Min-max Optimization Problem Research Articles

A Novel Method Designing Optimal Observer Gain for Robust Fault Detection

This paper considers the design of optimal gain of set-value observer for robust fault detection (FD) from a new perspective. Set-based robust FD can be implemented by real-timely check whether the measured output is contained in the corresponding output estimation set or not. Our proposed method focuses on designing observer gains at each step to maximize the exclusion tendency of the measured output from the output estimation zonotope to achieve the goal of robust FD. The key deign logic is formulated as a two-level min-max optimization problem, which is further equivalently transformed into a non-convex single-level programming based on duality theory. Finally, this non-convex single-level programming can be efficiently solved via linear programming and matrix decomposition. At the end, a circuit model is used to illustrate the effectiveness of the proposed method.

Polytopic Robust Distributed Kalman Consensus Filter for Sensor Networks

Distributed estimation is a fundamental task performed by sensor networks, often achieved through the Kalman consensus filter. However, it requires exact models, which is seldom possible in practice since parametric uncertainties are usually unavoidable. We address in this paper the robust distributed filtering problem regarding linear discrete-time systems subject to polytopic uncertainties. We consider uncertainties in all parameter matrices of the target system and sensing models. First, we introduce a centralized filter, obtained as the solution to a min-max optimization problem whose cost function collectively weights all the underlying polytope vertices. Then, we derive a fully distributed version of this filter by applying the hybrid consensus on measurements and information approach. The estimators are recursive and do not rely on numerical solvers, appealing features for real-time applications. We validate our approach and assess its performance with a numerical example, comparing it with other robust distributed strategies.

T-S fuzzy-model-based adaptive cruise control for longitudinal car-following considering vehicle lateral stability

Adaptive cruise control is one of the essential technologies of advanced driver assistance systems, which is used to maintain a safe distance between an ego vehicle and a preceding vehicle and has been extensively applied in the automotive industry and control community. Note that some vehicle manoeuvres may approach handling limits to prevent collisions under complex road conditions, which often leads to vehicle lateral instability while cruising. In this study, a T-S fuzzy model predictive control framework is applied to the problem of adaptive cruise control. Variations in the preceding vehicle velocity and road surface conditions are considered to formulate adaptive cruise control as a tracking control problem of a T-S fuzzy system subject to parameter uncertainties and external persistent perturbations. Then, a robust positively invariant set is introduced to derive an admissible T-S fuzzy controller by solving a min-max optimization problem under a series of linear matrix inequality constraints. Finally, a CarSim/MATLAB joint simulation is conducted to illustrate the effectiveness of the proposed method, which ensures longitudinal adaptive cruise control for a car-following scenario with lateral vehicle stability.

Solving Maxmin Optimization Problems via Population Games

Solving Maxmin Optimization Problems via Population Games

MIMO radar robust waveform-filter design for extended targets based on Lagrangian duality

We address the robust waveform-filter design problem for extended targets with a colocated Multiple-Input Multiple-Output (MIMO) radar . The goal is to maximize the worst-case Signal-to-Interference-pulse-Noise Ratio (SINR) at the receiver against the uncertain Target Impulse Response (TIR) with the Peak-to-Average Ratio (PAR) constraint imposed on the waveform, which results in a non-convex minimax optimization problem. Two kinds of uncertainty sets for the TIR, namely, the spherical set and the annular set, are considered. Combing the duality theory in optimization and the Semi-Definite Relaxation (SDR) technique, we devise the Lagrangian Duality Semi-Definite Relaxation (LDSDR) and the Lagrangian Duality Double SemiDefinite Relaxation (LDDSDR) algorithms to tackle the associated waveform-filter design problems against the spherical and the annular sets, respectively. The convergences of the proposed algorithms are proved theoretically. Numerical results verify the effectiveness of the proposed algorithms. Compared with the current algorithms for the spherical set, the proposed LDSDR algorithm achieves the highest worst-case SINR with a reduced computational complexity

Accelerated variance-reduced methods for saddle-point problems

We consider composite minimax optimization problems where the goal is to find a saddle-point of a large sum of non-bilinear objective functions augmented by simple composite regularizers for the primal and dual variables. For such problems, under the average-smoothness assumption, we propose accelerated stochastic variance-reduced algorithms with optimal up to logarithmic factors complexity bounds. In particular, we consider strongly-convex-strongly-concave, convex-strongly-concave, and convex-concave objectives. To the best of our knowledge, these are the first nearly-optimal algorithms for this setting. • First accelerated stochastic variance-reduced algorithms with optimal up to logarithmic factors complexity bounds for non-bilinear empirical minimax problems.

Robust optimal control of connected and automated vehicle platoons through improved particle swarm optimization

This paper addresses the robust optimal control problem for connected and automated vehicle platoons subject simultaneously to uncertain parasitic actuation lag and input delays. First, by adopting the constant time headway policy (CTHP) for spacing, the intervehicular spacing error propagation transfer function is derived between adjacent vehicles downstream under a decentralized control law. Second, the Hurwitz stability region for the derived spacing error propagation transfer function is obtained in a unified stability analysis framework for the time headway and controller gains by utilizing the signature conditions for real polynomials. The robust optimal control problem is formulated as a min–max optimization problem, and then an improved multi-agent based particle swarm optimization algorithm is developed to seek the optimal values of the design parameters in the derived stability region such that a weighted objective function is minimized. Finally, simulation experiments are performed on a numerical example to illustrate the effectiveness of the proposed design approach.

Fault detection and isolation for uncertain linear systems: A robust moving horizon estimation scheme using LMIs

In this paper, a Fault Detection and Isolation (FDI) scheme based on moving horizon state estimation (MHE) ideas is analyzed for plants modeled as uncertain linear systems subject to actuator fault occurrences. The key point is to exploit the MHE approach to ameliorate the FDI capabilities of the diagnostic filter unit. The design is numerically accomplished via a semidefinite programming min-max optimization problem subject to H∞ requirements so that the robust MHE is in charge to cope with any source of uncertainties. The effectiveness of the proposed approach is then verified by considering a multi-area power system subject to anomalies on the power generation units.

Pure characteristics demand models and distributionally robust mathematical programs with stochastic complementarity constraints

We formulate pure characteristics demand models under uncertainties of probability distributions as distributionally robust mathematical programs with stochastic complementarity constraints (DRMP-SCC). For any fixed first-stage variable and a random realization, the second-stage problem of DRMP-SCC is a monotone linear complementarity problem (LCP). To deal with ambiguity of probability distributions of the involved random variables in the stochastic LCP, we use the distributionally robust approach. Moreover, we propose an approximation problem with regularization and discretization to solve DRMP-SCC, which is a two-stage nonconvex-nonconcave minimax optimization problem. We prove the convergence of the approximation problem to DRMP-SCC regarding the optimal solution sets, optimal values and stationary points as the regularization parameter goes to zero and the sample size goes to infinity. Finally, preliminary numerical results for investigating distributional robustness of pure characteristics demand models are reported to illustrate the effectiveness and efficiency of our approaches.

Iterative learning control for consensus of second-order multi-agent systems with interval uncertain topologies

Abstract This paper is devoted to the robust consensus tracking problem of second-order nonlinear multi-agent systems (MASs) with the interval uncertain topologies. For the second-order MASs including one leader agent and multiple follower agents, a control protocol is proposed by combining the iterative learning control scheme with the sliding mode control method. By analyzing the convergence of sliding mode variables, the consensus conditions including the unknown eigenvalues and the undetermined weight coefficient are obtained. In order to deal with the difficulties of weight coefficient design caused by the unknown eigenvalues of graphs, a min-max optimization problem is formulated based on the fastest convergence of the λ-norm of sliding mode variables, then the optimal weight coefficient is obtained by solving the min-max optimization problem. Moreover, for the undirected and directed interval uncertain graphs, two algorithms about the optimal weight coefficients are proposed, respectively. Finally, three numerical simulation examples are presented to demonstrate the effectiveness of the proposed methods.

Channel power optimization in WDM systems using co-evolutionary genetic algorithm

In this work, we present a co-evolutionary genetic (CEGA) algorithm to adapt the optical launch powers and optimize the signal-to-noise ratio (SNR) values based on maximizing the minimum SNR margin. The introduced co-evolutionary algorithm provides lower computational complexity rather than convex optimization and linear programming techniques, applicable for both static and time-critical dynamic networking. The enhanced Gaussian noise nonlinear model is exploited to take the physical-layer impairments into account, considering networks with partial spectrum utilization. To optimize the minimum SNR margin, we formulate the power allocation problem as a minimax optimization problem. To this end, a two-space genetic algorithm (GA) is proposed to reduce the computational complexity. The obtained results demonstrate that the introduced co-evolutionary algorithm outperforms the common optimization methods in terms of run time. It is shown that the computational complexity of proposed co-evolutionary algorithm is significantly lower than convex and single-space evolutionary approaches by several orders of magnitude. Moreover, the minimum SNR margin is improved by about 2.4 dB compared to a flat launch power optimization.

A decentralized adaptive momentum method for solving a class of min-max optimization problems

Min-max saddle point games have recently been intensely studied, due to their wide range of applications, including training Generative Adversarial Networks (GANs). However, most of the recent efforts for solving them are limited to special regimes such as convex-concave games. Further, it is customarily assumed that the underlying optimization problem is solved either by a single machine or in the case of multiple machines connected in centralized fashion, wherein each one communicates with a central node. The latter approach becomes challenging, when the underlying communications network has low bandwidth. In addition, privacy considerations may dictate that certain nodes can communicate with a subset of other nodes. Hence, it is of interest to develop methods that solve min-max games in a decentralized manner. To that end, we develop a decentralized adaptive momentum (ADAM)-type algorithm for solving min-max optimization problem under the condition that the objective function satisfies a Minty Variational Inequality condition, which is a generalization to convex-concave case. The proposed method overcomes shortcomings of recent non-adaptive gradient-based decentralized algorithms for min-max optimization problems, that do not perform well in practice and require careful tuning. In this paper, we obtain non-asymptotic rates of convergence of the proposed algorithm (coined Dadam3 ) for finding a (stochastic) first-order Nash equilibrium point and subsequently evaluate its performance on training GANs. The extensive empirical evaluation shows that Dadam3 outperforms recently developed methods, including decentralized optimistic stochastic gradient for solving such min-max problems.

Garbage In, Garbage Out: Poisoning Attacks Disguised With Plausible Mobility in Data Aggregation

Location data are often aggregated to support a wide range of applications, such as mobility management, point-of-interest recommendation, map inferences, and so on. However, these aggregated results might suffer from poisoning attacks when adversaries deliberately send poisoned locations to the aggregator, namely, garbage in, garbage out. Most existing work focuses on poisoning attacks in anomaly detection, crowd sensing, recommendation, and machine learning models. One observation motivating our work is that poisoning attacks in the context of data aggregation introduce new challenges along with distinguished features compared to anomaly detection, recommendation systems and machine learning models. As such, in this paper, we concentrate on the inputs to data aggregation and introduce the first optimal attack framework that launches <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</u> oisoning <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</u> ttacks on <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</u> ocation <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</u> ata <underline xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</u> ggregation (PALDA) and disguises the attacking behaviors. Specifically, we first formalize the poisoning attack as a min-max optimization problem and then design an iterative algorithm to address the optimization problem. Moreover, we theoretically prove the convergence of the proposed optimization algorithm and extend the attack strategy optimization to more practical settings where PALDA is applicable to any linear decomposable aggregation model executed by an aggregator. Finally, simulations on six real-world mobility datasets have demonstrated the risks entailed by poisoning attacks with data aggregation.

Calmness and Calculus: Two Basic Patterns

We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.

Attribute-Guided Adversarial Training for Robustness to Natural Perturbations

While existing work in robust deep learning has focused on small pixel-level norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations --- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.

Decentralized Policy Gradient Descent Ascent for Safe Multi-Agent Reinforcement Learning

This paper deals with distributed reinforcement learning problems with safety constraints. In particular, we consider that a team of agents cooperate in a shared environment, where each agent has its individual reward function and safety constraints that involve all agents' joint actions. As such, the agents aim to maximize the team-average long-term return, subject to all the safety constraints. More intriguingly, no central controller is assumed to coordinate the agents, and both the rewards and constraints are only known to each agent locally/privately. Instead, the agents are connected by a peer-to-peer communication network to share information with their neighbors. In this work, we first formulate this problem as a distributed constrained Markov decision process (D-CMDP) with networked agents. Then, we propose a decentralized policy gradient (PG) method, Safe Dec-PG, to perform policy optimization based on this D-CMDP model over a network. Convergence guarantees, together with numerical results, showcase the superiority of the proposed algorithm. To the best of our knowledge, this is the first decentralized PG algorithm that accounts for the coupled safety constraints with a quantifiable convergence rate in multi-agent reinforcement learning. Finally, we emphasize that our algorithm is also novel in solving a class of decentralized stochastic nonconvex-concave minimax optimization problems, where both the algorithm design and corresponding theoretical analysis are of independent interest.

Minimax Estimation in Regression under Sample Conformity Constraints

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples.

A coordinated scheme of electricity-gas systems and impacts of a gas system FDI attacks on electricity system

The increased penetration of natural gas-fired generation units in electricity systems calls for coordination of natural gas and electricity systems. Moreover, concerns over security of coordinated gas-electricity systems are spreading in recent years. Notably, both energy systems are vulnerable to malicious cyber-attacks. Within this context, this article (1) proposes a coordination scheme for the day-ahead schedule of natural gas and electricity systems; (2) defines a new class of false data injection (FDI) cyber-attacks on the gas demand and scheduling information of the natural gas system; and (3) investigates the impacts of such attacks on the operation of both energy systems. The coordination mechanism consists of three nested optimization problems. The natural gas system scheduling step in the mechanism is modeled as a bi-level max–min optimization problem with consideration of attacks. We use a column and constraint generation (C&CG) algorithm to solve this bi-level problem. The coordination mechanism and bi-level attack problem are examined using a case study based on the IEEE 24-node test system and a 9-node gas system, as well as a case study based on the IEEE 118-node test system and a 25-node gas system.

Resolution of the Min-Max Optimization Problem Applied in the Agricultural Sector with the Estimation of Yields by Nonparametric Statistical Approaches

The ultimate objective of the problem under study is to apply the min-max tool, thus making it possible to optimize the default risks linked to several areas: the agricultural sector, for example, which requires the optimization of the default risk using the following elements: silage crops, annual consumption requirements, and crops produced for a given year. To minimize the default risk in the future, we start, in the first step, by forecasting the total budget of agriculture investment for the next 20 years, then distribute this budget efficiently between the irrigation and construction of silos. To do this, Bangladesh was chosen as an empirical case study given the availability of its data on the FAO website; it is considered a large agricultural country in South Asia. In this article, we give a detailed and original in-depth study of the agricultural planning model through a calculating algorithm suggested to be coded on the R software thereafter. Our approach is based on an original statistical modeling using nonparametric statistics and considering an example of a simulation involving agricultural data from the country of Bangladesh. We also consider a new pollution model, which leads to a vector optimization problem. Graphs illustrate our quantitative analysis.

Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications

Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications