding an optimal solution that is feasible for all realizations of the uncertain parameters. This paper presents a new feasibility robust optimization approach involving uncertain parameters defined on continuous domains without any known probability distributions. The proposed approach integrates a new sampling-based scenario generation scheme with a new scenario reduction approach in order to solve feasibility robust optimization problems. An analysis of the computational cost of the proposed approach was performed to provide worst case bounds on its computational cost. The new proposed approach was applied to three test problems and compared against other scenario-based robust optimization approaches. A test was conducted on one of the test problems to demonstrate that the computational cost of the proposed approach does not significantly increase as additional uncertain parameters are introduced. The results show that the proposed approach converges to a robust solution faster than conventional robust optimization approaches that discretize the uncertain parameters.

Variable-Fidelity Multidisciplinary Design Optimization Based on Analytical Target Cascading Framework

2012 ◽  
Vol 544 ◽  
pp. 49-54 ◽  
Author(s):  
Jun Zheng ◽  
Hao Bo Qiu ◽  
Xiao Lin Zhang
Keyword(s):  
Multidisciplinary Design Optimization ◽  
Model Building ◽  
Computational Cost ◽  
Simulation Models ◽  
Multidisciplinary Design ◽  
High Fidelity ◽  
Analytical Target Cascading ◽  
Approximation Model ◽  
Target Cascading ◽  
Variable Fidelity

ATC provides a systematic approach in solving decomposed large scale systems that has solvable subsystems. However， complex engineering system usually has a high computational cost , which result in limiting real-life applications of ATC based on high-fidelity simulation models. To address these problems, this paper aims to develop an efficient approximation model building techniques under the analytical target cascading (ATC) framework, to reduce computational cost associated with multidisciplinary design optimization problems based on high-fidelity simulations. An approximation model building techniques is proposed: approximations in the subsystem level are based on variable-fidelity modeling (interaction of low- and high-fidelity models). The variable-fidelity modeling consists of computationally efficient simplified models (low-fidelity) and expensive detailed (high-fidelity) models. The effectiveness of the method for modeling under the ATC framework using variable-fidelity models is studied. Overall results show the methods introduced in this paper provide an effective way of improving computational efficiency of the ATC method based on variable-fidelity simulation models.

An Enhanced Memetic Algorithm for Single-Objective Bilevel Optimization Problems

2017 ◽  
Vol 25 (4) ◽  
pp. 607-642 ◽  
Author(s):  
Md Monjurul Islam ◽  
Hemant Kumar Singh ◽  
Tapabrata Ray ◽  
Ankur Sinha
Keyword(s):  
Local Search ◽  
Memetic Algorithm ◽  
Optimization Problems ◽  
Computational Cost ◽  
Real Life ◽  
Practical Interest ◽  
Bilevel Optimization ◽  
Standard Test ◽  
Computational Effort ◽  
Test Problems

Bilevel optimization, as the name reflects, deals with optimization at two interconnected hierarchical levels. The aim is to identify the optimum of an upper-level  leader problem, subject to the optimality of a lower-level follower problem. Several problems from the domain of engineering, logistics, economics, and transportation have an inherent nested structure which requires them to be modeled as bilevel optimization problems. Increasing size and complexity of such problems has prompted active theoretical and practical interest in the design of efficient algorithms for bilevel optimization. Given the nested nature of bilevel problems, the computational effort (number of function evaluations) required to solve them is often quite high. In this article, we explore the use of a Memetic Algorithm (MA) to solve bilevel optimization problems. While MAs have been quite successful in solving single-level optimization problems, there have been relatively few studies exploring their potential for solving bilevel optimization problems. MAs essentially attempt to combine advantages of global and local search strategies to identify optimum solutions with low computational cost (function evaluations). The approach introduced in this article is a nested Bilevel Memetic Algorithm (BLMA). At both upper and lower levels, either a global or a local search method is used during different phases of the search. The performance of BLMA is presented on twenty-five standard test problems and two real-life applications. The results are compared with other established algorithms to demonstrate the efficacy of the proposed approach.

scholarly journals An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems

2021 ◽  
Author(s):  
Bennet Gebken ◽  
Sebastian Peitz
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Real Life ◽  
Practical Method ◽  
Descent Method ◽  
Bundle Method ◽  
Necessary Condition ◽  
Test Problems ◽  
Multiobjective Optimization Problems ◽  
Locally Lipschitz

AbstractWe present an efficient descent method for unconstrained, locally Lipschitz multiobjective optimization problems. The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth multiobjective optimization problems with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method with the multiobjective proximal bundle method by Mäkelä. The results indicate that our method is competitive while being easier to implement. Although the number of objective function evaluations is larger, the overall number of subgradient evaluations is smaller. Our method can be combined with a subdivision algorithm to compute entire Pareto sets of nonsmooth problems. Finally, we demonstrate how our method can be used for solving sparse optimization problems, which are present in many real-life applications.

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