Robust Optimization Problem Research Articles

Dynamic resource management for 6G vehicular networks: CORA-6G offloading and allocation strategies

Given the burgeoning demands of real-time applications and limited on-board computational resources, the rapid proliferation of 6G-enabled vehicular networks brings forth the critical challenge of efficient computation offloading. This paper introduces the CORA-6G algorithm, a novel solution specifically designed to address this issue by determining the optimal offloading ratio. Using a robust convex optimization problem, CORA-6G maximizes the computational utility, ensuring efficient resource allocation and enhanced responsiveness. Quantitative evaluations reveal that, when compared to conventional methods, CORA-6G improves offloading efficiency by up to 25%, thereby significantly boosting the overall performance of vehicular applications. This research underscores the potential of CORA-6G as a cornerstone for future 6G vehicular network advancements.

Stochastic First-Order Algorithms for Constrained Distributionally Robust Optimization

We consider distributionally robust optimization (DRO) problems, reformulated as distributionally robust feasibility (DRF) problems, with multiple expectation constraints. We propose a generic stochastic first-order meta-algorithm, where the decision variables and uncertain distribution parameters are each updated separately by applying stochastic first-order methods. We then specialize our results to the case of using two specific versions of stochastic mirror descent (SMD): (i) a novel approximate version of SMD to update the decision variables, and (ii) the bandit mirror descent method to update the distribution parameters in the case of [Formula: see text]-divergence sets. For this specialization, we demonstrate that the total number of iterations is independent of the dimensions of the decision variables and distribution parameters. Moreover, the cost per iteration to update both sets of variables is nearly independent of the dimension of the distribution parameters, allowing for high-dimensional ambiguity sets. Furthermore, we show that the total number of iterations of our algorithm has a logarithmic dependence on the number of constraints. Experiments on logistic regression with fairness constraints, personalized parameter selection in a social network, and the multi-item newsvendor problem verify the theoretical results and show the usefulness of the algorithm, in particular when the dimension of the distribution parameters is large. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: This work was supported by the National Science Foundation [Grant 2112533]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0167 .

A machine learning approach to two-stage adaptive robust optimization

We propose an approach based on machine learning to solve two-stage linear adaptive robust optimization (ARO) problems with binary here-and-now variables and polyhedral uncertainty sets. We encode the optimal here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the optimal wait-and-see decisions into what we denote as the strategy. We solve multiple similar ARO instances in advance using the column and constraint generation algorithm and extract the optimal strategies to generate a training set. We train machine learning models that predict high-quality strategies for the here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the wait-and-see decisions. The models can be applied to problems with varying dimensions. We also introduce novel methods to expedite training data generation and reduce the number of different target classes the machine learning algorithm needs to be trained on. We apply the proposed approach to the facility location, the multi-item inventory control and the unit commitment problems. Our approach solves ARO problems drastically faster than the state-of-the-art algorithms with high accuracy.

Sum‐rate maximization for downlink multiuser MISO URLLC system aided by IRS with discrete phase shifters

AbstractIntelligent reflecting surface (IRS) has recently been considered as a potential technology for realizing ultra‐reliable and low‐latency (URLLC) in wireless networks. This paper proposes a resource optimization scheme to maximize the sum‐rate for an IRS‐assisted downlink multiuser multi‐input single‐output (MISO) URLLC system. For the perfect CSI scenario, we jointly optimize each user's block‐length and packet‐error probability, the precoding vectors at the base station (BS), and the passive beamforming with discrete phase shifts at the IRS. Given the problem's complexity, we design a computationally efficient iterative algorithm using successive convex approximation (SCA) and semidefinite relaxation (SDR) techniques to obtain a locally optimal solution. Specifically, for the imperfect CSI scenario, we construct a robust resource optimization problem model and incorporate the S‐procedure to address the impact of channel uncertainty, proposing an iterative algorithm based on the alternating optimization (AO) method to achieve a locally optimal solution. Simulation results demonstrate that: 1) An IRS equipped with a 2‐bit quantized resolution phase shifter is sufficient to achieve a system sum‐rate comparable to that of an ideal phase shifter; 2) Compared to other Baseline schemes, Algorithm 2 exhibits better robustness and superior performance gains under imperfect CSI.

Robust combinatorial optimization problems under budgeted interdiction uncertainty

AbstractIn robust combinatorial optimization, we would like to find a solution that performs well under all realizations of an uncertainty set of possible parameter values. How we model this uncertainty set has a decisive influence on the complexity of the corresponding robust problem. For this reason, budgeted uncertainty sets are often studied, as they enable us to decompose the robust problem into easier subproblems. We propose a variant of discrete budgeted uncertainty for cardinality-based constraints or objectives, where a weight vector is applied to the budget constraint. We show that while the adversarial problem can be solved in linear time, the robust problem becomes NP-hard and not approximable. We discuss different possibilities to model the robust problem and show experimentally that despite the hardness result, some models scale relatively well in the problem size.

Optimality and Duality for Robust Optimization Problems Involving Intersection of Closed Sets

In this paper, we study a robust optimization problem whose constraints include nonsmooth and nonconvex functions and the intersection of closed sets. Using advanced variational analysis tools, we first provide necessary conditions for the optimality of the robust optimization problem. We then establish sufficient conditions for the optimality of the considered problem under the assumption of generalized convexity. In addition, we present a dual problem to the primal robust optimization problem and examine duality relations.

Robust topology and discrete fiber orientation optimization under principal material uncertainty

This paper introduces a formulation of the robust topology optimization problem that is tailored for designing fiber-reinforced composite structures with spatially varying principal mechanical properties. Specifically, a methodology is developed that incorporates the spatial variability in the engineering constants of the composite lamina into the concurrent topology (i.e., material distribution) and morphology (i.e., fiber orientation distribution) optimization problem for the minimization of the robust compliance function. The spatial variability in the mechanical properties of the lamina is modeled as a homogeneous random field within the design domain by means of the Karhunen-Loe´ve series expansion, and is thereafter intrusively propagated into the stochastic finite element analysis of the composite structure. To carry out the stochastic finite element analysis per iteration of the optimization cycle, the first-order perturbation method is utilized for approximating the current state variables of the physical system. The resulting robust topology and fiber orientation optimization problem is formulated step-by-step for the minimization of the robust compliance function. With the view of solving the optimization problem at hand by means of gradient-based solution algorithms, the first-order derivatives of the involved design functions w.r.t. the associated design variables are analytically derived. The present work concludes with a series of numerical examples, focusing on the benchmark academic case studies of the 2D cantilever and the half part of the Messerschmitt-Bölkow-Blohm beam, aiming to demonstrate the developed methodology as well as to explore the effect that different parameterization instances of the random field bear on the predicted topology and morphology of the beams.

A computational efficient approach for distributionally robust unit commitment with enhanced disjointed layered ambiguity set

AbstractTo achieve the sustainable development of the society, renewable energy dominated power systems are gradually formed. However, the uncertainty of renewable power poses challenges for power system operations, such as balancing the load and generation in day‐ahead unit commitment problems. To address this issue, an enhanced disjointed layered ambiguity set is proposed in this paper to effectively capture the uncertainty of renewable generation forecast errors. On this basis, a two‐stage distributionally robust unit commitment optimization problem is formulated, which is computational challenge. To this end, a novel reformation approach and a worst‐case support set identification method are proposed to derive the optimal solution in a computationally efficient way. Simulation results demonstrate that the proposed ambiguity set brings lower operation costs of unit commitment while maintaining similar reliability compared to existing methods. In detail, the proposed ambiguity set achieves a reduction in operating costs of approximately 10.01%. The proposed novel reformation approach and solution method are also shown to have higher computation efficiency than the existing Benders decomposition method, resulting in a reduction in computation time to less than 3.78% of the original duration.

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

Uncertainty in the organic Rankine cycle (ORC) system and the highly complex ORC models pose challenges to optimal operation. A data-driven robust parametric optimization for the ORC system is proposed to ensure high and stable performance under uncertainty. The uncertainty of the cyclic variables is estimated by their distribution parameters, and the average (expected) thermodynamic performance (the net output power of the ORC) is maximized as the optimization objective while minimizing its variance. An artificial neural network with the rectified linear unit is used as a surrogate model for optimization. Then the robust parametric optimization problem can be transformed into a mixed-integer linear optimization problem with a chance-constrained form, and the robust optimal solution can be solved quickly. Case studies show that the solution time of the robust optimization problem using the proposed method is about 1.4 s, which is much less than obtaining the optimal solution based on ORC mechanism models. Meanwhile, the optimal operating condition derived by the proposed robust optimization approach outperforms that obtained by the traditional deterministic strategy. The proposed approach not only improves the robustness of the system but also demonstrates the importance of considering uncertainty in parametric optimization.

Variational quantum circuit learning-enabled robust optimization for AI data center energy control and decarbonization

As the demand for artificial intelligence (AI) models and applications continues to grow, data centers that handle AI workloads are experiencing a rise in energy consumption and associated carbon footprint. This work proposes a variational quantum computing-based robust optimization (VQC-RO) framework for control and energy management in large-scale data centers to address the computational challenges and overcome limitations of conventional model-based and model-free strategies. The VQC-RO framework integrates variational quantum circuits (VQCs) with classical optimization to enable efficient and uncertainty-aware control of energy systems in AI data centers. Quantum algorithms executed on noisy intermediate-scale quantum (NISQ) devices are used for value function estimation trained with Q-learning, leading to the formulation of a robust optimization problem with uncertain coefficients. The quantum computing-based robust control strategy is designed to address uncertainties associated with weather conditions and renewable energy generation while optimizing energy consumption in AI data centers. This work also outlines the computational experiments conducted at various AI data center locations in the United States to analyze the reduction in power consumption and carbon emission levels associated with the proposed quantum computing-based robust control framework. This work contributes a novel approach to energy-efficient and sustainable data center operation, promising to reduce carbon emissions and energy consumption in large-scale data centers handling AI workloads by 9.8 % and 12.5 %, respectively.

Kernel-based Sensitivity Analysis for (Excursion) Sets

In this article, we aim to perform sensitivity analysis of set-valued models and, in particular, to quantify the impact of uncertain inputs on feasible sets, which are key elements in solving a robust optimization problem under constraints. While most sensitivity analysis methods deal with scalar outputs, this article introduces a novel approach to perform sensitivity analysis with set-valued outputs. Our innovative methodology is designed for excursion sets, but is versatile enough to be applied to set-valued simulators, including those found in viability fields, or when working with maps like pollutant concentration maps or flood zone maps. We propose to use the Hilbert-Schmidt Independence Criterion (HSIC) with a kernel designed for set-valued outputs. After setting a probabilistic framework for random sets, a first contribution is the proof that this kernel is characteristic, an essential property in a kernel-based sensitivity analysis context. To measure the contribution of each input, we then propose to use HSIC-ANOVA indices. With these indices, we can identify which inputs should be neglected (screening) and we can rank the others according to their influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally, we test the proposed method on three test cases of excursion sets.

Scenario based optimisation over uncertain system identification models

Models of dynamical systems are often built from observed data using system identification techniques, and the uncertainty in the model parameters is commonly described using confidence regions to which the true model parameters belong with a certain probability. In this paper we propose methods which use the scenario approach to solve robust optimisation problems for model based design where there is uncertainty in the model parameters. The methods draw samples (scenarios) of the uncertain model parameters and solve an optimisation problem only involving the drawn scenarios. Both a Bayesian system identification setting where the model parameters are stochastic, and a non-Bayesian system identification setting where the model parameters are treated as being deterministic are considered. The Bayesian and non-Bayesian settings require different treatment, and whereas standard scenario optimisation theory can be directly utilised in the Bayesian setting, this is not the case in the non-Bayesian setting. For that reason the model class is restricted to linear regression models with deterministic regressors in the non-Bayesian case. Algorithms are proposed for drawing samples of the parameters such that the uncertainties in the system identification models are reflected in the samples. Theoretical results are given bounding the probability that the found solution to the scenario optimisation problem will give a worse performance when applied to the true system than seen on the drawn scenarios. The methods are illustrated in simulation examples showing good performance.

Robust topology optimization for transient dynamic response minimization

This paper presents a novel framework for topology optimization (TO) of structures subjected to uncertain transient loading. Random transient uncertain but bounded loading is modeled in the form of an ellipsoid convex model. A transformation matrix is defined with uncertainty in input parameters based on orientation and lengths of semi-major and semi-minor axes of the ellipsoid. Latin hypercube sampling is utilized to generate samples inside the normalized space. These samples are then converted to a physical space using a transformation matrix. The generalized-α method is employed to compute the dynamic response of the system subjected to time-dependent loading. A density-based TO formulation is used to minimize dynamic response. The robust topology optimization problem is defined as bi-criteria optimization to minimize both the mean and variance across all deterministic cases within the ellipsoidal bounds. The solid isotropic material with penalization method is employed for the interpolation of material stiffness. Helmholtz filter is used to stabilize the problem numerically. The optimality criteria method is used to update the design variables. The proposed scheme’s effectiveness and the efficiency of the optimized designs are checked using a set of numerical examples. For most of the cases, robust design exhibits a substantial reduction of approximately 70%–75% in standard deviation when compared with deterministic design. The results demonstrate the proposed framework’s ability to reduce structure transient response under uncertain but bounded regions.

Robust topology optimization of multi-material structures based on the material-field series-expansion method

The performance of composite structures can be dramatically improved through the synergy between different constituent materials. However, the topology optimization of multi-material structures poses a heavy computational burden, which becomes more significant when load uncertainty is considered. This article proposes an efficient method to address the problem of robust topology optimization of multi-material structures. Employing the property that the material-field series-expansion method can significantly reduce the number of design variables, the proposed method improves the computational efficiency of the robust optimization problem. Meanwhile, the modification of the original multi-material interpolation model improves the quality of the optimized designs. Monte Carlo simulation is used to treat the load uncertainty, but it is decoupled from the optimization process based on linear elasticity theory to reduce computation cost. The effectiveness of the proposed method is demonstrated by numerical examples, which show that robust design results can be efficiently generated by this method.

E2E-AT: A Unified Framework for Tackling Uncertainty in Task-Aware End-to-End Learning

Successful machine learning involves a complete pipeline of data, model, and downstream applications. Instead of treating them separately, there has been a prominent increase of attention within the constrained optimization (CO) and machine learning (ML) communities towards combining prediction and optimization models. The so-called end-to-end (E2E) learning captures the task-based objective for which they will be used for decision making. Although a large variety of E2E algorithms have been presented, it has not been fully investigated how to systematically address uncertainties involved in such models. Most of the existing work considers the uncertainties of ML in the input space and improves robustness through adversarial training. We extend this idea to E2E learning and prove that there is a robustness certification procedure by solving augmented integer programming. Furthermore, we show that neglecting the uncertainty of COs during training causes a new trigger for generalization errors. To include all these components, we propose a unified framework that covers the uncertainties emerging in both the input feature space of the ML models and the COs. The framework is described as a robust optimization problem and is practically solved via end-to-end adversarial training (E2E-AT). Finally, the performance of E2E-AT is evaluated by a real-world end-to-end power system operation problem, including load forecasting and sequential scheduling tasks.

REGLO: Provable Neural Network Repair for Global Robustness Properties

We present REGLO, a novel methodology for repairing pretrained neural networks to satisfy global robustness and individual fairness properties. A neural network is said to be globally robust with respect to a given input region if and only if all the input points in the region are locally robust. This notion of global robustness also captures the notion of individual fairness as a special case. We prove that any counterexample to a global robustness property must exhibit a corresponding large gradient. For ReLU networks, this result allows us to efficiently identify the linear regions that violate a given global robustness property. By formulating and solving a suitable robust convex optimization problem, REGLO then computes a minimal weight change that will provably repair these violating linear regions.

Frequency regulation with storage: On losses and profits

Low-carbon societies will need to store vast amounts of electricity to balance intermittent generation from wind and solar energy, for example, through frequency regulation. Here, we derive an analytical solution to the decision-making problem of storage operators who sell frequency regulation power to grid operators and trade electricity on day-ahead markets. Mathematically, we treat future frequency deviation trajectories as functional uncertainties in a receding horizon robust optimization problem. We constrain the expected terminal state-of-charge to be equal to some target to allow storage operators to make good decisions not only for the present but also the future. Thanks to this constraint, the amount of electricity traded on day-ahead markets is an implicit function of the regulation power sold to grid operators. The implicit function quantifies the amount of power that needs to be purchased to cover the expected energy loss that results from providing frequency regulation. We show how the marginal cost associated with the expected energy loss decreases with roundtrip efficiency and increases with frequency deviation dispersion. We find that the profits from frequency regulation over the lifetime of energy-constrained storage devices are roughly inversely proportional to the length of time for which regulation power must be committed.

The perturbation method applied to a robust optimization problem with constraint

The perturbation method applied to a robust optimization problem with constraint

ERGO-II: An Improved Bayesian Optimization Technique for Robust Design With Multiple Objectives, Failed Evaluations, and Stochastic Parameters

Abstract In this work, the efficient robust global optimization (ERGO) method is revisited with the aim of enhancing and expanding its existing capabilities. The original objective of ERGO was to address the computational challenges associated with optimization-under-uncertainty through the use of Bayesian optimization (BO). ERGO tackles robust optimization problems which are characterized by sensitivity in the objective function due to stochasticity in the design space. It does this by concurrently minimizing the mean and variance of the objective in a multi-objective setting. To handle the computational complexity arising from the uncertainty propagation, ERGO exploits the analytical expression of the surrogate model underlying BO. In this study, ERGO is extended to accommodate multiple objectives, incorporate an improved predictive error estimation approach, investigate the treatment of failed function evaluations, and explore the handling of stochastic parameters next to stochastic design variables. To evaluate the effectiveness of these improvements, the enhanced ERGO scheme is compared with the original method using an analytical test problem with varying dimensionality. Additionally, the novel optimization technique is applied to an aerodynamic design problem to validate its performance.

A multi-factor evolutionary algorithm for solving the multi-tasking robust optimization problem on networked systems

Realistic networks constantly face complex environments and are vulnerable to external disturbances that can damage their structure and disrupt information flow. These challenges give rise to two critical optimization problems: network robustness optimization and robust influence maximization. These problems hold significant theoretical and practical importance, garnering increased attention in recent research. Models and optimizers have been developed to tackle related typological design and seed determination tasks, which provide rational candidates for practical scenarios. However, existing approaches have limitations. Firstly, while structural disturbances significantly impact network robustness, their effects on information propagation remain understudied. The quest for robust seed determination in the face of topological failures remains unresolved. Secondly, both network robustness optimization and robust influence maximization focus on leveraging structural network information, but the potential synergy between solutions for these problems has been overlooked in current research. There is a pressing need for highly efficient optimizers and versatile candidate solutions capable of withstanding various scenarios comprehensively. In response to these limitations, this study introduces multi-task optimization theory to address the network correlation optimization problem described above. We first systematically analyze the correlation between the two problems, demonstrating a positive relationship through experimental results. Building on this insight, we propose MFEA-Net, a multifactor evolutionary algorithm equipped with problem-specific operators. MFEA-Net concurrently tackles the challenges of network robustness optimization and robust influence maximization, considering multiple optimization scenarios simultaneously. It aims to uncover synergistic insights across different tasks during the optimization process. Our experimental results indicate that MFEA-Net surpasses existing methods in terms of performance. This research bridges a gap in the fields of robust influence maximization and network robust cooperative optimization, offering valuable guidance for addressing real-world network system design and optimization challenges.