Max Optimization Research Articles

A Scalable Convex Min–Max Optimization Algorithm for Robust Control of Polytopic Positive Systems

ABSTRACTIn this paper, we investigate the robust control of a class of bilinear positive systems. The main objective is to minimize the norm of the controlled system while dealing with polytopic uncertainties. In contrast to traditional Lyapunov‐based robust control problems with polytopic uncertainties, we establish that minimizing the objective function over the vertices of the polytopic set, rather than considering its infinite elements, is adequate for achieving the globally optimal solution. Following this, we present a customized algorithm that relies on partial cutting plane to address the min–max problem. The introduced robust control strategy exhibits global convergence to the optimal solution and demonstrates scalability with the number of states and the number of vertices in the uncertain family. Simulation results validate the effectiveness of the proposed strategy in handling uncertainties.

Robust min-max (regret) optimization using ordered weighted averaging

In decision-making under uncertainty, several criteria have been studied to aggregate the performance of a solution over multiple possible scenarios. This paper introduces a novel variant of ordered weighted averaging (OWA) for optimization problems. It generalizes the classic OWA approach, which includes the robust min–max optimization as a special case, as well as the min–max regret optimization. We derive new complexity results for this setting, including insights into the inapproximability and approximability of this problem. In particular, we provide stronger positive approximation results that asymptotically improve the previously best-known bounds for the classic OWA approach.

Group decision making method for third-party logistics management: An interval rough cloud optimization model

Group decision making in third-party logistics service selection plays an essential role for improving service quality, increasing efficiency and reducing the net cost. Fuzzy and uncertain linguistic variables are commonly used to represent experts‘rankings in optimization problems. To recognize the limits of human cognition and subjectivity of human evaluations, several optimization approaches have been studied to select remanufacturing alternatives in decision making processes, however these methods have certain deficiencies such as lacking manipulation tools of diverse information, randomness, use of predefined parameters increasing uncertainty, interpersonal relations among evaluation criteria. The integration of interval numbers, rough approximations, and cloud model theory plays a significant role to model incomplete and inadequate information occurring in decision making problems. This research paper focuses on the integration of dual interval rough integrated cloud model with best-worst optimization technique, Multi-Attributive Border Approximation area Comparison (MABAC) and Weighted Aggregated Sum Product Assessment (WASPAS) approaches. A novel min–max optimization model based dual interval rough integrated cloud values is designed to compute the weight coefficients and consistency ratio for each criteria. The consistency of proposed optimization model is checked using a consistency ratio test. Secondly, the alternatives are ranked using the proposed DIRI cloud based MABAC and WASPAS approaches using interval clouds based weighted sum and weighted product coefficients, approximation area values and distance formulae. The significance of the proposed model is highlighted with a case study of third-party logistics service management of an electronic firm. The rationality and out-performance of the proposed methodology is studied by a comparative analysis with existing approaches and detailed sensitivity analysis on different variations of criteria weights and parameter.

Inverse Adversarial Diversity Learning for Network Ensemble.

Network ensemble aims to obtain better results by aggregating the predictions of multiple weak networks, in which how to keep the diversity of different networks plays a critical role in the training process. Many existing approaches keep this kind of diversity either by simply using different network initializations or data partitions, which often requires repeated attempts to pursue a relatively high performance. In this article, we propose a novel inverse adversarial diversity learning (IADL) method to learn a simple yet effective ensemble regime, which can be easily implemented in the following two steps. First, we take each weak network as a generator and design a discriminator to judge the difference between the features extracted by different weak networks. Second, we present an inverse adversarial diversity constraint to push the discriminator to cheat generators that all the resulting features of the same image are too similar to distinguish each other. As a result, diverse features will be extracted by these weak networks through a min-max optimization. What is more, our method can be applied to a variety of tasks, such as image classification and image retrieval, by applying a multitask learning objective function to train all these weak networks in an end-to-end manner. We conduct extensive experiments on the CIFAR-10, CIFAR-100, CUB200-2011, and CARS196 datasets, in which the results show that our method significantly outperforms most of the state-of-the-art approaches.

Simultaneous robust model predictive control and state estimation for nonlinear systems with state‐ and input‐dependent uncertainties

AbstractThe convergence and stability of uncertain nonlinear systems is a challenging problem in the nonlinear control area. Besides, in many practical cases, all states are not measurable and are affected by measurement noise. Based on this motivation, the first objective of this paper is to design a novel output feedback robust model predictive control approach for nonlinear systems with state‐ and input‐dependent uncertainties and measurement noise. This approach combines state estimation and robust model predictive control (MPC) into one min–max optimization and by solving the optimization, these two tasks are performed simultaneously. The studied nonlinear system comprises a linear part, a nonlinear part, and a function that denotes the state‐ and input‐dependent uncertainties. Therefore, the other objective is to reduce the computational complexity; thus, the system's nonlinear term and the aforementioned uncertainties are converted into additional disturbances. Subsequently, the optimization problem becomes a quadratic form, which leads to global convergence with the appropriate selection of objective function weights. Besides, this paper explores the convergence of the closed‐loop system states and the sufficient synthesis conditions to guarantee input‐to‐state stability. The implementation on a numerical example and a CSTR process demonstrate the applicability and reliability of the proposed approach.

Min–max optimization of node‐targeted attacks in service networks

AbstractThis article considers resilience of service networks that are composed of service and control nodes to node‐targeted attacks. Two complementary problems of selecting attacked nodes and placing control nodes reflect the interaction between the network operator and the network attacker. This interaction can be analyzed within the framework of game theory. Considering the limited performance of the previously introduced iterative solution algorithms based on non‐compact problem models, new compact integer programming formulations of the node attack optimization problem are proposed, which are based on the notion of pseudo‐components and on a bilevel model. The efficiency of the new formulations is illustrated by the numerical study that uses two reference networks (medium‐size and large‐size), and a wide range of the sizes of attacks and controllers placements.

An extended three-field principle to scale-bridge the granular micromechanics of polymer-bonded particulate materials

Observed phenomena of damage, viscoelasticity, and viscoplasticity in polymer-bonded particulate materials (PBPM), like polymer-bonded explosive (PBX), are accommodated in a scale-bridging constitutive model based on granular micromechanics. An extended three-field variational principle with min−max optimization problem is introduced to homogenize the bonded aggregates’ grain-scale contact mechanics. Similarly to prior applications of a three-field principle, volume constraints are invoked to embed microscale-to-macroscale correspondences for kinematic and static volumetric measures, like mean strain and its energy-conjugate stress invariant. Unlike prior applications, our three-field principle realizes a strain-driven model for effective constitutive behavior due to bonded intergranular contact. The resultant micro-sphere integration-based homogenization is demonstrated in comparison to existing publicly-available measurements to capture the extreme tension/compression constitutive asymmetry exhibited by PBPMs (specifically PBX-9501) during material softening.

Deterministic-based robust design optimization of composite structures under material uncertainty

We propose a new deterministic robust design optimization method for composite laminate structures under worst-case material uncertainty. The method is based on a simultaneous parametrization of topology and material and combines a design problem and a material uncertainty problem into a single min–max optimization problem which provides an efficient approach to handle variation of material properties in stiffness driven design optimization problems. An analysis is performed using a design problem based on a failure criterion formulation to evaluate the ability of the proposed method to generate robust composite designs. The design problem is solved using various loads, boundary conditions and manufacturing constraints. The designs generated with the proposed method have improved objective responses compared to the worst-case response of designs generated with nominal material properties and are less sensitive to the variation of material properties. The analysis indicates that the proposed method can be efficiently applied in a robust structural optimization framework.

A delayed subgradient method for nonsmooth convex-concave min–max optimization problems

In this paper, we aim to solve a convex-concave min–max optimization problem, where the convex-concave coupling function is nonsmooth in both variables. We propose a simple subgradient method which is simultaneously updating the iterates through their delayed subgradients at stale iterates of their own variable together with the current iterates of the other variable. We investigate the convergence behavior of the proposed method by providing an upper bound of the function value at the coupling averaged iterates to the saddle value. Specifically, the obtained result refers to the function value at the coupling averaged iterates as the function value of an approximate saddle point at the current iteration. The numerical results demonstrate that, with a suitable strategy of delays’ selections, the delayed subgradient method can achieve better convergence behaviors than its non-delayed counterpart.

Robust Multi-Criteria Traffic Network Equilibrium Problems with Path Capacity Constraints

With the progress of society and the diversification of transportation modes, people are faced with more and more complicated travel choices, and thus, multi-criteria route choosing optimization problems have drawn increased attention in recent years. A number of multi-criteria traffic network equilibrium problems have been proposed, but most of them do not involve data uncertainty nor computational methods. This paper focuses on the methods for solving robust multi-criteria traffic network equilibrium problems with path capacity constraints. The concepts of the robust vector equilibrium and the robust vector equilibrium with respect to the worst case are introduced, respectively. For the robust vector equilibrium, an equivalent min–max optimization problem is constructed. A direct search algorithm, in which the step size without derivatives and redundant parameters, is proposed for solving this min–max problem. In addition, we construct a smoothing optimization problem based on a variant version of ReLU activation function to compute the robust weak vector equilibrium flows with respect to the worst case and then find robust vector equilibrium flows with respect to the worst case by using the heaviside step function. Finally, extensive numerical examples are given to illustrate the excellence of our algorithms compared with existing algorithms. It is shown that the proposed min–max algorithm may take less time to find the robust vector equilibrium flows and the smoothing method can more effectively generate a subset of the robust vector equilibrium with respect to the worst case.

Covariance Matrix Adaptation Evolutionary Strategy with Worst-Case Ranking Approximation for Min–Max Optimization and Its Application to Berthing Control Tasks

In this study, we consider a continuous min–max optimization problem minx∈ 𝕏maxy∈ 𝕐f(x, y) whose objective function is a black-box. We propose a novel approach to minimize the worst-case objective functionF(x) = maxy∈ 𝕐f(x, y) directly using a covariance matrix adaptation evolution strategy in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation mechanism. We develop two variants of worst-case ranking approximation combined with a covariance matrix adaptation evolution strategy and approximate gradient ascent as numerical solvers for the inner maximization problem. Numerical experiments show that our proposed approach outperforms several existing approaches when the objective function is a smooth strongly convex–concave function and the interaction betweenxandyis strong. We investigate the advantages of the proposed approach for problems where the objective function is not limited to smooth strongly convex–concave functions. The effectiveness of the proposed approach is demonstrated in the robust berthing control problem with uncertainty.

Nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic modelling of saturated soils under large deformation

This paper presents the nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic analysis of saturated soils subject to large deformation, utilising the generalised Hellinger-Reissner variational principle to reformulate the governing equations for saturated soil dynamics into a min–max optimisation problem. With finite element discretisation and nodal integration over cells, the problem is transformed into a standard second-order cone programming problem, efficiently resolved using the advanced primal–dual interior point method. The N-PFEM method has several advantages, including the use of linear triangular elements without volumetric locking issues, the avoidance of regularisation techniques, and the elimination of tedious variable mapping after remeshing. The numerical model is validated for large deformation analysis of saturated soils with a series of benchmarks against available analytical and numerical solutions, with a case study of the deformation of an embankment considering stone column reinforcement also carried out. This N-PFEM framework offers an effective and efficient simulation approach for the evolutionary behaviour of saturated soils with large deformation in complex geotechnical configurations of practical relevance.

Distributionally robust unsupervised domain adaptation

Obtaining ground-truth label information from real-world data along with uncertainty quantification can be difficult or even infeasible. In the absence of labeled data for a certain task, unsupervised domain adaptation (UDA) techniques have shown great accomplishment by learning transferable knowledge from labeled source domain data and applying it to unlabeled target domain data, yet scarce studies consider addressing uncertainties under domain shifts to improve the model robustness. Distributionally robust learning (DRL) is emerging as a high-potential technique for building reliable learning systems that are robust to distribution shifts. In this study, we propose a distributionally robust unsupervised domain adaptation (DRUDA) method to enhance the machine learning model generalization ability under input space perturbations. The DRL-based UDA learning scheme is formulated as a min–max optimization problem by optimizing worst-case perturbations of the training source data. Our Wasserstein distributionally robust framework can reduce the shifts in the joint distributions across domains. The proposed DRUDA has been tested on digit datasets and the Office-31 dataset, compared with other state-of-the-art domain adaptation techniques. Our experimental results show that the proposed DRUDA leads to improvements in domain adaptation accuracy performance on target domains.

Resource allocation and BER performance analysis of NOMA based cooperative networks

Resource sharing and management can significantly improve the performance and spectral efficiency of wireless systems. In power domain non-orthogonal multiple access (NOMA) systems, users share a common bandwidth for simultaneous data transmission and therfore, the spectral efficiency of the system will be improved at the expense of increased complexity. Provided proper power allocation, it is possible to detect symbols at receivers. In this paper, the downlink of a cellular system is considered where a base station (BS) communicates with two users using NOMA with the assistance of a decode-and-forward (DF) relay. In the proposed scheme, receivers combine received signals from direct and cooperative links, and decode symbols by employing successive interference cancellation (SIC). The bit error rate (BER) performance of the proposed scheme is analyzed and the optimal power allocation is also derived through a Min–Max optimization, and a closed form approximate solution is proposed for binary phase shift keying (BPSK) modulation. Furthermore, it is proved that the proposed receiver achieves full diversity order for all users for proper choice of power allocation. Simulation results also corroborate BER and diversity analysis.

Distributed robust moving horizon estimation for multisensor systems with stochastic and norm‐bounded uncertainties

SummaryIn this study, a distributed robust moving horizon estimation (DRMHE) approach is proposed for time‐varying multisensor systems affected by stochastic and norm‐bounded uncertainties. First, by considering the uncertainties in all model matrices, stochastic min–max optimization problems in local and collective forms are presented to achieve the proposed estimator. Then, by converting the problems to robust regularized least‐squares problems with uncertain parameters, closed‐form solutions are obtained for estimations in both local and collective forms. Furthermore, the stability of the proposed estimator is investigated under some appropriate conditions. At last, two different examples are employed to show the robust performance and superiority of the proposed DRMHE approach compared to the existing methods.

Optimal control of inventory level for perishable goods with uncertain decay factor and uncertain forecast information: a new robust MPC approach

This paper deals with the inventory control in supply chains under the following assumptions: (1) perishable goods with uncertain deteriorating factor, (2) a future uncertain customer demand that, over a limited prediction horizon, belongs to a known compact set. The problem is to define a smooth control policy maximising the fulfilled customer demand and minimising the inventory level. This problem is here solved through a new Robust Model Predictive Control (RMPC) approach. This implies solving a min–max optimisation problem with hard constraints on the control effort (i.e. the sequence of replenishment orders). To drastically reduce the numerical complexity of this problem, the control signal is sought in the space of B-spline functions, which are known to be universal approximators admitting a parsimonious parametric representation. This allows us: (1) to reduce the number of both decision variables and constraints involved in the optimisation procedure, (2) to reformulate the numerically demanding minimisation of the worst case cost functional as a simpler Weighted Constrained Robust Least Squares (WCRLS) estimation problem. The WCRLS algorithm can be efficiently solved using interior point methods. A rigorous analysis of stability and feasibility conditions is provided.

A mixed formulation of limit analysis for seismic slope stability

This paper presents a mixed formulation based on the node-based smoothed finite-element method for estimating seismic slope stability. The final form of Hellinger–Reissner is cast as the min–max optimisation problem with conic constraints. Solving the second-order cone programming directly determines both master fields, stress and displacement. It is shown from the numerical results that the stability numbers obtained are almost equal to the exact collapse loads for the relatively flat slopes. Moreover, the effects of the dilation angle on the seismic stability number are studied by incorporating a simple iterative technique in the algorithm cast as max-min programming, arriving at the observation that both the static and the seismic stability numbers are susceptible to the dilation angle. Although the magnitude of the seismic stability number largely depends on the vertical seismic acceleration for a broad range of typical slope angles, the seismic stability number for the relatively flat slopes in cohesive-frictional soils is likely independent of the seismic acceleration in the vertical direction.

Information and sensing beamforming optimization for multi-user multi-target MIMO ISAC systems

Integrated sensing and communication (ISAC) has been envisioned as a key enabler in the next-generation wireless networks. In this paper, we consider the joint information and sensing beamforming design in a multi-user and multi-target multi-input multi-output ISAC system, where a transmit BS and a sensing BS collaborate to sense targets, and the transmit BS sends information streams to communication users at the same time. To optimize the sensing performance and guarantee the communication throughput, we formulate a joint beamforming design problem to minimize the trace of the weighted Cramer–Rao bound of target parameters subject to the sum-rate constraint. The problem is challenging to solve due to the intricate non-convex objective function and constraints. We firstly exploit the weighted mean square error minimization (WMMSE) and semidefinite relaxation (SDR) techniques to devise a WMMSE–SDR algorithm that can achieve a KKT point of the problem. The SDR can be shown to be tight for a subproblem in the WMMSE–SDR algorithm, which implies zero duality for the subproblem. Based on this property and fractional programming techniques, we further reformulate the beamforming problem as a min–max form with simple constraints which then can be efficiently solved by first-order min–max optimization algorithms. Finally, the proposed algorithms are evaluated extensively in simulations. Numerical results show that both proposed algorithms can achieve promising performance in sensing and communication, and the low-complexity algorithm has a significantly reduced computation time.

Controlling interstory drift ratio profiles via topology optimization strategies

An approach to control the profiles of interstory drift ratios along the height of building structures via topology optimization is proposed herein. The theoretical foundation of the proposed approach involves solving a min–max optimization problem to suppress the maximum interstory drift ratio among all stories. Two formulations are suggested: one inherits the bound formulation and the other utilizes a p-norm function to aggregate all individual interstory drift ratios. The proposed methodology can shape the interstory drift ratio profiles into inverted triangular or quadratic patterns because it realizes profile control using a group of shape weight coefficients. The proposed formulations are validated via a series of numerical examples. The disparity between the two formulations is clear. The optimization results show the optimal structural features for controlling the interstory drift ratios under different requirements.

Model Dynamic Facility Location in Post-Disaster Areas in Uncertainty

Indonesia has many disaster-prone areas, natural disasters that occur in Indonesia in 2021 are 5,402 disasters. For disaster management in post-disaster areas, logistical planning is needed in the distribution of logistical assistance, it is estimated that the logistics costs of disaster assistance reach approximately 80% of the total costs in disaster management so that logistical assistance is an expensive activity of disaster relief. However, so far the process of distributing logistical assistance to disaster posts has not been evenly distributed. One of the causes of the unequal distribution is the inappropriate selection of distribution post locations. The facility location model is dynamic and has the objective function of minimizing the distance between emergency posts and refugee posts in terms of distribution of disaster relief goods in one cluster group. For grouping unsupervised learning data using a machine learning clustering algorithm, k-means. Model validation has been carried out using max run and max optimization 1000 times with results reaching 90%. This proves that the emergency facility location model can be used to determine the location of the emergency center, where the determination of the location of the emergency center has the closest distance to the request point/post shelter for disaster victims