Surrogate duality for robust optimization

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 .

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.

In Chapter 4, several polynomially solvable cases of robust discrete optimization problems are discussed in detail. However, by results presented in Chapter 3 we also know that most robust discrete optimization problems belong to the NP-hard class. In this chapter, we present our approach for solving these difficult robust discrete optimization problems. We are in this chapter restricting our attention to robust discrete optimization problems with equivalent single scenario problems that can be efficiently solved with a polynomial or pseudo-polynomial procedure. The solution procedures are based on branch-and-bound with both upper and lower bounds generated by surrogate relaxation. To be exact, the upper bound (for a maximization problem) is obtained from surrogate relaxation and the lower bound is obtained as a by-product via a heuristic based on the surrogate relaxation result. In the case when input data satisfies bounded percentage deviation condition (to be defined in Section 5.2), the heuristic is shown to provide a constant approximation. Computational results in Section 5.3 demonstrate the effectiveness of the bounds and the solution procedure.

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.

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.

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

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.

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

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.

A Reinforcement Learning Variable Neighborhood Search for the Robust Dynamic Bayesian Network Optimization Problem under the Supply Chain Ripple Effect

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.

In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is that the minimizers and the optimal value of the sub-problems converge, in some sense, to the minimizers and the optimal value of the initial problem, respectively. Our result particularly implies the consistency of the scenario approach for nonconvex optimization problems. Finally, we show that our method can also be used to solve infinite programming problems.