Robust Optimum Solution Research Articles

Bi-Objective Surrogate Feasibility Robust Design Optimization Utilizing Expected Non-Dominated Improvement With Relaxation

Abstract Engineering design optimization problems often have two competing objectives as well as uncertainty. For these problems, quite often there is interest in obtaining feasibly robust optimum solutions. Feasibly robust here refers to solutions that are feasible under all uncertain conditions. In general, obtaining bi-objective feasibly robust solutions can be computationally expensive, even more so when the functions to evaluate are themselves computationally intensive. Although surrogates have been utilized to decrease the computational costs of such problems, there is limited usage of Bayesian frameworks on problems of multi-objective optimization under interval uncertainty. This article seeks to formulate an approach for the solution of these problems via the expected improvement Bayesian acquisition function. In this paper, a method is developed for iteratively relaxing the solutions to facilitate convergence to a set of non-dominated, robust optimal solutions. Additionally, a variation of the bi-objective expected improvement criterion is proposed to encourage variety and density of the robust bi-objective non-dominated solutions. Several examples are tested and compared against other bi-objective robust optimization approaches with surrogate utilization. It is shown that the proposed method performs well at finding robustly optimized feasible solutions with limited function evaluations.

Surrogate feasibility testing–cutting for single-objective robust optimization under interval uncertainty

A robust optimum solution is defined as one that is optimum and maintains its feasibility regardless of the values taken by uncertain parameters within a bounded interval. Solving for a robust optimum solution, however, can be computationally costly. To reduce the computational cost, a new robust optimization method, surrogate feasibility testing–cutting robust optimization (SFTC-RO), is proposed that solves two optimization subproblems. In the first subproblem, a scenario robust optimization problem is solved using a local optimization technique. In the second subproblem, a feasibility check and constraint cut of the feasible domain are performed via operations on a surrogate model of the constraints. The two subproblems are iteratively solved until convergence at a robust optimum solution. Results from numerical and engineering examples show that on average the proposed method can obtain robust optimal solutions at a lower computational cost and with better scalability than an existing method from the literature.

Studying the Impact of Initialization for Population-Based Algorithms with Low-Discrepancy Sequences

To solve different kinds of optimization challenges, meta-heuristic algorithms have been extensively used. Population initialization plays a prominent role in meta-heuristic algorithms for the problem of optimization. These algorithms can affect convergence to identify a robust optimum solution. To investigate the effectiveness of diversity, many scholars have a focus on the reliability and quality of meta-heuristic algorithms for enhancement. To initialize the population in the search space, this dissertation proposes three new low discrepancy sequences for population initialization instead of uniform distribution called the WELL sequence, Knuth sequence, and Torus sequence. This paper also introduces a detailed survey of the different initialization methods of PSO and DE based on quasi-random sequence families such as the Sobol sequence, Halton sequence, and uniform random distribution. For well-known benchmark test problems and learning of artificial neural network, the proposed methods for PSO (TO-PSO, KN-PSO, and WE-PSO), BA (BA-TO, BA-WE, and BA-KN), and DE (DE-TO, DE-WE, and DE-KN) have been evaluated. The synthesis of our strategies demonstrates promising success over uniform random numbers using low discrepancy sequences. The experimental findings indicate that the initialization based on low discrepancy sequences is exceptionally stronger than the uniform random number. Furthermore, our work outlines the profound effects on convergence and heterogeneity of the proposed methodology. It is expected that a comparative simulation survey of the low discrepancy sequence would be beneficial for the investigator to analyze the meta-heuristic algorithms in detail.

Approach to robust multi-objective optimization and probabilistic analysis: the ROPAR algorithm

AbstractThis paper considers the problem of robust optimization, and presents the technique called Robust Optimization and Probabilistic Analysis of Robustness (ROPAR). It has been developed for finding robust optimum solutions of a particular class in model-based multi-objective optimization (MOO) problems (i.e. when the objective function is not known analytically), where some of the parameters or inputs to this model are assumed to be uncertain. A Monte Carlo simulation framework is used. It can be straightforwardly implemented in a distributed computing environment which allows the results to be obtained relatively fast. The technique is exemplified in the two case studies: (a) a benchmark problem commonly used to test MOO algorithms (a version of the ZDT1 function); and (b) a design problem of a simple storm drainage system, where the uncertainty is associated with design rainfall events. It is shown that the design found by ROPAR can adequately cope with these uncertainties. The approach can be useful for assisting in a wide range of risk-based decisions.

Robust Optimum and Near-Optimum Beamformers for Decode-and-Forward Full-Duplex Multi-Antenna Relay With Self-Energy Recycling

In this paper, we consider the full-duplex decode-and-forward wireless-powered relaying system, which employs energy harvesting protocol with power splitting. The robust joint optimum relay transmit beamformer and power splitting factor are obtained for the quality of service (QoS)-aware problem for the first time in the literature. The optimum solution is found by analyzing the Karush–Kuhn–Tucker conditions, thanks to the effective reformulation of the problem in an equivalent and simplified manner. In addition, the signal-to-interference-plus-noise ratio (SINR) maximization problem is investigated in order to find the robust optimum solution. The simulation results verify the optimality of the proposed method compared with the sub-optimum one which is presented by Zhao et al. . In the next part of this paper, the considered system is generalized by employing multiple receive antennas at the relay. Both QoS-aware and SINR maximization problems are considered. The near-optimum relay transmit and receive beamformers as well as power splitting factor are found by optimizing the variables alternately. First, transmit beamformer and power splitting factor are found optimally for a given initial receive beamformer. Then, the optimum receive beamformer is obtained. Relay with multiple-receive antennas is shown to perform better than the single receive antenna relay in terms of SINR and transmission power.

Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel

Deterministic optimization has been successfully applied to a range of design problems involving foam-filled thin-walled structures, and to some extent gained significant confidence for the applications of such structures in automotive, aerospace, transportation and defense industries. However, the conventional deterministic design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, a robust design methodology is presented in this paper to address the effects of parametric uncertainties of foam-filled thin-walled structure on design optimization, in which different sigma criteria are adopted to measure the variations. The Kriging modeling technique is used to construct the corresponding surrogate models of mean and standard deviation for different crashworthiness criteria. A sequential sampling approach is introduced to improve the fitness accuracy of these surrogate models. Finally, a gradient-based sequential quadratic program (SQP) method is employed from 20 different initial points to obtain a quasi-global robust optimum solution. The optimal solutions were verified by using the Monte Carlo simulation. The results show that the presented robust optimization method is fairly effective and efficient, the crashworthiness and robustness of the foam-filled thin-walled structure can be improved significantly.

Adaptive gradient-assisted robust design optimization under interval uncertainty

Robust optimization techniques attempt to find a solution that is both optimum and relatively insensitive to input uncertainty. In general, these techniques are computationally more expensive than their deterministic counterparts. In this article two new robust optimization methods are presented. The first method is called gradient-assisted robust optimization (GARO). In GARO, a robust optimization problem is first converted to a deterministic one by using a gradient-based approximation technique. After solving this deterministic problem, the solution robustness and the accuracy of the approximation are checked. If the accuracy meets a threshold, a robust optimum solution is found; otherwise, the approximation is adaptively modified until the threshold is met and a solution, if it exists, is obtained. The second method is a faster version of GARO called quasi-concave gradient-assisted robust optimization (QC-GARO). QC-GARO is for problems with quasi-concave objective and constraint functions. The difference between GARO and QC-GARO is in the way that they check the approximation accuracy. Both GARO and QC-GARO methods are applied to a set of six engineering design test problems and the results are compared with a few related previous methods. It was found that, compared to the methods considered, GARO could solve all test problems but with a higher computational effort compared to QC-GARO. However, QC-GARO was computationally much faster when it was able to solve the problems.

Interval Uncertainty-Based Robust Optimization for Convex and Non-Convex Quadratic Programs with Applications in Network Infrastructure Planning

Planning infrastructure networks such as roads, pipelines, waterways, power lines and telecommunication systems, require estimations on the future demand as well as other uncertain factors such as operating costs, degradation rates, or the like. When trying to construct infrastructure that is either optimal from a social welfare or profit perspective (depending on a public or private sector focus), typically researchers treat the uncertainties in the problem by using robust optimization methods. The goal of robust optimization is to find optimal solutions that are relatively insensitive to uncertain factors. This paper presents an efficient and tractable approach for finding robust optimum solutions to linear and, more importantly, quadratic programming problems with interval uncertainty using a worst case analysis. For linear, mixed-integer linear, and mixed-integer problems with quadratic objective and constraint functions, our robust formulations have the same complexity and tractability as their deterministic counterparts. Numerous examples with differing difficulties and complexities, especially with selected ones on network planning/operations problems, have been tested to demonstrate the viability of the proposed approach. The results show that the computational effort of the proposed approach, in terms of the number of function calls, for the robust problems is comparable to or even better than that of deterministic problems in some cases.

New Method for Uncertainty Analysis in Multidisciplinary System—Collaborative Uncertainty Analysis

Uncertainties in multidisciplinary design optimization (MDO) have a significant influence on the whole design process othe engineering system. The existing definitions of uncertainties are compared and classified. The uncertainties in multidisciplinarsystem and their propagation processes are described in detail. On the basis of the existing methods for uncertainty analysis, a newmethod for uncertainty analysis in multidisciplinary, i.e. collaborative uncertainty analysis (CUA), is proposed, and its particulacomputational process is described in detail. The CUA is an approach that utilizes Taylor approximation to evaluate the variance othe coupled variables and the system outputs, and then obtains the robust optimum solution through quadratic optimization. Thcomputational results of the example indicate that the collaborative uncertainty analysis method (CUA) is not only feasible but alshas very high accuracy, and it is an effective method to analyze the uncertainties in multidisciplinary system.

A response surface based hierarchical approach to multidisciplinary robust optimization design

A multidisciplinary robust optimization design framework, concurrent subsystem robust design optimization, is proposed to obtain robust optimum solution in the large-scaled and coupled system. In this framework, response surfaces in the form of artificial neural networks provide information pertaining to system performance characteristics, and individual subsystems engage in performing robust optimization design in parallel while communicating with the system level. This optimization approach incorporates uncertainty analysis and generates a global robust optimum solution in an iterative fashion. Two applications are considered, and the results demonstrate that the approach yields a reasonable robust optimum solution, and it is a potential and efficient multidisciplinary robust optimization approach .