Remarks on Sufficiency of Constraint-Bound Solutions in Optimal Design

1993 ◽  
Vol 115 (3) ◽  
pp. 374-379
Author(s):  
P. Y. Papalambros
Keyword(s):  
Nonlinear Programming ◽  
Optimal Design ◽  
Active Set ◽  
Design Problems ◽  
Local Optima ◽  
Active Set Strategy ◽  
Active Constraints ◽  
Trade Offs ◽  
Design Variables ◽  
Monotonicity Analysis

Early preliminary models for optimal design problems in nonlinear programming formulations often have solutions that are constraint-bound points, i.e., the number of active constraints equals the number of design variables. Models leading to such solutions will typically offer little insight to design trade-offs, and it is desirable to identify them early in order to revise the model or to exclude the points from an active set strategy. Application of monotonicity analysis can quickly identify constraint-bound candidate solutions but not always prove their optimality. This article discusses some conditions under which these points are in fact global or local optima.

Remarks on Sufficiency of Constraint-Bound Solutions in Optimal Design

1988 ◽  
Author(s):  
P. Y. Papalambros
Keyword(s):  
Nonlinear Programming ◽  
Optimal Design ◽  
Design Problems ◽  
Local Optima ◽  
Solution Strategies ◽  
Active Constraints ◽  
Active Sets ◽  
Trade Offs ◽  
Design Variables ◽  
Monotonicity Analysis

Abstract Solution strategies for optimal design problems in nonlinear programming formulations may require verification of optimality for constraint-bound points. These points are candidate solutions where the number of active constraints is equal to the number of design variables. Models leading to such solutions will typically offer little insight to design trade-offs and it would be desirable to identify them early, or exclude them in a strategy using active sets. Potential constrained-bound solutions are usually identified based on the principles of monotonicity analysis. This article discusses some cases where these points are in fact global or local optima.

PRIMA: A Production-Based Implicit Elimination System for Monotonicity Analysis of Optimal Design Models

1991 ◽  
Vol 113 (4) ◽  
pp. 408-415 ◽  
Author(s):  
J. R. Rao ◽  
P. Y. Papalambros
Keyword(s):  
Optimal Design ◽  
Heuristic Search ◽  
Programming Environment ◽  
Initial Model ◽  
Active Set ◽  
Design Models ◽  
Active Set Strategy ◽  
Knowledge Based ◽  
Reasoning System ◽  
Monotonicity Analysis

Monotonicity analysis is a useful method for analyzing optimal design models prior to numerical computation. Much of the information required for such analysis is represented in the monotonicity table. Rigorous procedures using the monotonicity principles and the implicit function theorem have been combined with heuristics, to extract additional constraint activity knowledge based only on the information contained in the monotonicity table. PRIMA is a production system implemented in the OPS5 programming environment. The system receives as input the monotonicity table of the initial model and derives global facts about boundedness and constraint activity by heuristic search of sequences of successively reduced models. Such reduction is obtained by implicit elimination of active constraints. Global facts generated automatically by this reasoning system can be used either for a global solution, or for a combined local-global active set strategy.

Optimal Design of Product Family Throughout Commonalization, Customization and Lineup Arrangement

2008 ◽  
Author(s):  
Kikuo Fujita ◽  
Ryota Akai
Keyword(s):  
Optimal Design ◽  
Design Problem ◽  
Product Family ◽  
Feasible Region ◽  
Product Family Design ◽  
Active Set ◽  
Active Set Strategy ◽  
Design Variables ◽  
Design Paradigm

Product family design is a framework for effectively and efficiently meeting with spread customers’ needs by sharing components or modules across a series of products. This paper systematizes product family design toward its extension to throughout consideration of commonalization, customization and lineup arrangement under the optimal design paradigm. That is, commonalization is viewed as the operation that restricts the feasible region by fixing a set of design variables related to commonalized components or modules against later customization and final lineup offered to customers. Customization is viewed as the operation that arranges lineup by adjusting another set of design variables related to reserved freedom for customers’ needs. Their mutual and bi-directional relationships must be a matter of optimal design. This paper discusses the mathematical fundamentals of optimal product family design throughout commonalization, customization and lineup arrangement under active set strategy, and demonstrates a case study with a design problem of centrifugal compressors for showing the meaning of throughout optimal design.

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.

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.

scholarly journals Tailoring of Composite Links for Optimal Damped Elasto-Dynamic Performance

1989 ◽  
Author(s):  
D. A. Saravanos ◽  
C. C. Chamis
Keyword(s):  
Optimal Design ◽  
Dynamic Stiffness ◽  
Dynamic Performance ◽  
Performance Criteria ◽  
Modal Damping ◽  
Trade Offs ◽  
Design Variables ◽  
Material Systems ◽  
Structural Composite ◽  
Selective Minimization

Abstract A method is developed for the optimal design of composite links based on dynamic performance criteria directly related to structural modal damping and dynamic stiffness. An integrated mechanics theory correlates structural composite damping to the parameters of basic composite material systems, laminate parameters, link shape, and modal deformations. The inclusion of modal properties allows the selective minimization of vibrations associated with specific modes. Ply angles and fiber volumes are tailored to obtain optimal combinations of damping and stiffness. Applications to simple composite links indicate wide margins for trade-offs and illustrate the importance of various design variables to the optimal design.

Implementation of Semi-Heuristic Reasoning for Boundedness Analysis of Design Optimization Models

1987 ◽  
Author(s):  
J. R. J. Rao ◽  
P. Y. Papalambros
Keyword(s):  
Optimal Design ◽  
Heuristic Search ◽  
Optimization Models ◽  
Programming Environment ◽  
Initial Model ◽  
Active Set ◽  
Design Models ◽  
Active Set Strategy ◽  
Reasoning System ◽  
Global Boundedness

Abstract A production system performing global boundedness analysis of optimal design models has been implemented in the OPS5 programming environment. The system receives as input an initial model monotonicity table and derives global facts about boundedness and constraint activity using monotonicity principles. Additional facts may be discovered by heuristic search of implicit elimination sequences that examine boundedness of reduced models with active constraints eliminated. The global facts generated automatically by this reasoning system can be used either for a global solution, or for a combined local-global active set strategy.

A Framework for Algorithms in Globally Optimal Design

1989 ◽  
Vol 111 (3) ◽  
pp. 353-360 ◽  
Author(s):  
P. Hansen ◽  
B. Jaumard ◽  
S. H. Lu
Keyword(s):  
Combinatorial Optimization ◽  
Optimal Design ◽  
Branch And Bound ◽  
General Framework ◽  
Ad Hoc ◽  
Design Problems ◽  
Branch And Bound Algorithms ◽  
Monotonicity Analysis

Many problems of globally optimal design have been solved in the literature using monotonicity analysis and a variety of tests, often applied in an ad hoc way. These tests are developed here, expressed mathematically and classified according to the conclusions they yield. Moreover, many new tests, similar to those used in combinatorial optimization, are presented. Finally, a general framework is proposed in which branch-and-bound algorithms for globally optimal design problems can be expressed.

Robust Optimal Design for Worst-Case Tolerances

1994 ◽  
Vol 116 (4) ◽  
pp. 1019-1025 ◽  
Author(s):  
G. Emch ◽  
A. Parkinson
Keyword(s):  
Nonlinear Programming ◽  
Optimal Design ◽  
Worst Case ◽  
New Approach ◽  
Robust Designs ◽  
Design Variables ◽  
Robust Optimal Design ◽  
Programming Techniques ◽  
Engineering Models ◽  
Analytical Tools

Engineering models can and should be used to understand the effects of variability on a design. When variability is ignored, brittle designs can result that will not function properly or that will fail in service. By contrast, robust designs function properly even when subjected to off-nominal conditions. There is a need for better analytical tools to help engineers develop robust designs. In this paper we present a new approach for developing designs that are robust to variability induced by worst-case tolerances. An advantage of this approach is that tolerances may be placed on any or all model inputs, whether design variables or parameters. The method adapts nonlinear programming techniques in order to determine how a design should be modified to account for variability. We tested the method under relatively severe conditions on 13 problems, with excellent results. Using this approach, a designer can account for the effects of worst-case tolerances, making it possible to build robustness into an engineering design.

Activity Analysis: Simplifying Optimal Design Problems Through Qualitative Partitioning

1995 ◽  
Author(s):  
Brian C. Williams ◽  
Jonathan Cagan
Keyword(s):  
Numerical Analysis ◽  
Optimal Design ◽  
Design Problem ◽  
Optimization Problem ◽  
Activity Analysis ◽  
Design Problems ◽  
Gradient Based ◽  
Monotonicity Analysis

Abstract Activity analysis is introduced as a means to strategically cut away subspaces of a design problem that can quickly be ruled out as suboptimal. This results in focused regions of the space in which additional symbolic or numerical analysis can take place. Activity analysis is derived from a qualitative abstraction of the Karush-Kuhn-Tucker conditions of optimality, used to partition an optimization problem into regions which are nonstationary and qualitatively stationary (qstationary). Activity analysis draws from the fields of gradient-based optimization, conflict-based approaches of combinatorial satisficing search, and monotonicity analysis.

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