Minimax Problem Research Articles

Fenton type minimax problems for sum of translates functions

Following an insightful work of P. Fenton, we investigate sum of translates functions F(x,t):=J(t)+∑j=1nνjK(t−xj), where J:[0,1]→R_:=R∪{−∞} is a “sufficiently non-degenerate” and upper-bounded “field function”, and K:[−1,1]→R_ is a fixed “kernel function”, concave both on (−1,0) and (0,1), x:=(x1,…,xn) with 0≤x1≤…≤xn≤1, and ν1,…,νn>0 are fixed. We analyze the behavior of the local maxima vector m:=(m0,m1,…,mn), where mj:=mj(x):=supxj≤t≤xj+1⁡F(x,t), with x0:=0, xn+1:=1; and study the optimization (minimax and maximin) problems infx⁡maxj⁡mj(x) and supx⁡minj⁡mj(x). The main result is the equality of these quantities, and provided J is upper semicontinuous, the existence of extremal configurations and their description as equioscillation points w, i.e., w satisfying m0(w)=m1(w)=⋯=mn(w). In our previous papers we obtained results for the case of singular kernels, i.e., when K(0)=−∞ and the field was assumed to be upper semicontinuous. In this work we get rid of these assumptions and prove common generalizations of Fenton's and our previous results, and arrive at the greatest generality in the setting of concave kernel functions.

EXPRESS: Distributionally Robust Appointment Scheduling That Can Deal with Independent Service Times

Consider a single server that should serve a given number of customers during a fixed period. The Appointment Scheduling Problem (ASP) determines the schedule of planned appointments that minimizes some cost function that accounts for both the cost of idle times and the cost of waiting. When service time distributions are fully specified, the ASP presents a much investigated computationally challenging stochastic program. When service time distributions are only partially specified, one can apply distributionally robust optimization (DRO) to find the schedule that minimizes costs in worst-case circumstances. We assume that only the mean, mean absolute deviation and range of the service times are known, and develop a DRO method that finds the optimal (minimax) schedule. For independent service times, the minmax problem becomes nonlinear and difficult, if not impossible, to solve exactly. Existing DRO methods for ASP with partial information (such as mean and variance) therefore consider relaxations that allow correlations between service times. Such relaxations have major repercussions, as the worst-case scenario will then be highly correlated. Our method thus deals with independent service times and finds the robust schedule as the solution to a linear program. We identify several new structural features of optimal robust schedules. We also apply the method to model extensions including sequencing and alternative objective functions.

Primal characterizations of stability of error bounds for semi-infinite convex constraint systems in Banach spaces

This article focuses on the stability of error bounds, both local and global, for semi-infinite convex constraint systems in Banach spaces. We present primal characterizations of the stability of these error bounds under small perturbations. These characterizations are expressed through the directional derivatives of the functions that define the systems. It is shown that ensuring stability of the error bounds is closely tied to verifying that the optimal values of several minimax problems, formulated using the directional derivatives, remain outside a certain neighbourhood of zero. Furthermore, this stability condition only requires that all component functions of the system share the same linear perturbation. When applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, these stability results yield primal criteria that guarantee the uniform boundedness of Hoffman's constants under perturbations in the problem data.

An integrated distributionally robust model for two-echelon patient appointment scheduling

We develop a distributionally robust optimization (DRO) model for the outpatient appointment scheduling problem of a set of patients served in two serial stages, consultation and examination. The arrival sequence of patients is known, and the problem of scheduling is to assign appointment time for each patient to minimize total cost with random service time for two serial stages. A max–min problem is formulated for the two-stage appointment scheduling as a whole, in which the waiting time exhibits a high degree of coupling due to the continuous two-stage process. To address this, we devise a two-stage network maximum flow model that provides an equivalent linear expression for the waiting time. For the inner maximum problem, we employ a conic programming approach for equivalent representation, incorporate the scheduling decision of the outer minimum problem, and convert the model to its equivalent copositive programming by taking the conic duality. We conduct numerical experiments and sensitivity analysis using real and simulated data, and the results verify the effectiveness of our proposed DRO model.

The Forecasting of the Spread of Infectious Diseases Based on Conditional Generative Adversarial Networks

New epidemics encourage the development of new mathematical models of the spread and forecasting of infectious diseases. Statistical epidemiology data are characterized by incomplete and inexact time series, which leads to an unstable and non-unique forecasting of infectious diseases. In this paper, a model of a conditional generative adversarial neural network (CGAN) for modeling and forecasting COVID-19 in St. Petersburg is constructed. It takes 20 processed historical statistics as a condition and is based on the solution of the minimax problem. The CGAN builds a short-term forecast of the number of newly diagnosed COVID-19 cases in the region for 5 days ahead. The CGAN approach allows modeling the distribution of statistical data, which allows obtaining the required amount of training data from the resulting distribution. When comparing the forecasting results with the classical differential SEIR-HCD model and a recurrent neural network with the same input parameters, it was shown that the forecast errors of all three models are in the same range. It is shown that the prediction error of the bagging model based on three models is lower than the results of each model separately.

Neural Operator Variational Inference Based on Regularized Stein Discrepancy for Deep Gaussian Processes.

Deep Gaussian process (DGP) models offer a powerful nonparametric approach for Bayesian inference, but exact inference is typically intractable, motivating the use of various approximations. However, existing approaches, such as mean-field Gaussian assumptions, limit the expressiveness and efficacy of DGP models, while stochastic approximation can be computationally expensive. To tackle these challenges, we introduce neural operator variational inference (NOVI) for DGPs. NOVI uses a neural generator to obtain a sampler and minimizes the regularized Stein discrepancy (RSD) between the generated distribution and true posterior in L2 space. We solve the minimax problem using Monte Carlo estimation and subsampling stochastic optimization techniques and demonstrate that the bias introduced by our method can be controlled by multiplying the Fisher divergence with a constant, which leads to robust error control and ensures the stability and precision of the algorithm. Our experiments on datasets ranging from hundreds to millions demonstrate the effectiveness and the faster convergence rate of the proposed method. We achieve a classification accuracy of 93.56 on the CIFAR10 dataset, outperforming state-of-the-art (SOTA) Gaussian process (GP) methods. We are optimistic that NOVI possesses the potential to enhance the performance of deep Bayesian nonparametric models and could have significant implications for various practical applications.

An Inexact Bundle Method for Semi-Infinite Minimax Problems

Semi-infinite minimax problems are widely utilized in various fields; however, there is a scarcity of algorithms that can directly tackle convex-convex and convex-concave semi-infinite minimax problems. An inexact algorithm based on the bundle method is introduced in this paper, which can be directly applied to solve both types of semi-infinite minimax problems. The novel algorithm offers the advantage of not requiring exact solutions for the inner maximization problem but only necessitates optimal solution with a certain level of precision. Additionally, the augmentation function method is employed to address nonconvergence issues encountered in traditional bundle method when dealing with convex-convex minimax problems. Global convergence of our algorithm is proven under reasonable assumptions. Numerical results from several examples demonstrate the effectiveness and practicality of our proposed approach.

Continuous Dependence of Sets in a Space of Measures and a Program Minimax Problem

Continuous Dependence of Sets in a Space of Measures and a Program Minimax Problem

About the Subgradient Method for Equilibrium Problems

Convergence results of the subgradient algorithm for equilibrium problems were mainly obtained using a Lipschitz continuity assumption on the given bifunctions. In this paper, we first provide a complexity result for monotone equilibrium problems without assuming Lipschitz continuity. Moreover, we give a convergence result of the value of the averaged sequence of iterates beyond Lipschitz continuity. Next, we derive a rate convergence in terms of the distance to the solution set relying on a growth condition. Applications to convex minimization and min–max problems are also stated. These ideas and results deserve to be developed and further refined.

Distributed Online Planning for Min-Max Problems in Networked Markov Games

Distributed Online Planning for Min-Max Problems in Networked Markov Games

An optimal algorithm for minimax problems with smooth components

An optimal algorithm for minimax problems with smooth components

Robust co-planning of transmission expansion and merchant energy storage investments considering long- and short-term uncertainties

The decreasing cost of energy storage technologies and the increasing utilization of renewable energy sources have made distributed transmission-scale energy storage economically viable. Merchant energy storage investors can collect significant profits through spatiotemporal energy arbitrage from congested transmission lines. This paper formulates a three-stage, four-level min–min–max–min problem from the viewpoint of power system operators and planners such that they will build new transmission lines and identify potential locations for profitable merchant energy storage investments. The first level determines transmission expansion decisions from the system operator’s perspective. The second level plans potential investments in merchant energy storage to ensure profitability at a desired rate of return. The third level realizes the worst-case situations of long-term uncertainties related to the expansion of wind farms and peak load growth in each regional zone, while the fourth level simulates market clearing for representative days, modeling short-term uncertainties. A novel iterative nested column and constraint generation (C&CG) technique is adopted to cope with numerical issues caused by binary variables of transmission lines and storage blocks and the enormous size of the problem. The case study of a modified IEEE 24-bus test system was used to assess the effectiveness of the proposed algorithm.

Optimality conditions for differentiable linearly constrained pseudoconvex programs

AbstractThe aim of this paper is to study optimality conditions for differentiable linearly constrained pseudoconvex programs. The stated results are based on new transversality conditions which can be used instead of complementarity ones. Necessary and sufficient optimality conditions are stated under suitable generalized convexity properties. Moreover, two different pairs of dual problems are proposed and weak and strong duality results proved. Finally, it is shown how transversality conditions can be applied to characterize optimality of convex quadratic problems and to efficiently solve a particular class of Max-Min problems

Constrained tensorial total variation problem based on an alternating conditional gradient algorithm

The present paper aims to regularize ill-posed high-dimensional problems using a constrained total variation minimization approach. The analysis of the continuous model via the dual formulation converts this minimization problem into a constrained minimax problem. Three main issues stand before the resolution of this problem: The minimization over a constraint, the high dimensional representation of the model, and the non-linearity and non-differentiability of the cost function. On the one hand, multidimensional data analysis will be performed using the tensor representation. On the other hand, a new alternating approach based on a proposed pseudo-Lagrangian operator will be developed to deal with the non-linearity and the non-differentiability of this constrained optimization problem. The proposed technique aims to decompose the main tensorial problem into feasible subproblems that can be solved using the conditional gradient method. The convergence of the proposed algorithm is proved and some numerical results are given to illustrate the effectiveness of the presented approach in different fields of image and video processing.

Robust cooperative control strategy for a platoon of connected and autonomous vehicles against sensor errors and control errors simultaneously in a real-world driving environment

In a real-world driving environment, a platoon of connected and autonomous vehicles (CAVs) is subject to many internal and external disturbances, resulting in uncertain vehicle dynamics. In general, the disturbances can be categorized into two types: disturbances due to vehicle sensor errors (e.g., GPS error) and disturbances due to vehicle control errors (e.g., actuator delay). In the literature, many control strategies have been proposed to improve the robustness of the CAV platoon against uncertain vehicle dynamics induced by these disturbances. However, most of these strategies only consider one type of disturbance and cannot tackle both types of disturbances simultaneously. Furthermore, they are designed to maximize the benefits of each vehicle in the platoon independently, which can deteriorate the performance of the platoon. To address these problems, this study proposes a robust cooperative control (RCC) strategy to maneuver the vehicles in the platoon cooperatively to counteract the impacts of both types of disturbances. The RCC strategy is developed based on a minimax problem, where the maximization subproblem seeks to find the worst inputs for the uncertainty terms in the vehicle dynamics equation to minimize the platoon performance, while the minimization subproblem seeks to find the optimal control decisions for all subsequent vehicles to maximize the platoon performance in the worst case. To solve the minimax problem, this study proposes a globally convergent solution algorithm. It can solve the minimax problem very efficiently to enable real time deployment of the RCC strategy. Numerical application indicates that compared to the existing methods, the RCC strategy can dramatically improve the robustness of the CAV platoon against the uncertain vehicle dynamics induced by both vehicle state detection errors and vehicle control errors. Therefore, it can maneuver the CAV platoon safely and efficiently in a real-world driving environment.

Robust Design Optimization of Electric Machines with Isogeometric Analysis

In electric machine design, efficient methods for the optimization of the geometry and associated parameters are essential. Nowadays, it is necessary to address the uncertainty caused by manufacturing or material tolerances. This work presents a robust optimization strategy to address uncertainty in the design of a three-phase, six-pole permanent magnet synchronous motor (PMSM). The geometry is constructed in a two-dimensional framework within MATLAB®, employing isogeometric analysis (IGA) to enable flexible shape optimization. The main contributions of this research are twofold. First, we integrate shape optimization with parameter optimization to enhance the performance of PMSM designs. Second, we use robust optimization, which creates a min–max problem, to ensure that the motor maintains its performance when facing uncertainties. To solve this bilevel problem, we work with the maximal value functions of the lower-level maximization problems and apply a version of Danskin’s theorem for the computation of generalized derivatives. Additionally, the adjoint method is employed to efficiently solve the lower-level problems with gradient-based optimization. The paper concludes by presenting numerical results showcasing the efficacy of the proposed robust optimization framework. The results indicate that the optimized PMSM designs not only perform competitively compared to their non-robust counterparts but also show resilience to operational and manufacturing uncertainties, making them attractive for industrial applications.

Boosting sharpness-aware training with dynamic neighborhood

Learning algorithms motivated by minimizing the sharpness of loss surface is a hot research topic in improving generalization. The existing methods usually solve a constrained min–max problem to minimize sharpness and find flat minima. However, most constraints (i.e., the neighborhood of the sharpness) are inappropriate, leading to sub-optimal results. This paper theoretically explores the optimal neighborhood from the view of Probably Approximately Correct-Bayesian (PAC-Bayesian) framework. A closed form of the optimal neighborhood is provided. This neighborhood is determined by the Hessian matrix and the scales of parameters. Then a generalization bound is derived that serves as a guiding principle in the design of the sharpness minimization algorithm. The Dynamic neighborhood-based Sharpness-Aware Minimization algorithm is proposed, which can adaptively adjust the neighborhood during the training process to gain better performance. Also, the algorithm is proved can convergent at the rate O(logT/T). Experimental results demonstrate that the proposed algorithm outperforms the other methods (e.g., accuracy ＋2.86% over baseline on CIFAR-100 for VGG-16).

Boosting generalization of fine-tuning BERT for fake news detection

Using deep neural networks to detect fake news has gradually become an extremely important field of research. Recent studies, which adopt fine-tuning BERT to detect fake news, have yielded promising results. Unfortunately, fine-tuning BERT is prone to overfitting the training data, which may reduce its generalization ability. The problem remains challenging when one tries to generalize the detector to unseen fake news. First, we investigate the effects of several fine-tuning strategies on the generalization of BERT models used in fake news detection. Specifically, we explore three fine-tuning strategies, which include initialization of model parameters, freezing of model parameters, and training iterations for fine-tuning. Besides, we propose an adversarial fine-tuning strategy to boost the generalization of fine-tuning BERT for fake news detection. In detail, we define the BERT fine-tuning procedure as a mini-max optimization problem. We also propose a novel adversarial fine-tuning training strategy based on feature regularization, called FGM-FRAT, to further improve the generalization. Specifically, we propose to adopt the model representation to guide the generation of adversarial examples during adversarial training on the word vector space. The proposed method outperforms state-of-the-art fake news detection methods. Extensive experiments on several benchmark databases, which include FakeNewsNet, BuzzFeedNews, LIAR, WELFake, and KaggleFakeNews, show that using the proposed method improves the generalization accuracy from 61% to 73% for BERT. It indicates that our FGM-FRAT can greatly improve the generalization of fine-tuning BERT for fake news detection. Moreover, the proposed method also can be extended to other pre-trained language models and other text classification tasks.

Using the Ellipsoid Method for Sylvester's Problem and its Generalization

Sylvester's problem or the problem of the smallest bounding circle is the problem of constructing a circle of the smallest radius that contains a finite set of points on the plane. In n-dimensional space, it corresponds to the problem of the smallest bounding hypersphere, which can be formulated as the problem of minimizing a convex piecewise quadratic function. The article is dedicated to study of the ellipsoid method application for solving this problem and the minimax convex programming problem, which is equivalent to the generalized problem of the smallest bounding hypersphere. The generalized problem consists in finding the center of a sphere in an n-dimensional space that has a minimal radius and contains a finite set of n-dimensional spheres given by their centers and radii. The article consists of 3 sections. Section 1 describes the emshor algorithm – the algorithm of the ellipsoid method for problem of minimization of an arbitrary convex function, proves its convergence theorems, gives a geometric interpretation of the algorithm, which is based on the use of a minimum volume ellipsoid. Octave implementation of the emshor algorithm is presented here, which can be successfully applied for non-smooth convex functions minimization if the number of variables is n = 2 - 30. In Section 2, the sylvester1 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimizing a convex piecewise quadratic function, which is equivalent to the problem of finding a sphere of minimum radius for a finite set of points. In Section 3, the sylvester2 algorithm is built, which is an application of the emshor algorithm for solving the problem of minimization of a convex function, which is equivalent to the generalized problem of finding a sphere of minimum radius for a finite set of spheres with given centers and radii. The results of testing the sylvester1 and sylvester2 algorithms demonstrate high working speed for modern computers and high accuracy in terms of optimal value of the objective function when solving problems in n-dimensional spaces for small values n = 2 - 30. Keywords: ellipsoid method, convex function, Sylvester's problem, piecewise smooth function, minimax problem.

Incorporating local uncertainty management into distribution system planning: An adaptive robust optimization approach

As the penetration of renewable energy sources (RES) significantly increases, their inherent uncertainty has brought about a paradigm shift in distribution system planning and operation. The existing planning methodologies usually consider the uncertainty during the investment and day-ahead stages while neglecting the interactive power fluctuations during the intraday stage, resulting in planning schemes with a risk of inadequate regional stability. In light of this, this paper proposes a novel adaptive robust distribution system planning approach that integrates local uncertainty management (LUM) to provide a comprehensive and refined consideration of uncertainty. First, we introduce the concept of LUM that can effectively address the deviations between predicted and actual power of RES and load. Second, we propose the adaptive robust planning model for the distribution system, in which the investment and day-ahead operation are considered in the first stage, while LUM is considered in the second stage to maintain the local stability of interactive power. The proposed model can actively manage uncertainties of RES and load demand, reaching the optimal balance between the investment cost, operational cost, and LUM cost. Finally, by employing the column-and-constraint generation (C&CG), the adaptive robust planning model is decoupled into a master min problem and a slaver max-min problem, after which the max-min problem is converted into a single mixed-integer linear programming (MILP) problem through the strong duality theory and big-M method. The effectiveness of the proposed model is demonstrated using a real-world distribution system located in Fujian Province, China. The simulation results indicate that the proposed model achieves a quantitative decrease of approximately 10% in total cost compared to deterministic planning. Sensitivity analysis is also conducted to reveal the trade-off between economic cost and robustness level.