Minimax Optimization Research Articles

Newton and interior-point methods for (constrained) nonconvex–nonconcave minmax optimization with stability and instability guarantees

We address the problem of finding a local solution to a nonconvex–nonconcave minmax optimization using Newton type methods, including primal-dual interior-point ones. The first step in our approach is to analyze the local convergence properties of Newton’s method in nonconvex minimization. It is well established that Newton’s method iterations are attracted to any point with a zero gradient, irrespective of it being a local minimum. From a dynamical system standpoint, this occurs because every point for which the gradient is zero is a locally asymptotically stable equilibrium point. We show that by adding a multiple of the identity such that the Hessian matrix is always positive definite, we can ensure that every non-local-minimum equilibrium point becomes unstable (meaning that the iterations are no longer attracted to such points), while local minima remain locally asymptotically stable. Building on this foundation, we develop Newton-type algorithms for minmax optimization, conceptualized as a sequence of local quadratic approximations for the minmax problem. Using a local quadratic approximation serves as a surrogate for guiding the modified Newton’s method toward a solution. For these local quadratic approximations to be well-defined, it is necessary to modify the Hessian matrix by adding a diagonal matrix. We demonstrate that, for an appropriate choice of this diagonal matrix, we can guarantee the instability of every non-local-minmax equilibrium point while maintaining stability for local minmax points. Using numerical examples, we illustrate the importance of guaranteeing the instability property. While our results are about local convergence, the numerical examples also indicate that our algorithm enjoys good global convergence properties.

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.

A robust design strategy for active control of scattered sound based on virtual sensing.

Combining virtual sensing (VS) with scattered sound control enables active acoustic cloaking when there are limitations in sensor configurations. The remote microphone method and additional filter method (AFM) are two common VS methods, and both can be divided into the training and control stages; the consistency of the environments in these two stages is essential for the control system. This paper investigates the effects of uncertainties in the incidence angle of the detection wave on these two VS-based scattered sound control methods. Following the analysis, we propose a robust design strategy based on the optimal layout of physical sensors, and the placement scheme is chosen by minimax optimization. The feasibility of the proposed strategy is verified by numerical simulations of a finite-length cylindrical scattering model. The results demonstrate that the proposed strategy can effectively reduce the degradation of system performance over the examined range of variations in the detection waves. In particular, the AFM-based method, combined with optimal placement, shows a remarkable improvement in robustness. It improves the worst noise reduction by approximately 14.5 and 10.8 dB on average, respectively, compared with the uniform placement and the direct control method based on the Wiener solution.

Adversarial Heterogeneous Graph Neural Network for Robust Recommendation

Recommendation systems play a vital role in identifying the hidden interactions between users and items in online social networks. Recently, graph neural networks (GNNs) have exhibited significant performance gains by modeling the information propagation process in graph-structured data for a recommendation. However, existing GNN-based methods do not have broad applicability to heterogeneous graphs that integrate auxiliary data with diverse types. Moreover, graph structures are susceptible to noise and even unnoticed malicious perturbations, as perturbations from connected nodes can create cumulative effects on a target node in the graph. To enhance the robustness and generalization of GNN-based recommendations, we propose a new optimization model named Adversarial Heterogeneous Graph Neural Network for RECommendation (AHGNNRec). First, AHGNNRec learns user and item embeddings by exploring the distinct contributions of various types of interactions between users and items using a hierarchical heterogeneous graph neural network (HGNN). Second, to produce more robust embeddings for recommendations, we employ the adversarial training (AT) method to optimize the HGNN layers. AT is a min-max optimization training process where the generated adversarial fake nodes from normal nodes with intentional perturbations try to maximally deteriorate the recommendation performance. Following this, we learn about these adversarial user or item nodes by minimizing the impact of an additional regularization term for the recommendation. The experimental outcomes on two real-world benchmark datasets demonstrate the effectiveness of AHGNNRec.

Minimax estimation of distances on a surface and minimax manifold learning in the isometric-to-convex setting

Abstract We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction—the tangential Delaunay complex—for that purpose. We then turn to manifold learning and argue that a variant of Isomap where the distances are instead computed on a reconstructed surface is minimax optimal for the isometric variant of the problem.

Energy Management Study of Interconnected Microgrids Considering Pricing Strategy Under the Stochastic Impacts of Correlated Renewables

The exponential augmentation of electricity consumption and the restructuring of the conventional power industry resulted in the emergence of microgrids (MGs). The nondispatchability and random attributes of the integrated renewable energy resources (RERs) challenge MGs scheduling operation, for which the multi-MGs system's energy management (EM) study has been gaining paramount importance and studied in this article. This article contemplates both internal and external markets for MGs’ effective participation in energy trading, which involves the energy exchange among MGs and that with the utility grid (UG). The energy pricing considers two conflicting objectives: MGs’ goal to improve their economy by reducing their purchasing prices and reliance on UG; the distribution network operator's aim to maximize its profit from the deployed market. Also, this article implements hybrid scenario and copula-based Monte Carlo techniques to assess the intermittencies associated with load demands, plug-in hybrid vehicle charging demands, and correlated RERs generations. The EM framework is formulated as a max–min optimization problem solved by a metaheuristic fuzzified jellyfish search optimization algorithm. Several simulation outcomes under different charging scenarios are reported, suggesting a 1.5% and 3.2% reduction in the multi-MG system's operational cost compared with the particle swarm optimization algorithm considering the best charging strategy during the summer and winter seasons.

Post-Processed Posteriors for Banded Covariances

We consider Bayesian inference of banded covariance matrices and propose a post-processed posterior. The post-processing of the posterior consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior, which does not satisfy any structural restrictions. In the second step, the posterior samples are transformed to satisfy the structural restriction through a post-processing function. The conceptually straightforward procedure of the post-processed posterior makes its computation efficient and can render interval estimators of functionals of covariance matrices. We show that it has nearly optimal minimax rates for banded covariances among all possible pairs of priors and post-processing functions. Additionally, we provide a theorem on the credible set of the post-processed posterior under the finite dimension assumption. We prove that the expected coverage probability of the 100(1−α)% highest posterior density region of the post-processed posterior is asymptotically 1−α with respect to any conventional posterior distribution. It implies that the highest posterior density region of the post-processed posterior is, on average, a credible set of conventional posterior. The advantages of the post-processed posterior are demonstrated by a simulation study and a real data analysis.

Method for control by orbital spacecraft magnetic cleanliness based on multiple magnetic dipole models with consideration of their uncertainty

Aim. Development of method for control by orbital spacecraft magnetic cleanliness based on multiple magnetic dipole models using compensation of the initial magnetic field with consideration of magnetic characteristics uncertainty. Methodology. Orbital spacecraft multiple magnetic dipole models calculated as solution of nonlinear minimax optimization problem based on near field measurements for prediction orbital spacecraft far magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field. Weight matrix calculated as inverse covariance matrix of random errors vector. Values of magnetic moments and coordinates of placement of compensating magnetic dipoles for compensation of the orbital spacecraft initial magnetic field also calculated as solution of nonlinear minimax optimization problem. Both solutions of this nonlinear minimax optimization problems calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction spacecraft far magnetic field magnitude based on orbital spacecraft multiple magnetic dipole models using near field measurements and compensation of the initial magnetic field with consideration of orbital spacecraft magnetic characteristics uncertainty for ensuring the orbital spacecraft magnetic cleanliness. Originality. The method for control by orbital spacecraft magnetic cleanliness based on multiple magnetic dipole models using compensation of the initial magnetic field with consideration of magnetic characteristics uncertainty is developed. Practical value. An important practical problem of ensuring orbital spacecraft magnetic cleanliness based on orbital spacecraft multiple magnetic dipole models using near field measurements and compensation of the initial magnetic field with consideration of orbital spacecraft magnetic characteristics uncertainty solved.

On the Linear Convergence of Extragradient Methods for Nonconvex–Nonconcave Minimax Problems

Recently, minimax optimization has received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods. However, the nonconvexities and nonconcavities present in these problems, prevents the application of typical gradient descent/ascent, which is known to diverge even in bilinear problems. Recently, it was shown that the proximal point method (PPM) converges linearly for a family of nonconvex–nonconcave problems. In this paper, we study the convergence of a damped version of the extragradient method (EGM), which avoids potentially costly proximal computations, relying only on gradient evaluation. We show that the EGM converges linearly for smooth minimax optimization problems satisfying the same nonconvex–nonconcave condition needed by the PPM. Funding: H. Lu was supported by The University of Chicago Booth School of Business Benjamin Grimmer was supported by Johns Hopkins Applied Mathematics and Statistics Department.

Riemannian Hamiltonian Methods for Min-Max Optimization on Manifolds

Riemannian Hamiltonian Methods for Min-Max Optimization on Manifolds

Enhancing adversarial robustness of quantum neural networks by adding noise layers

Abstract The rapid advancements in machine learning and quantum computing have given rise to a new research frontier: quantum machine learning. Quantum models designed for tackling classification problems possess the potential to deliver speed enhancements and superior predictive accuracy compared to their classical counterparts. However, recent research has revealed that quantum neural networks (QNNs), akin to their classical deep neural network-based classifier counterparts, are vulnerable to adversarial attacks. In these attacks, meticulously designed perturbations added to clean input data can result in QNNs producing incorrect predictions with high confidence. To mitigate this issue, we suggest enhancing the adversarial robustness of quantum machine learning systems by incorporating noise layers into QNNs. This is accomplished by solving a Min-Max optimization problem to control the magnitude of the noise, thereby increasing the QNN’s resilience against adversarial attacks. Extensive numerical experiments illustrate that our proposed method outperforms state-of-the-art defense techniques in terms of both clean and robust accuracy.

Sensor network design for post-combustion CO[formula omitted] capture plants: Computational efficiency and robustness

State estimation is crucial for the monitoring and control of post-combustion CO2 capture plants (PCCPs). The performance of state estimation is highly reliant on the configuration of sensors. In this work, we consider the problem of sensor selection for PCCPs and propose a computationally efficient method to determine an appropriate number of sensors and the corresponding placement of the sensors. The objective is to find the (near-)optimal set of sensors that provides the maximum degree of observability for state estimation while satisfying the budget constraint. Specifically, we resort to the information in the sensitivity matrix calculated around the operating region of a PCCP to quantify the degree of observability of the entire system corresponding to the placed sensors. The sensor selection problem is converted to an optimization problem and is efficiently solved by a one-by-one removal approach through sensitivity analysis. Next, we extend our approach to study the fault tolerance of the selected sensors to sensor malfunction. The robust sensor selection problem is to find a sensor network that gives good estimation performance even when some of the sensors fail, thereby improving the overall system robustness. The robust sensor selection problem is formulated as a max–min optimization problem. We show how the proposed approach can be adapted to solve the sensor selection max–min optimization problem. By implementing the proposed approaches, the sensor network is configured for the PCCP efficiently.

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.

The global convergence properties of an adaptive QP-free method without a penalty function or a filter for minimax optimization.

In this paper, we proposed an adaptive QP-free method without a penalty function or a filter for minimax optimization. In each iteration, solved two linear systems of equations constructed from Lagrange multipliers and KKT-conditioned NCP functions. Based on the work set, the computational scale is further reduced. Instead of the filter structure, we adopt a nonmonotonic equilibrium mechanism with an adaptive parameter adjusted according to the result of each iteration. Feasibility of the algorithm are given, and the convergence under some assumptions is demonstrated. Numerical results and practical application are reported at the end.

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.

Large-Scale Rate-Splitting Multiple Access in Uplink UAV Networks: Effective Secrecy Throughput Maximization Under Limited Feedback Channel

Unmanned aerial vehicles (UAVs) are capable of improving the performance of next generation wireless systems. However, their communication performance is prone to both channel estimation errors and potential eavesdropping. Hence, we investigate the effective network secrecy throughput (ENST) of the uplink UAV network, in which rate-splitting multiple access (RSMA) is employed by each legitimate user for secure transmission under the scenario of massive access. To maximize the ENST, the transmission rate versus power allocation relationship is formulated as a max-min optimization problem, relying on realistic imperfect channel state information (CSI) of both the legitimate users and passive eavesdroppers ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Eves$</tex-math></inline-formula> ). In the model considered, each user transmits a superposition of two messages to a UAV base-stations (UAV-BS), each having different transmit power and the UAV-BS adopts a successive interference cancellation (SIC) technique to decode the received messages. Given the non-convexity of the problem, it is decoupled into a pair of sub-problems. In particular, we derive a closed form expression for the optimal rate-splitting fraction of each user. Then, given the optimal rate-splitting fraction of each user, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -constrainted transmit power of each user is calculated by harnessing sequential parametric convex approximation (SPCA) programming. Our simulation results confirm that the scheme conceived significantly improves the ENST compared to both the existing orthogonal and non-orthogonal benchmarks.

Post-processed posteriors for sparse covariances

We consider Bayesian inference of sparse covariance matrices and propose a post-processed posterior. This method consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior without considering the sparse structural assumption. The posterior samples are transformed in the second step to satisfy the sparse structural assumption through a generalized thresholding function. This non-traditional Bayesian procedure is justified by showing that the post-processed posterior attains the optimal minimax rates under the spectral norm loss in high-dimensional settings. We also propose the post-processed posterior for contaminated data and apply it to the estimation of the sparse idiosyncratic covariance of the approximate factor model. The advantages of our method are demonstrated via a simulation study and a real data analysis with S&P 500 data.

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.

Optimization of spatial arrangement of magnetic field sensors of closed loop system of overhead power lines magnetic field active silencing

Aim. Development of a method for optimization of spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active silencing canceling of the magnetic field generated by overhead power lines. Methodology. Spatial arrangement and angular position of magnetic field sensors of closed loop system of overhead power lines magnetic field active silencing determined based on binary preference relations of local objective for multi-objective minimax optimization problem, in whith the vector objective function calculated based on Biot–Savart law. The solution of this vector minimax optimization problem calculated based on nonlinear Archimedes algorithm of multi-swarm multi-agent optimization. Results. Results of simulation and experimental research of optimal spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active silencing of the magnetic field generated by overhead power lines with a barrel-type arrangement of wires. Originality. The method for optimization of spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active shielding of the magnetic field generated by overhead power lines is developed. Practical value. An important practical problem optimization of spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active silencing of the magnetic field generated by overhead power lines has been solved.