Algorithmic Developments for Difficult Robust Discrete Optimization Problems

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

Surrogate duality for robust optimization

In this paper, we design a Branch and Bound algorithm based on interval arithmetic to address nonconvex robust optimization problems. This algorithm provides the exact global solution of such difficult problems arising in many real life applications. A code was developed in MatLab and was used to solve some robust nonconvex problems with few variables. This first numerical study shows the interest of this approach providing the global solution of such difficult robust nonconvex optimization problems.

Mars entry trajectory planning using robust optimization and uncertainty quantification

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

In this paper, we define a distributionally robust k-sum optimization problem as the problem of finding a solution that minimizes the worst-case expected sum of up to the k largest costs of the elements in the solution. The costs are random with a joint probability distribution that is not completely specified but rather assumed to be known to lie in a set of probability distributions. For k=1, this reduces to a distributionally robust bottleneck optimization problem while for k=n, this reduces to distributionally robust minimum sum optimization problem. Our main result is that for a Fréchet class of discrete marginal distributions with finite support, the distributionally robust k-sum combinatorial optimization problem is solvable in polynomial time if the deterministic minimum sum problem is solvable in polynomial time. This extends the result of Punnen and Aneja [On k-sum optimization, Oper. Res. Lett. 18(5)(1996), pp. 233–236] from the deterministic to the robust case. We show that this choice of the set of distributions helps preserve the submodularity of the k-sum objective function which is an useful structural property for optimization problems.

Fuzzy systems (FSs) have been extensively applied to automatic control, pattern recognition, and decision analysis. However, a common bottleneck is encountered in the derivation of fuzzy rules, which is often difficult, time consuming, and relies on expert knowledge. To automate the design of FSs, many metaheuristic learning algorithms have been proposed. One major optimization category uses Swarm Intelligence (SI) model (Kennedy et al., 2001). The SI technique studies collective behavior in decentralized systems. Its development was based on mimicking the social behavior of animals or insects in an effort to find the optima in the problem space. SI models are initialized with a population of random solutions. One well-known SI model is particle swarm optimization (PSO) (Kennedy & Eberhart, 1995). Many modified PSO models have been proposed and successfully applied to different optimization problems (Clerc & Kennedy, 2002; Bergh & Engelbrecht, 2004; Ratnaweera et al., 2004; Juang, 2004; Kennedy & Mendes, 2006; Parrott & Li, 2006; Chen & Li, 2007). FS design using PSO has also been proposed in several studies (Juang, 2004; Chatterjee et al., 2005; Juang et al., 2007; Araujo & Coelho, 2008; Sharma et al., 2009). Another well-known SI is ant colony optimization (ACO) (Dorigo & Stutzle, 2004). The ACO technique is inspired by real ant colony observations. It is a multi-agent approach that was originally proposed to solve difficult discrete combinatorial optimization problems, such as the traveling salesman problem (TSP) (Dorigo et al., 1996; Dorigo & Gambardella, 1997). In the original ACO meta-heuristic, artificial ant colonies cooperate to find good solutions for difficult discrete optimization problems. Different ACO models have been applied to FS design problems (Cassillas et al., 2000; Cassillas et al., 2005; Mucientes & Casillas; 2007; Juang & Lo, 2007; Juang et al., 2008; Juang & Lu; 2009). In (Cassillas et al.,2000; Mucientes & Casillas; 2007; Juang et al., 2008; Juang & Lu; 2009), the FS input space was partitioned in grid type with antecedent part parameters of an FS manually assigned in advance. In (Juang & Lo, 2007), the FS input space was flexibly partitioned using a fuzzy clustering-like algorithm in order to reduce the total number of rules. For all of these studies, the consequent part parameters were optimized in discrete space using ACO. Since only the consequent part parameters are optimized, and the optimization space is restricted to be discrete, the designed FSs are unsuitable for problems where high accuracy is a major concern. Source: Fuzzy Systems, Book edited by: Ahmad Taher Azar, ISBN 978-953-7619-92-3, pp. 216, February 2010, INTECH, Croatia, downloaded from SCIYO.COM

We propose a dynamical system-based approach for solving robust optimization problems. The well-known continuous-time dynamical system for solving deterministic optimization problems arises in the form of primal-dual gradient dynamics where the vector field is derived as the gradient of the Lagrangian. The new continuous-time dynamical system we introduce for solving robust optimization problems differs from the primal-dual dynamics in the sense that the vector field is not derived as the gradient of the Lagrangian function. We call this new dynamical system as saddle point dynamics. In the saddle point dynamics, the uncertain variable arises as a dynamical state. For a general class of robust optimization problem, where the cost function is convex in decision variable and concave in uncertain variable, we show that the robust optimal solution can be recovered as a globally asymptotically stable equilibrium point of the saddle point dynamical system. Simulation results are presented to demonstrate the capability of this new dynamical system to solve various robust optimization problems. We also compare our proposed approach with existing methods based on robust counterpart and scenario-based random sampling.

Quadratically adjustable robust linear optimization with inexact data via generalized S-lemma: Exact second-order cone program reformulations

Model-based design principles have received considerable attention in biotechnology and the chemical industry over the last two decades. However, parameter uncertainties of first-principle models are critical in model-based design and have led to the development of robustification concepts. Various strategies have been introduced to solve the robust optimization problem. Most approaches suffer from either unreasonable computational expense or low approximation accuracy. Moreover, they are not rigorous and do not consider robust optimization problems where parameter correlation and equality constraints exist. In this work, we propose a highly efficient framework for solving robust optimization problems with the so-called point estimation method (PEM). The PEM has a fair trade-off between computational expense and approximation accuracy and can be easily extended to problems of parameter correlations. From a statistical point of view, moment-based methods are used to approximate robust inequality and equality constraints for a robust process design. We also apply a global sensitivity analysis to further simplify robust optimization problems with a large number of uncertain parameters. We demonstrate the performance of the proposed framework with two case studies: (1) designing a heating/cooling profile for the essential part of a continuous production process; and (2) optimizing the feeding profile for a fed-batch reactor of the penicillin fermentation process. According to the derived results, the proposed framework of robust process design addresses uncertainties adequately and scales well with the number of uncertain parameters. Thus, the described robustification concept should be an ideal candidate for more complex (bio)chemical problems in model-based design.

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.

Interval‐based multi‐objective robust optimization aims to achieve high‐performance solutions insensitive to uncertainty, has garnered significant attention. However, efficiently analyzing the maximum fluctuations of solutions in both objective and constraint functions to assess their robustness remains challenging. To address this issue, this article proposes an adaptive coevolution method, which can evaluate the maximum fluctuations of multiple candidate solutions with respect to a function in a single run. This method is integrated with the multi‐objective evolutionary algorithm (MOEA) to develop a framework termed adaptive coevolution‐based multi‐objective robust optimization (AC‐MORO) for solving multi‐objective robust optimization problems. To evaluate the performance of AC‐MORO, it is compared with MODE‐RO on a set of benchmark problems; meanwhile, a performance metric is proposed to test the accuracy of the adaptive coevolution method in analyzing the robustness of solutions. The impact of various parameter settings on the efficiency of the proposed method is also investigated. Subsequently, a variant of the adaptive coevolution method is explored to further enhance the performance of AC‐MORO. Finally, AC‐MORO is applied to address robust optimization problems in practical engineering.

In this paper, we study the following robust optimization problem. Given a set family representing feasibility and candidate objective functions, we choose a feasible set, and then an adversary chooses one objective function, knowing our choice. The goal is to find a randomized strategy (i.e., a probability distribution over the feasible sets) that maximizes the expected objective value in the worst case. This problem is fundamental in wide areas such as artificial intelligence, machine learning, game theory, and optimization. To solve the problem, we provide a general framework based on the dual linear programming problem. In the framework, we utilize the ellipsoid algorithm with the approximate separation algorithm. We prove that there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for our robust optimization problem if there exists an α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-approximation algorithm for finding a (deterministic) feasible set that maximizes a nonnegative linear combination of the candidate objective functions. Using our result, we provide approximation algorithms for the max–min fair randomized allocation problem and the maximum cardinality robustness problem with a knapsack constraint.

We examine the robust mean-VaR portfolio optimization problem when a parametric approach is used for estimating VaR. A robust optimization formulation is used to accommodate estimation risk, and we obtain an analytic solution when there is a risk-free asset and short-selling is allowed. This renders the model computationally tractable. Further, to avoid the conservatism of robust optimal portfolios, we suggest an adjusted robust optimization approach. Empirically, we evaluate the out-of-sample performance of the new approach, the robustness of obtained solutions and level of conservatism of the resulting portfolios. The empirical results highlight some benefits of our approach.

In active distribution networks, topology change is an important scheme to cope with the rapid fluctuations of distributed renewable generations. This paper studies the real-time distribution network reconfiguration (DNR) problem while ensuring a reliable power supply under set-based uncertainty. Considering the intertemporal constraint on the maximum number of action times for each switchable line, real-time DNR is formulated as a multi-stage robust optimization (MRO) problem. To solve this problem, a customized implicit policy method is proposed. First, the non-anticipative constraints of each switchable line are specifically devised to generate the upper and lower bounds for line status, ensuring that the intertemporal constraint is always satisfied as long as the bounds are met. Second, by augmenting the stage to optimize the above bounds, the original MRO problem is reformulated as a two-stage robust optimization problem with mixed-integer recource by relaxing the intertemporal constraints and aggregating stages, thereby suggesting an OPF-based real-time DNR policy that is robustly feasible and parameterized in the bounds of line status. Finally, these bounds are obtained by solving the two-stage robust optimization problem through the nested column-and-constraint generation algorithm. Tests on IEEE 33-, 69-, and 123-bus distribution systems comprehensively verify the effectiveness of the proposed method, as well as its advantages in scalability and optimality over state-of-the-art methods.