Robust Optimization Problem Research Articles

A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0306 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Robust linear algebra

We propose a robust optimization (RO) framework that immunizes some of the central linear algebra problems in the presence of data uncertainty. Namely, we formulate linear systems, matrix inversion, eigenvalues–eigenvectors and matrix factorization under uncertainty, as robust optimization problems using appropriate descriptions of uncertainty. The resulting optimization problems are computationally tractable and scalable. We show in theory that RO improves the relative error of the linear system by reducing the condition number of the underlying matrix. Moreover, we provide empirical evidence showing that the proposed approach outperforms state of the art methods for linear systems and matrix inversion, when applied on ill-conditioned matrices. We show that computing eigenvalues–eigenvectors under RO, corresponds to solving linear systems that are better conditioned than the nominal and illustrate with numerical experiments that the proposed approach is more accurate than the nominal, when perturbing ill-conditioned matrices. Finally, we demonstrate empirically the benefit of the robust Cholesky factorization over the nominal.

Cutting Plane Approaches for the Robust Kidney Exchange Problem

Renal patients who have a willing but incompatible donor can decide to participate in a kidney exchange program (KEP). The goal of a KEP is to identify sets of incompatible pairs that can exchange donors, leading to compatible transplants for each recipient. Significant uncertainty is involved in this process, as planned transplants may be canceled for many reasons. It is therefore crucial to take into account failures while planning exchanges. In this paper, we consider a robust variant of this problem with recourse studied in the literature that takes into account failures of donors or recipients. This problem belongs to the class of defender–attacker–defender (DAD) models. We propose a cutting plane method for solving the attacker–defender subproblem based on two commonly used mixed-integer programming formulations for kidney exchange problems. Our results imply a running time improvement of one order of magnitude compared to the state-of-the-art. Moreover, our cutting plane methods can solve many previously unsolved instances. Furthermore, we propose a new practical policy for recourse in KEPs and show that the robust optimization problem concerning this policy is tractable for small to mid-size KEPs in practice.

On the complexity of robust multi-stage problems with discrete recourse

We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known.In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove Σ3p-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant difference in computational complexity between column-wise and row-wise uncertainty.

Adaptive weighted robust data recovery with total variation for hyperspectral image

This paper proposes an adaptive weighted robust data recovery method with total variation (TV) regularization for hyperspectral image (HSI). In the proposed method, the HSI recovery problem is modeled as a tensor robust principal component analysis optimization problem, which decomposes the received data into low-rank HSI data, outliers, and noise component, with tensor nuclear norm (TNN), ℓ1 norm, and ℓF norm constrained, respectively. Then, an adaptive weighted strategy is defined to impose on the TNN and outliers to flexibly use the priori information of singular values (SVs) and strengthen the sparsity of outliers, where the weights are adaptively conducted by the elements in SVs and outliers. Specifically, the weighted strategy in TNN retains larger SVs to determine the main information for data recovery, while penalizing smaller SVs to eliminate the influence of interference. And the weighted strategy on outliers enables the proposed method to more effectively measure the sparsity of outliers. Furthermore, TV regularization is introduced to extract the local information for data recovery by a difference operation along different modes, which also can help the proposed method avoid the detail loss caused by the weighted strategy. Experiments confirm that the proposed method significantly outperforms the existing methods.

Location Information Assisted Beamforming Design for Reconfigurable Intelligent Surface Aided Communication Systems

The large overhead arising from conventional channel estimations in reconfigurable intelligent surface (RIS) aided millimeter-wave communication systems, may offset the performance gain brought by the RIS. To tackle this issue, we propose a location information assisted beamforming design without the requirement of the channel training process. First, we establish the geometrical relationship between the channel model and the user location, and mathematically derive an approximate channel state information (CSI) error bound based on the user location error region. Then, for combating the negative impact of the location error on the communication performance, we formulate a worst-case robust beamforming optimization problem to optimize the beamformer at the base station (BS) and the phase-shift matrix at the RIS. To solve this non-convex problem, we develop a novel relaxed alternating optimization process (RAOP) by utilizing various optimization tools, such as the Lagrange multiplier, the matrix inversion lemma, the semidefinite relaxation (SDR), as well as the branch-and-bound (BnB). Additionally, we prove sufficient conditions for the SDR to yield rank-one solutions, and modify the BnB to acquire the phase-shift solution under an arbitrary constraint of possible phase-shift values. Finally, we analyse the convergence and complexity of the proposed RAOP, and carry out simulations for performance evaluations. Compared to the conventional non-robust beamforming, our method performs better and shows strong robustness against the location-error-related CSI uncertainty. Compared to the robust beamforming based on the S-procedure and penalty convex-concave procedure (CCP), our method with BnB shows the advantages of being able to converge faster and handle arbitrary phase-shift argument sets.

Uncertainty Injection: A Deep Learning Method for Robust Optimization

This paper proposes a paradigm of uncertainty injection for training deep learning model to solve robust optimization problems. The majority of existing studies on deep learning focus on the model learning capability, while assuming the quality and accuracy of the inputs data can be guaranteed. However, in realistic applications of deep learning for solving optimization problems, the accuracy of inputs, which are the problem parameters in this case, plays a large role. This is because, in many situations, it is often costly or sometime impossible to obtain the problem parameters accurately, and correspondingly, it is highly desirable to develop learning algorithms that can account for the uncertainties in the input and produce solutions that are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">robust</i> against these uncertainties. This paper presents a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uncertainty injection</i> scheme for training machine learning models that are capable of implicitly accounting for the uncertainties and producing statistically robust solutions. We further identify the wireless communications as an application field where uncertainties are prevalent in problem parameters such as the channel coefficients. We show the effectiveness of the proposed training scheme in two applications: the robust power loading for multiuser multiple-input-multiple-output (MIMO) downlink transmissions; and the robust power control for device-to-device (D2D) networks.

Two-stage robust optimal capacity configuration of a wind, photovoltaic, hydropower, and pumped storage hybrid energy system

The hybrid energy system of hydro-powers, pumped storages and renewable energies has become a new topic direction in modern power system developments. Consequently, it is essential to realize a rational and efficient allocation of different energy source capacities. Nevertheless, there is still a gap between the available studies and the requirement for further hybrid energy system development. This paper focuses on the optimal capacity configuration of a wind, photovoltaic, hydropower, and pumped storage power system. In this direction, a bi-level programming model for the optimal capacity configuration of wind, photovoltaic, hydropower, pumped storage power system is derived. To model the operating mode of a pumped storage power station, two 0-1 variables are introduced. To handle the nonlinear and nonconvex lower level programing problem caused by the two 0-1 variables, it is proposed that the 0-1 variables are treated as some uncertain parameters. Also, by treating the 0-1 variables as some uncertain parameters, a two-stage robust optimization problem to decompose the original bi-level programing one into a master problem and a subproblem is finally introduced. The Karush-Kuhn-Tucker (KKT) conditions are then applied to simplify and linearize the min-max problem and nonlinear terms in the master problem. This results in both the master problem and the subproblem being formulated as mixed integer linear programming (MILP) problems. By utilizing the powerful Column-and-Constraint Generation (C&amp;amp;CG) algorithm, the two-stage robust optimization model is decomposed into an iterative procedure of solving the master problem and the subproblem sequentially. This approach eliminates the need for intricate optimization algorithms as commonly used in existing bi-level planning problems in hybrid energy systems. Finally, the effectiveness and advantages of the proposed model is verified by the numerical results on a case study.

A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains.

Robust LNG sales planning under demand uncertainty: A data-driven goal-oriented approach

This paper addresses the liquefied natural gas (LNG) sales planning problem over a pipeline network with a focus on uncertain demands. Generically, the total profit is maximized by seeking optimal transportation and inventory decisions, and robust optimization (RO) has been a viable decision-making strategy to this end, which is however known to suffer from over-conservatism. To circumvent this, a new goal-oriented data-driven RO approach is proposed. First, we adopt data-driven polytopic uncertainty sets based on kernel learning, which yields a compact high-density region from data and assures tractability of RO problems. Based on this, a new goal-oriented RO formulation is put forward to satisfy to the greatest extent the target profit while tolerating slight constraint violations. In contrast to traditional min–max RO scheme, the proposed scheme not only ensures a flexible trade-off but also yields parameters with clear interpretation. The resulting optimization problem turns out to be equivalent to a mixed-integer linear program that can be effectively handled using off-the-shelf solvers. We illustrate the merit of the proposed method in satisfying a prescribed goal with optimized robustness by means of a case study.

Randomized Strategies for Robust Combinatorial Optimization with Approximate Separation

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.

Distributionally Robust Optimization Under Distorted Expectations

Optimal Decision Making Under Distorted Expectation with Partial Distribution Information Decision makers who are not risk neutral may evaluate expected values by distorting objective probabilities to reflect their risk attitudes, a phenomenon known as distorted expectations. This concept is widely applied in behavioral economics, insurance, finance, and other business domains. In “Distributionally Robust Optimization Under Distorted Expectations,” Cai, Li, and Mao study how decision makers using distorted expectations can optimize their decisions when only partial information about objective probabilities is available. They show that decision makers who are ambiguity averse can optimize their decisions as if they are risk averse with their risk attitudes characterized by a convex distortion function. This finding demonstrates why even non–risk-averse decision makers, such as those studied in the celebrated cumulative prospect theory, may consider it optimal to take risk-averse decisions when facing uncertainty about objective probabilities. Leveraging this finding, the authors show that a large class of distributionally robust optimization problems involving the use of distorted expectations can be tractably solved as convex programs.

A mixed-integer approximation of robust optimization problems with mixed-integer adjustments

In the present article we propose a mixed-integer approximation of adjustable-robust optimization problems, that have both, continuous and discrete variables on the lowest level. As these trilevel problems are notoriously hard to solve, we restrict ourselves to weakly-connected instances. Our approach allows us to approximate, and in some cases exactly represent, the trilevel problem as a single-level mixed-integer problem. This allows us to leverage the computational efficiency of state-of-the-art mixed-integer programming solvers. We demonstrate the value of this approach by applying it to the optimization of power systems, particularly to the control of smart converters.

Joint Optimization of Secure Over-the-Air Computation and Reliable Multicasting Assisted by a MIMO Untrusted Two-Way Relay

In this paper, we investigate joint optimization of secure AirComp and reliable multicasting assisted by a multiple-input multiple-output untrusted two-way relay, where artificial noise is employed at the access point (AP) to interfere the relay for ensuring secure AirComp. We aim to minimize the computation distortion at the AP by jointly designing the transmit variables of all the nodes and the aggregation variables at the AP and relay, under the secure AirComp constraint, the reliable multicasting constraint, and the transmit power constraints of all the nodes. We consider two scenarios that perfect and imperfect channel state information (CSI) are available. For the former, the formulated optimization problem is highly nonconvex, and we propose an effective block coordinate descent (BCD)-penalty successive convex approximation (penaltySCA) method to solve the nonconvex problem. For the latter, we model the imperfect CSI by using worst-case criterion and the formulated robust optimization problem is much more challenging than its counterpart with perfect CSI. To solve the robust problem effectively, we first transform it to a deterministic optimization problem by employing some powerful mathematical lemmas, and then apply the proposed BCD-penaltySCA method to solve the reformulated deterministic problem. The proposed methods are shown by simulations to significantly reduce the computation distortion compared with other benchmarks under considering secure AirComp and reliable multicasting.

A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle

A Primal-Worst-Equals-Dual-Best Perspective on Robust and Distributionally Robust Optimization In the paper “A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle,” Jianzhe Zhen, Daniel Kuhn, and Wolfram Wiesemann develop a generalized “primal-worst-equals-dual-best” principle that establishes strong duality between semi-infinite primal worst and nonconvex dual best formulations of robust and distributionally robust nonlinear optimization problems. Their theory offers an alternative characterization of (distributionally) robust optimization problems that bypasses the need to mobilize the machinery of abstract semi-infinite duality theory. The paper will be of interest to researchers and practitioners in the field of optimization under uncertainty.

Improved Decision Rule Approximations for Multistage Robust Optimization via Copositive Programming

Improved decision rule approximations for multistage robust optimization via copositive programming Previous research in the field has proposed several approaches to tackle multistage robust optimization problems, but they are often limited in their applicability. These existing methods either fail to handle cases where recourse matrices are uncertain or struggle to handle large-scale problems effectively. In their paper titled “Improved decision rule approximations for multistage robust optimization via copositive programming,” Guanglin Xu and Grani A. Hanasusanto contribute to the robust optimization literature by presenting a novel solution method. Their approach utilizes convex conic techniques and aims to address the general case of multistage robust optimization, where uncertainty exists in the recourse matrices. One significant advantage of their proposed method is its ability to scale well with large-sized instances, overcoming a common limitation faced by previous methods. Through numerical experiments on various simulated applications, Xu and Hanasusanto demonstrate the superiority of their algorithm over existing state-of-the-art methods.

Secure Multi-Path Selection with Optimal Controller Placement Using Hybrid Software-Defined Networks with Optimization Algorithm

The Internet's growth in popularity requires computer networks for both agility and resilience. Recently, unable to satisfy the computer needs for traditional networking systems. Software Defined Networking (SDN) is known as a paradigm shift in the networking industry. Many organizations are used SDN due to their efficiency of transmission. Striking the right balance between SDN and legacy switching capabilities will enable successful network scenarios in architecture networks. Therefore, this object grand scenario for a hybrid network where the external perimeter transport device is replaced with an SDN device in the service provider network. With the moving away from older networks to SDN, hybrid SDN includes both legacy and SDN switches. Existing models of SDN have limitations such as overfitting, local optimal trapping, and poor path selection efficiency. This paper proposed a Deep Kronecker Neural Network (DKNN) to improve its efficiency with a moderate optimization method for multipath selection in SDN. Dynamic resource scheduling is used for the reward function the learning performance is improved by the deep reinforcement learning (DRL) technique. The controller for centralised SDN acts as a network brain in the control plane. Among the most important duties network is selected for the best SDN controller. It is vulnerable to invasions and the controller becomes a network bottleneck. This study presents an intrusion detection system (IDS) based on the SDN model that runs as an application module within the controller. Therefore, this study suggested the feature extraction and classification of contractive auto-encoder with a triple attention-based classifier. Additionally, this study leveraged the best performing SDN controllers on which many other SDN controllers are based on OpenDayLight (ODL) provides an open northbound API and supports multiple southbound protocols. Therefore, one of the main issues in the multi-controller placement problem (CPP) that addresses needed in the setting of SDN specifically when different aspects in interruption, ability, authenticity and load distribution are being considered. Introducing the scenario concept, CPP is formulated as a robust optimization problem that considers changes in network status due to power outages, controller’s capacity, load fluctuations and changes in switches demand. Therefore, to improve network performance, it is planned to improve the optimal amount of controller placements by simulated annealing using different topologies the modified Dragonfly optimization algorithm (MDOA).

Robust Multiplexing Waveform Design for Integrated OFDM Radar and Communication via Complex Weight Optimization

Compared to traditional single orthogonal frequency division multiplexing (OFDM) radar and OFDM communication systems, integrated OFDM radar and communication systems have the advantages of improved sharing of scarce spectrum resources, a simple hardware structure and a reduced interference between signals. In this paper, a constraint relaxation-based robust OFDM multiplexing waveform (ROW-CR) design method for integrated radar and communication systems is proposed. Considering the influence of correlated clutter and jamming signals, transmission and reception models of different system platforms are established by allocating subcarrier complex weights of different antenna signals so as to determine radar conditional mutual information and communication channel capacity parameters. Meanwhile, a restricted closed model is introduced through the limit range of the target frequency response. Then, a robust OFDM waveform optimization problem for integrated radar and communication systems is constructed by the minimax criterion, and the closed-form solution is obtained by adopting an improved method based on trace function properties and constraint relaxation, resulting in a better radar and communication performance trade-off. In addition, a parameter hierarchical optimization-based robust OFDM waveform (ROW-PHO) design method is further explored to reduce the computational complexity, and this method can also ensure a low system performance loss. Finally, the numerical simulation results verify the effectiveness of the proposed methods.

Distributed Robust Optimization for Networked Agent Systems With Unknown Nonlinearities

The distributed robust optimization problem is investigated in this paper for a class of networked agent systems under stochastically switching communication graphs. Unlike most existing results on distributed optimization of networked agent systems, the dynamics of each agent in the present model are allowed to be subjected to matched linearly-parameterized unknown nonlinearities while the underlying communication graph is stochastically switching among some matrix-weighted communication graphs. The objective here is to cooperatively optimize a holistic cost function, which is a sum of all the convex local objective functions, under the mild condition that each agent may only have access to the gradient of its local objective function and the relative local information between itself and its neighbors. To solve such an outstanding optimization problem, a neuro-adaptive optimization protocol is developed based upon parameterizations of the unknown nonlinearities where the neuro-adaptive learning algorithm for estimating the weight matrices therein acts as the key role to compensate for the unknown nonlinearities. Furthermore, the signum function-based feedback law is subtly integrated into the proposed protocol to deal with the effect of the gradient term of the local objective function. Efficient criteria are provided to guide selections of proper control parameters under which the robust optimization problem is successfully resolved in the mean-square sense. Lastly, theoretical results are validated by presenting several illustrative simulation examples.

Resilient Expansion Planning of Electricity Grid Under Prolonged Wildfire Risk

Public Safety Power Shut-off (PSPS) during potentially dangerous weather conditions is an emergency measure within areas prone to wildfires. While balancing the risk of wildfire ignition and the continuation of energy supply is a short-term operation challenge, this paper explores the optimal long-term resilient expansion planning strategies. With the quantified risk of wildfire ignition, the proposed expansion planning problem maximizes the supplied power to the end-users while limiting the risk of wildfire ignition. The presented scheme provides utility decision-makers with three groups of network expansion decisions, namely addition of new lines, modification of existing lines, and installation of Distributed Energy Resources (DERs). Given the uncertainty of DERs and wildfire risk, a two-stage robust optimization problem is proposed which ensures power system resilience against unfavorable events. The case studies illustrate how the presented model balances between shutting off customers, DER installation, and line addition/modification while inhibiting wildfire ignition risk. Besides, the results suggest that, with appropriate transmission switching, DER installation can be the optimal choice for meeting the growing demand while limiting wildfire ignition risk. The implications of different risk level choices are illustrated, and the impacts of uncertainties on the expansion decisions are explored. Finally, the results of applying the proposed model to the IEEE 118-bus test network are illustrated.