Monotonicity and Active Set Strategies in Probabilistic Design Optimization

2006 ◽  
Vol 128 (4) ◽  
pp. 893-900 ◽  
Author(s):  
Kuei-Yuan Chan ◽  
Steven Skerlos ◽  
Panos Y. Papalambros
Keyword(s):  
Design Optimization ◽  
Computational Cost ◽  
Inequality Constraints ◽  
Probabilistic Constraints ◽  
Programming Algorithm ◽  
Probabilistic Design ◽  
Active Set ◽  
Probability Metrics ◽  
Active Set Strategies ◽  
Monotonicity Analysis

Probabilistic design optimization addresses the presence of uncertainty in design problems. Extensive studies on reliability-based design optimization, i.e., problems with random variables and probabilistic constraints, have focused on improving computational efficiency of estimating values for the probabilistic functions. In the presence of many probabilistic inequality constraints, computational costs can be reduced if probabilistic values are computed only for constraints that are known to be active or likely active. This article presents an extension of monotonicity analysis concepts from deterministic problems to probabilistic ones, based on the fact that several probability metrics are monotonic transformations. These concepts can be used to construct active set strategies that reduce the computational cost associated with handling inequality constraints, similarly to the deterministic case. Such a strategy is presented as part of a sequential linear programming algorithm along with numerical examples.

Monotonicity and Active Set Strategies in Probabilistic Design Optimization

2005 ◽  
Author(s):  
Kuei-Yuan Chan ◽  
Steven J. Skerlos ◽  
Panos Y. Papalambros
Keyword(s):  
Design Optimization ◽  
Computational Cost ◽  
Inequality Constraints ◽  
Probabilistic Constraints ◽  
Programming Algorithm ◽  
Probabilistic Design ◽  
Active Set ◽  
Probability Metrics ◽  
Active Set Strategies ◽  
Monotonicity Analysis

Probabilistic design optimization addresses the presence of uncertainty in design problems. Extensive studies on Reliability-Based Design Optimization (RBDO), i.e., problems with random variables and probabilistic constraints, have focused on improving computational efficiency of estimating values for the probabilistic functions. In the presence of many probabilistic inequality constraints, computational costs can be reduced if probabilistic values are computed only for constraints that are known to be active or likely active. This article presents an extension of monotonicity analysis concepts from deterministic problems to probabilistic ones, based on the fact that several probability metrics are monotonic transformations. These concepts can be used to construct active set strategies that reduce the computational cost associated with handling inequality constraints, similarly to the deterministic case. Such a strategy is presented as part of a sequential linear programming algorithm along with a numerical example.

Probabilistic Designs of Air-Bearing Surface on Manufacturing Tolerances

2004 ◽  
Author(s):  
Sang-Joon Yoon ◽  
Dong-Hoon Choi
Keyword(s):  
Design Optimization ◽  
Reliability Analysis ◽  
Computation Time ◽  
Probabilistic Constraints ◽  
Air Bearing ◽  
Probabilistic Design ◽  
Bearing Surface ◽  
Deterministic Optimization ◽  
Design Variables ◽  
Conventional Optimization

The focus in this paper is to automatically design the air-bearing surface (ABS) considering the randomness of its geometry as an uncertainty of design variables. Designs determined by the conventional optimization could only provide a low level of confidence in practical products due to the existence of uncertainties in either engineering simulations or manufacturing processes. This calls for a reliability-based approach to the design optimization, which increases product or process quality by addressing randomness or stochastic properties of design problems. In this study, a probabilistic design problem is formulated considering the reliability analysis which is employed to estimate how the fabrication tolerances of individual slider parameters affect the final flying attitude tolerances. The proposed approach first solves the deterministic optimization problem. Beginning with this solution, the reliability-based design optimization (RBDO) is continued with the probabilistic constraints affected by the random variables. Probabilistic constraints overriding the constraints of the deterministic optimization attempt to drive the design to a reliability solution with minimum increase in the objective. The simulation results of the probabilistic design are directly compared with the values of the initial design and the results of the deterministic optimum design, respectively. In order to show the effectiveness of the proposed approach, the reliability analyses by the Monte Carlo simulation are carried out. And the results demonstrate how efficient the proposed approach is, considering the enormous computation time of the reliability analysis.

Structural Topology Design Considering Reliability

2005 ◽  
Vol 297-300 ◽  
pp. 1901-1906 ◽  
Author(s):  
Seung Jae Min ◽  
Seung Hyun Bang
Keyword(s):  
Topology Optimization ◽  
Design Optimization ◽  
Optimization Problem ◽  
Performance Measure ◽  
Topology Design ◽  
Optimization Process ◽  
Programming Algorithm ◽  
Probabilistic Design ◽  
Most Probable Point ◽  
Design Variables

In the design optimization process design variables are selected in the deterministic way though those have uncertainties in nature. To consider variances in design variables reliability-based design optimization problem is formulated by introducing the probability distribution function. The concept of reliability has been applied to the topology optimization based on a reliability index approach or a performance measure approach. Since these approaches, called double-loop singlevariable approach, requires the nested optimization problem to obtain the most probable point in the probabilistic design domain, the time for the entire process makes the practical use infeasible. In this work, new reliability-based topology optimization method is proposed by utilizing single-loop singlevariable approach, which approximates searching the most probable point analytically, to reduce the time cost and dealing with several constraints to handle practical design requirements. The density method in topology optimization including SLP (Sequential Linear Programming) algorithm is implemented with object-oriented programming. To examine uncertainties in the topology design of a structure, the modulus of elasticity of the material and applied loadings are considered as probabilistic design variables. The results of a design example show that the proposed method provides efficiency curtailing the time for the optimization process and accuracy satisfying the specified reliability.

Probabilistic Design Optimization of Frequency Dispersion for Rotating Blades

2010 ◽  
Author(s):  
Liqiang An ◽  
G. Gary Wang ◽  
Zhangqi Wang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Design Optimization ◽  
Optimization Problem ◽  
Frequency Dispersion ◽  
Probabilistic Constraints ◽  
Design Parameters ◽  
Probabilistic Design ◽  
Rotating Blades ◽  
Element Method

In this paper, a probabilistic design optimization method based on finite element method is proposed to calculate the variability of design parameters subject to a specified dispersion of natural frequencies of rotating blades. The element stiffness and mass matrices are derived using a two-stage finite element method and numerical integration. Based on the perturbation technology, the sensitivity of the frequencies, as well as relationship between the frequency dispersion and the coefficient of variability (CV) of the design parameters can be obtained. Such sensitivity information is then used to convert the probabilistic design optimization problem into a deterministic optimization problem. Two case studies are given to illustrate the proposed method. From the results, it is concluded that rotation of blade changes the sensitivity of CV to the design parameters considered, and using the proposed method can transform the probabilistic constraints to deterministic constraints.

An Automated Procedure for Local Monotonicity Analysis

1984 ◽  
Vol 106 (1) ◽  
pp. 82-89 ◽  
Author(s):  
S. Azarm ◽  
P. Papalambros
Keyword(s):  
Design Optimization ◽  
Active Set ◽  
Optimization Program ◽  
Active Set Strategy ◽  
Active Constraints ◽  
Automated Algorithm ◽  
Active Sets ◽  
Local Monotonicity ◽  
Monotonicity Analysis ◽  
Selection Of

A strategy for selecting active constraints in a design optimization program is implemented computationally. The strategy uses local monotonicity information to iterate on the active set. A fully automated algorithm is developed with the aid of constrained derivatives and conventional search methods. Four design examples are presented, one of which demonstrates how global rules derived from monotonicity analysis can be included in the active set strategy to enhance the performance of the algorithm. The procedure is flexible, so that any available rules that can bias the selection of active sets may be included in the strategy.

Probabilistic Designs of Air-Bearing Surface on Manufacturing Tolerances

2005 ◽  
Vol 127 (1) ◽  
pp. 149-154 ◽  
Author(s):  
Sang-Joon Yoon ◽  
Dong-Hoon Choi
Keyword(s):  
Design Optimization ◽  
Reliability Analysis ◽  
Computation Time ◽  
Probabilistic Constraints ◽  
Air Bearing ◽  
Probabilistic Design ◽  
Bearing Surface ◽  
Deterministic Optimization ◽  
Design Variables ◽  
Conventional Optimization

The focus in this paper is to automatically design the air-bearing surface (ABS) considering the randomness of its geometry as an uncertainty of design variables. Designs determined by the conventional optimization could only provide a low level of confidence in practical products due to the existence of uncertainties in either engineering simulations or manufacturing processes. This calls for a reliability-based approach to the design optimization, which increases product or process quality by addressing randomness or stochastic properties of design problems. In this study, a probabilistic design problem is formulated considering the reliability analysis which is employed to estimate how the fabrication tolerances of individual slider parameters affect the final flying attitude tolerances. The proposed approach first solves the deterministic optimization problem. Beginning with this solution, the reliability-based design optimization (RBDO) is continued with the probabilistic constraints affected by the random variables. Probabilistic constraints overriding the constraints of the deterministic optimization attempt to drive the design to a reliability solution with a minimum increase in the objective. The simulation results of the probabilistic design are directly compared with the values of the initial design and the results of the deterministic optimum design, respectively. In order to show the effectiveness of the proposed approach, the reliability analyses by the Monte Carlo simulation are carried out. And the results demonstrate how efficient the proposed approach is, considering the enormous computation time of the reliability analysis.

scholarly journals An active set sequential quadratic programming algorithm for nonlinear optimisation

2006 ◽  
Vol 74 (1) ◽  
pp. 69-83
Author(s):  
Qing-Jie Hu ◽  
Yun-Hai Xiao ◽  
Y. Chen
Keyword(s):  
Quadratic Programming ◽  
Sequential Quadratic Programming ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Least Square ◽  
Programming Algorithm ◽  
Active Set ◽  
Nonlinear Inequality ◽  
Nonlinear Inequality Constraints ◽  
Sequential Quadratic Programming Algorithm

In this paper, we have proposed an active set feasible sequential quadratic programming algorithm for nonlinear inequality constraints optimization problems. At each iteration of the proposed algorithm, a feasible direction of descent is obtained by solving a reduced quadratic programming subproblem. To overcome the Maratos effect, a higher-order correction direction is obtained by solving a reduced least square problem. The algorithm is proved to be globally convergent and superlinearly convergent under some mild conditions without strict complementarity.

Optimal Design With Non-Normally Distributed Random Parameters, Conditional Probability, and Joint Constraint Reliabilities: A Case Study in Vehicle Emissions Regulations to Achieve Ambient Air Quality Standards

2006 ◽  
Author(s):  
Kuei-Yuan Chan ◽  
Steven J. Skerlos ◽  
Panos Y. Papalambros
Keyword(s):  
Design Optimization ◽  
High Reliability ◽  
Ambient Air ◽  
Probabilistic Constraints ◽  
Model Complexity ◽  
Bayesian Probability ◽  
Vehicle Speed ◽  
Random Parameters ◽  
Active Set ◽  
Regulatory Constraints

Making appropriate environmental policy decisions requires considering various sources of uncertainty. An air pollution example is formulated as a design optimization problem with probabilistic constraints, also referred to as reliability-based design optimization (RBDO). Environmental applications with a large number of constraints and significant model complexity present special challenges. In this paper an efficient active set strategy is integrated with a reliability contour surface approach to solve probabilistic problems with non-normal variable probability distributions. Discrete random parameters, which result in Bayesian probability, are also present and they are incorporated using delta function approximations. Joint constraint reliability that considers satisfying all regulatory constraints is also discussed. A demonstration example of setting the optimal vehicle speed limit while maintaining high reliability for CO and NOx standards of a residential area near two highway systems is included.

An Active Set Sequential Linearization Algorithm for Nonlinear Design Optimization

1987 ◽  
Author(s):  
N. Tzannetakis ◽  
P. Y. Papalambros
Keyword(s):  
Design Optimization ◽  
Optimization Problems ◽  
Programming Algorithm ◽  
Linear Programs ◽  
Chemical Process Design ◽  
Active Set ◽  
Active Set Strategy ◽  
Nonlinear Design ◽  
Working Set ◽  
Sequential Linearization

Abstract Solution of nonlinear design optimization problems via a sequence of linear programs is regaining attention for solving certain model classes, such as in structural design and chemical process design. An active set strategy modification of an algorithm by Palacios-Gomez is presented. A special interior linear programming algorithm with active set strategy is used also for solving the subproblem and generating the working set of the outer iterations. Examples are included.

A Case for a Knowledge-Based Active Set Strategy

1984 ◽  
Vol 106 (1) ◽  
pp. 77-81 ◽  
Author(s):  
S. Azarm ◽  
P. Papalambros
Keyword(s):  
Design Optimization ◽  
Global Knowledge ◽  
Active Set ◽  
Optimization Program ◽  
Active Set Strategy ◽  
Active Constraints ◽  
Knowledge Based ◽  
The One ◽  
Monotonicity Analysis

A strategy for selecting active constraints in a design optimization program is proposed. The strategy differs from previous ones in that it suggests a combination of local and global knowledge. This knowledge may be analytical in nature, such as the one provided by monotonicity analysis. But it may also be provided by an expert. The strategy is proposed as a first attempt toward development of knowledge-based iteration procedures for optimization.

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