scholarly journals Visualizing Exact and Approximated 3D Empirical Attainment Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Tea Tušar ◽  
Bogdan Filipič
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Low Cost ◽  
Optimal Solution ◽  
Maximum Intensity Projection ◽  
Maximum Intensity ◽  
Engineering Optimization ◽  
Trade Off ◽  
Visualization Techniques ◽  
Casting Optimization

Most real-world engineering optimization problems are inherently multiobjective, for example, searching for trade-off solutions of high quality and low cost. As no single optimal solution exists for such problems, multiobjective optimizers provide sets of optimal (or near-optimal) trade-off solutions to choose from. The empirical attainment function (EAF) is a powerful tool that can be used to analyze and compare the performance of these optimizers. While the visualization of EAFs is rather straightforward in two objectives, the three-objective case presents a great challenge as we need to visualize a large number of 3D cuboids. This paper addresses the visualization of exact as well as approximated 3D EAF values and differences in these values provided by two competing multiobjective optimizers. We show that the exact EAFs can be visualized using slicing and maximum intensity projection (MIP), while the approximated EAFs can be visualized using slicing, MIP, and direct volume rendering. In addition, the paper demonstrates the use of the proposed visualization techniques on a steel casting optimization problem.

scholarly journals Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 12 ◽  
Author(s):  
Xiangkai Sun ◽  
Hongyong Fu ◽  
Jing Zeng
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Optimal Solution ◽  
Sufficient Optimality Conditions ◽  
Convex Constraint ◽  
Approximate Optimality Conditions ◽  
Necessary And Sufficient

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

An Interactive Neural Network for Constrained Multi-Objective Optimization with Application to the Design of Digital Filters

2012 ◽  
Vol 433-440 ◽  
pp. 2808-2816
Author(s):  
Jian Jin Zheng ◽  
You Shen Xia
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Computational Complexity ◽  
Digital Filters ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Single Objective

This paper presents a new interactive neural network for solving constrained multi-objective optimization problems. The constrained multi-objective optimization problem is reformulated into two constrained single objective optimization problems and two neural networks are designed to obtain the optimal weight and the optimal solution of the two optimization problems respectively. The proposed algorithm has a low computational complexity and is easy to be implemented. Moreover, the proposed algorithm is well applied to the design of digital filters. Computed results illustrate the good performance of the proposed algorithm.

scholarly journals Application Research on the Maintainability Allocation Method of a Certain Shooter Seat

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Liying Jin ◽  
Wensheng Wang ◽  
HouYong Shu ◽  
Xuemei Ma ◽  
Chenxing Liang ◽  
...  
Keyword(s):  
Optimization Problem ◽  
Optimal Solution ◽  
Pareto Optimal Solution ◽  
Allocation Scheme ◽  
Application Research ◽  
Product Lifecycle ◽  
Nsga Ii ◽  
Trade Off ◽  
Allocation Method ◽  
Mean Time

In view of the traditional maintainability allocation method for a certain shooter seat for maintainability allocation did not consider the lifecycle expense problem, the improved NSGA-II algorithm (iNSGA-II, for short) is adopted to establish a multiobjective comprehensive trade-off model for a certain shooter seat product lifecycle maintenance-related expenses and mean time to repair (MTTR, for short) and construct multiobjective optimization problem. The experimental results show that the Pareto optimal solution effectively solves the limitation of the traditional maintainability allocation method and then provides a basis for a certain shooter seat to obtain a reasonable maintainability allocation scheme. The superiority of the iNSGA-II algorithm to optimize the maintainability allocation of a certain shooter seat was verified by comparing it with the traditional maintainability allocation method.

Robust Structural Design Optimization Under Non-Probabilistic Uncertainties

2011 ◽  
Author(s):  
Jiantao Liu ◽  
Hae Chang Gea ◽  
Ping An Du
Keyword(s):  
Structural Optimization ◽  
Design Optimization ◽  
Structural Design ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Global Optimality ◽  
Worst Case ◽  
Probabilistic Uncertainty ◽  
Structural Design Optimization

Robust structural design optimization with non-probabilistic uncertainties is often formulated as a two-level optimization problem. The top level optimization problem is simply to minimize a specified objective function while the optimized solution at the second level solution is within bounds. The second level optimization problem is to find the worst case design under non-probabilistic uncertainty. Although the second level optimization problem is a non-convex problem, the global optimal solution must be assured in order to guarantee the solution robustness at the first level. In this paper, a new approach is proposed to solve the robust structural optimization problems with non-probabilistic uncertainties. The WCDO problems at the second level are solved directly by the monotonocity analysis and the global optimality is assured. Then, the robust structural optimization problem is reduced to a single level problem and can be easily solved by any gradient based method. To illustrate the proposed approach, truss examples with non-probabilistic uncertainties on stiffness and loading are presented.

Ultimate and Practical Limits for Counterweight Balancing of Six-Bar Linkages

2006 ◽  
Author(s):  
Bram Demeulenaere ◽  
Jan Swevers ◽  
Joris De Schutter
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Global Optimum ◽  
Great Efficiency ◽  
Convex Programs ◽  
Local Optima ◽  
Trade Off ◽  
Convex Constraint ◽  
Main Challenge ◽  
Dynamic Forces

The designer’s main challenge when counterweight balancing a linkage is to determine the counterweights that realize an optimal trade-off between the dynamic forces of interest. This problem is often formulated as an optimization problem that is generally nonlinear and therefore suffers from local optima. It has been shown earlier, however, that, through a proper parametrization of the counterweights, a convex program can be obtained. Convex programs are nonlinear optimization problems of which the global optimum is guaranteed to be found with great efficiency. The present paper extends this previous work in two respects: (i) the methodology is generalized from four-bar to planar N-bar (rigid) linkages and (ii) it is shown that requiring the counterweights to be realizable in practice can be cast as a convex constraint. Numerical results for a Watt six-bar linkage suggest much more balancing potential for six-bar linkages than for four-bar linkages.

scholarly journals A Fixed-Point Subgradient Splitting Method for Solving Constrained Convex Optimization Problems

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 377
Author(s):  
Nimit Nimana
Keyword(s):  
Fixed Point ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Splitting Method ◽  
Optimal Solution ◽  
Bilevel Optimization ◽  
Convergence Properties ◽  
Nonlinear Operators ◽  
Convex Optimization Problems ◽  
Feasible Points

In this work, we consider a bilevel optimization problem consisting of the minimizing sum of two convex functions in which one of them is a composition of a convex function and a nonzero linear transformation subject to the set of all feasible points represented in the form of common fixed-point sets of nonlinear operators. To find an optimal solution to the problem, we present a fixed-point subgradient splitting method and analyze convergence properties of the proposed method provided that some additional assumptions are imposed. We investigate the solving of some well known problems by using the proposed method. Finally, we present some numerical experiments for showing the effectiveness of the obtained theoretical result.

A Constraint Satisfaction Approach for Multi-Attribute Design Optimization Problems

1997 ◽  
Author(s):  
Joseph D’Ambrosio ◽  
Timothy Darr ◽  
William Birmingham
Keyword(s):  
Constraint Satisfaction ◽  
Value Function ◽  
Necessary Conditions ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Compact Representation ◽  
Constrained Optimization Problems ◽  
Design Problems ◽  
Trade Off ◽  
The Value Function

Abstract In this paper, we describe a multi-attribute domain CSP approach for solving a class of discrete, constrained, optimization problems. The multi-attribute domain CSP formulation provides a compact representation for design problems characterized by multiple, conflicting attributes. Design trade-off information is represented by a multi-attribute value function. Necessary conditions for an optimal solution, defined in terms of the value function, are represented as constraints. This provides a uniform problem-solving approach (constraint satisfaction) for identifying solutions that are both feasible and of high value. We present and characterize a consistency algorithm for this type of CSP.

scholarly journals Bayesian Optimization Based on K-Optimality

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 594 ◽  
Author(s):  
Liang Yan ◽  
Xiaojun Duan ◽  
Bowen Liu ◽  
Jin Xu
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Information Gain ◽  
Latin Hypercube Sampling ◽  
Bayesian Optimization ◽  
Expected Improvement ◽  
Gram Matrix ◽  
Trade Off ◽  
Posterior Variance ◽  
The Stability

Bayesian optimization (BO) based on the Gaussian process (GP) surrogate model has attracted extensive attention in the field of optimization and design of experiments (DoE). It usually faces two problems: the unstable GP prediction due to the ill-conditioned Gram matrix of the kernel and the difficulty of determining the trade-off parameter between exploitation and exploration. To solve these problems, we investigate the K-optimality, aiming at minimizing the condition number. Firstly, the Sequentially Bayesian K-optimal design (SBKO) is proposed to ensure the stability of the GP prediction, where the K-optimality is given as the acquisition function. We show that the SBKO reduces the integrated posterior variance and maximizes the hyper-parameters’ information gain simultaneously. Secondly, a K-optimal enhanced Bayesian Optimization (KO-BO) approach is given for the optimization problems, where the K-optimality is used to define the trade-off balance parameters which can be output automatically. Specifically, we focus our study on the K-optimal enhanced Expected Improvement algorithm (KO-EI). Numerical examples show that the SBKO generally outperforms the Monte Carlo, Latin hypercube sampling, and sequential DoE approaches by maximizing the posterior variance with the highest precision of prediction. Furthermore, the study of the optimization problem shows that the KO-EI method beats the classical EI method due to its higher convergence rate and smaller variance.

scholarly journals Applying Hybrid PSO to Optimize Directional Overcurrent Relay Coordination in Variable Network Topologies

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Ming-Ta Yang ◽  
An Liu
Keyword(s):  
Power Systems ◽  
Constrained Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Pso Algorithm ◽  
Test System ◽  
Dial Setting ◽  
Overcurrent Relays ◽  
Network Topologies

In power systems, determining the values of time dial setting (TDS) and the plug setting (PS) for directional overcurrent relays (DOCRs) is an extremely constrained optimization problem that has been previously described and solved as a nonlinear programming problem. Optimization coordination problems of near-end faults and far-end faults occurring simultaneously in circuits with various topologies, including fixed and variable network topologies, are considered in this study. The aim of this study was to apply the Nelder-Mead (NM) simplex search method and particle swarm optimization (PSO) to solve this optimization problem. The proposed NM-PSO method has the advantage of NM algorithm, with a quicker movement toward optimal solution, as well as the advantage of PSO algorithm in the ability to obtain globally optimal solution. Neither a conventional PSO nor the proposed NM-PSO method is capable of dealing with constrained optimization problems. Therefore, we use the gradient-based repair method embedded in a conventional PSO and the proposed NM-PSO. This study used an IEEE 8-bus test system as a case study to compare the convergence performance of the proposed NM-PSO method and a conventional PSO approach. The results demonstrate that a robust and optimal solution can be obtained efficiently by implementing the proposal.

scholarly journals An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

2021 ◽  
Author(s):  
Riley Badenbroek ◽  
Etienne de Klerk
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Cutting Plane ◽  
Mixed Integer ◽  
Cutting Plane Method ◽  
Analytic Center ◽  
Completely Positive ◽  
Plane Method ◽  
Copositive Cone

We propose an analytic center cutting plane method to determine whether a matrix is completely positive and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman et al. [Berman A, Dur M, Shaked-Monderer N (2015) Open problems in the theory of completely positive and copositive matrices. Electronic J. Linear Algebra 29(1):46–58]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia et al. [Xia W, Vera JC, Zuluaga LF (2020) Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1):40–56]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like [Formula: see text] for d × d matrices. The method is implemented in Julia and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl . Summary of Contribution: Completely positive matrices play an important role in operations research. They allow many NP-hard problems to be formulated as optimization problems over a proper cone, which enables them to benefit from the duality theory of convex programming. We propose an analytic center cutting plane method to determine whether a matrix is completely positive by solving an optimization problem over the copositive cone. In fact, we can use our method to solve any copositive optimization problem, provided we know the radius of a ball containing an optimal solution. We emphasize numerical performance and stability in developing this method. A software implementation in Julia is provided.

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