Traditional Robust Optimization Research Articles

Distribution-robust loss-averse optimization

Distribution-robust loss-averse optimization optimizes a nominal value with some protection against downside loss, under the assumption that only partial information on the underlying distribution is available. We herein present a general modeling framework for the distribution-robust loss-averse optimization problem. We provide an equivalent simpler formulation that usually permits a tractable solution procedure. We then explore the modeling framework’s relations with traditional robust optimization and mean-variance optimization. Additionally, we discuss extensions to stochastic linear programming.

Robust design with Variability Response Functions; an alternative approach

In this work a different Robust Design Optimization (RDO) approach is proposed implementing the concept of Variability Response Function (VRF), which is a function that when combined with the power spectral density of the stochastic field that models the system’s uncertainty, formulates an integral expression for the variance of the system’s response. The basic idea is to exploit a very well-known property of the VRF, which is its independence of the stochastic system parameters, in order to obtain global optima that depend only on the deterministic parameters of the problem. This way, optimal structural designs are achieved which are globally insensitive to uncertainties, that is to say they are free of the spectral-distribution characteristics of the stochastic fields modeling the uncertainties. This is achieved by setting in addition to the total material cost, the maximum VRF value as an objective function. The advantages of using the proposed methodology over traditional Robust Design Optimization are illustrated through an application to a frame-type structure where it is demonstrated that the designs achieved through classical RDO for a given stochastic field description are not optimal for a variation on the spectral properties of the random field modeling the system uncertainty, while optimal designs obtained with the VRF-based RDO remain optimum for the worst case scenario stochastic fields.

Robust optimization under correlated uncertainty: Formulations and computational study

The uncertainty set-induced robust optimization framework has received considerable attention in the past decades. It has been extensively studied in literature and applied to address various decision-making problems. However, existing robust optimization methods generally assume that the uncertain parameters are independent. As a result, the traditional robust optimization methods may lead to a conservative solution in practice when correlations between uncertain parameters exist. In this work, we present novel results on robust optimization under correlated uncertainties that appear in a single constraint. Robust counterpart optimization formulations are derived based on various types of uncertain sets. Numerical and application examples are studied to compare the performance of robust optimization by incorporating various levels of correlation. The results demonstrate that incorporating more accurate correlation into the robust optimization formulation can lead to less conservative robust solution.

Robust optimization of structural dynamic characteristics based on adaptive Kriging model and CNSGA

Dynamic characteristics greatly influence the comprehensive performance of a structure. But they are rarely included as objectives in traditional robust optimization of structures. In this study, a robust optimization model including both means and standard deviations of dynamic characteristic indices in the objective and constraint functions is constructed for improving the structural dynamic characteristics and reducing their fluctuations under uncertainty. Adaptive Kriging models are employed for the efficient computation of dynamic characteristics. An intelligent resampling technology is proposed to save computational costs and accelerate convergence of Kriging models, which takes full advantage of the test points for precision verification, the sample points within the local region of the biggest relative maximum absolute error and the near-optimal point to improve the global and local precision of Krigings. The high efficiency of proposed intelligent resampling technology is demonstrated by a numerical example. Finally, an efficient algorithm integrating adaptive Kriging models, Monte Carlo (MC) method, constrained non-dominated sorting genetic algorithm (CNSGA) is proposed to solve the robust optimization model of structural dynamic characteristics. Kriging models are interfaced with MC method to efficiently compute the fitness of individuals during CNSGA. The implementation of proposed methodology is explained in detail and highlighted by the robust optimization of a cone ring fixture with lots of circumferentially distributed holes in a large turbo generator aimed at moving its natural frequencies away from the exciting one. The comparison of the optimized design with the initial one demonstrates that the proposed methodology is feasible and applicable in engineering practice.

A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: III. Improving the Quality of Robust Solutions.

In this paper, we study the solution quality of robust optimization problems when they are used to approximate probabilistic constraints and propose a novel method to improve the quality. Two solution frameworks are first compared: (1) the traditional robust optimization framework which only uses the a priori probability bounds and (3) the approximation framework which uses the a posteriori probability bound. We illustrate that the traditional robust optimization method is computationally efficient but its solution is in general conservative. On the other hand, the a posteriori probability bound based method provides less conservative solution but it is computationally more difficult because a nonconvex optimization problem is solved. Based on the comparative study of the two methods, we propose a novel iterative solution framework which combines the advantage of the a priori bound and the a posteriori probability bound. The proposed method can improve the solution quality of traditional robust optimization framework without significantly increasing the computational effort. The effectiveness of the proposed method is illustrated through numerical examples and applications in planning and scheduling problems.

Adjustable Decisions for Reducing the Price of Robustness of Capacity Expansion Planning

This paper proposes and implements robust optimization methodologies for making investment decisions in the capacity expansion planning (CEP) of power systems in an uncertain environment. Uncertainties of fuel prices, demand, and transmission capacity are captured in an uncertainty set. With adjustable robust optimization (ARO), we represent all the decision variables as affine functions of multiple uncertain data. This adjustability of decisions provides that the ARO solution has significant less price of robustness than in traditional robust optimization (RO). ARO models uncertainty in terms of parameter ranges, called “uncertainty sets.” An attractive attribute of utilizing uncertainty sets is that they facilitate computational tractability when simulating scenarios with multiple uncertainties. We study the 40-year planning of a 5-region, 13-technology US energy portfolio. Results show that 1) by appropriately selecting the decision rules in ARO, the price of robustness can be significantly reduced while maintaining the same levels of robustness; and 2) the RO-based models maintain high levels of robustness even under operational conditions provided by data coming from larger sizes of the uncertainty sets.

Waterflooding optimization in uncertain geological scenarios

In conventional waterflooding of an oil field, feedback based optimal control technologies may enable higher oil recovery than with a conventional reactive strategy in which producers are closed based on water breakthrough. To compensate for the inherent geological uncertainties in an oil field, robust optimization has been suggested to improve and robustify optimal control strategies. In robust optimization of an oil reservoir, the water injection and production borehole pressures (bhp) are computed such that the predicted net present value (NPV) of an ensemble of permeability field realizations is maximized. In this paper, we both consider an open-loop optimization scenario, with no feedback, and a closed-loop optimization scenario. The closed-loop scenario is implemented in a moving horizon manner and feedback is obtained using an ensemble Kalman filter for estimation of the permeability field from the production data. For open-loop implementations, previous test case studies presented in the literature, show that a traditional robust optimization strategy (RO) gives a higher expected NPV with lower NPV standard deviation than a conventional reactive strategy. We present and study a test case where the opposite happen: The reactive strategy gives a higher expected NPV with a lower NPV standard deviation than the RO strategy. To improve the RO strategy, we propose a modified robust optimization strategy (modified RO) that can shut in uneconomical producer wells. This strategy inherits the features of both the reactive and the RO strategy. Simulations reveal that the modified RO strategy results in operations with larger returns and less risk than the reactive strategy, the RO strategy, and the certainty equivalent strategy. The returns are measured by the expected NPV and the risk is measured by the standard deviation of the NPV. In closed-loop optimization, we investigate and compare the performance of the RO strategy, the reactive strategy, and the certainty equivalent strategy. The certainty equivalent strategy is based on a single realization of the permeability field. It uses the mean of the ensemble as its permeability field. Simulations reveal that the RO strategy and the certainty equivalent strategy give a higher NPV compared to the reactive strategy. Surprisingly, the RO strategy and the certainty equivalent strategy give similar NPVs. Consequently, the certainty equivalent strategy is preferable in the closed-loop situation as it requires significantly less computational resources than the robust optimization strategy. The similarity of the certainty equivalent and the robust optimization based strategies for the closed-loop situation challenges the intuition of most reservoir engineers. Feedback reduces the uncertainty and this is the reason for the similar performance of the two strategies.

Dynamic Pricing Through Sampling Based Optimization

In this paper we develop an approach to dynamic pricing that combines ideas from data-driven and robust optimization to address the uncertain and dynamic aspects of the problem. In our setting, a firm offers multiple products to be sold over a fixed discrete time horizon. Each product sold consumes one or more resources, possibly sharing the same resources among different products. The firm is given a fixed initial inventory of these resources and cannot replenish this inventory during the selling season. We assume there is uncertainty about the demand seen by the firm for each product and seek to determine a robust and dynamic pricing strategy that maximizes revenue over the time horizon. While the traditional robust optimization models are tractable, they give rise to static policies and are often too conservative. The main contribution of this paper is the exploration of closed-loop pricing policies for different robust objectives, such as MaxMin, MinMax Regret and MaxMin Ratio. We introduce a sampling based optimization approach that can solve this problem in a tractable way, with a confidence level and a robustness level based on the number of samples used. We will show how this methodology can be used for data-driven pricing or adapted for a random sampling optimization approach when limited information is known about the demand uncertainty. Finally, we compare the revenue performance of the different models using numerical simulations, exploring the behavior of each model under different sample sizes and sampling distributions.

Development of a variance prioritized multiresponse robust design framework for quality improvement

PurposeRobust design is a well‐known quality improvement method that focuses on building quality into the design of products and services. Yet, most well established robust design models only consider a single performance measure and their prioritization schemes do not always address the inherent goal of robust design. This paper aims to propose a new robust design method for multiple quality characteristics where the goal is to first reduce the variability of the system under investigation and then attempt to locate the mean at the desired target value.Design/methodology/approachThe paper investigates the use of a response surface approach and a sequential optimization strategy to create a flexible and structured method for modeling multiresponse problems in the context of robust design. Nonlinear programming is used as an optimization tool.FindingsThe proposed methodology is demonstrated through a numerical example. The results obtained from this example are compared to that of the traditional robust design method. For comparison purposes, the traditional robust design optimization models are reformulated within the nonlinear programming framework developed here. The proposed methodology provides enhanced optimal robust design solutions consistently.Originality/valueThis paper is perhaps the first study on the prioritized response robust design with the consideration of multiple quality characteristics. The findings and key observations of this paper will be of significant value to the quality and reliability engineering/management community.

Fast numerical methods for robust optimal design

A fast numerical approach for robust design optimization is presented. The core of the method is based on the state-of-the-art fast numerical methods for stochastic computations with parametric uncertainty. These methods employ generalized polynomial chaos (gPC) as a high-order representation for random quantities and a stochastic Galerkin (SG) or stochastic collocation (SC) approach to transform the original stochastic governing equations to a set of deterministic equations. The gPC-based SG and SC algorithms are able to produce highly accurate stochastic solutions with (much) reduced computational cost. It is demonstrated that they can serve as efficient forward problem solvers in robust design problems. Possible alternative definitions for robustness are also discussed. Traditional robust optimization seeks to minimize the variance (or standard deviation) of the response function while optimizing its mean. It can be shown that although variance can be used as a measure of uncertainty, it is a weak measure and may not fully reflect the output variability. Subsequently a strong measure in terms of the sensitivity derivatives of the response function is proposed as an alternative robust optimization definition. Numerical examples are provided to demonstrate the efficiency of the gPC-based algorithms, in both the traditional weak measure and the newly proposed strong measure.