Minimax Formulation Research Articles

Nonconvex robust programming via value-function optimization

Convex programming based robust optimization is an active research topic in the past two decades, partially because of its computational tractability for many classes of optimization problems and uncertainty sets. However, many problems arising from modern operations research and statistical learning applications are nonconvex even in the nominal case, let alone their robust counterpart. In this paper, we introduce a systematic approach for tackling the nonconvexity of the robust optimization problems that is usually coupled with the nonsmoothness of the objective function brought by the worst-case value function. A majorization-minimization algorithm is presented to solve the penalized min-max formulation of the robustified problem that deterministically generates a “better” solution compared with the starting point (that is usually chosen as an unrobustfied optimal solution). A generalized saddle-point theorem regarding the directional stationarity is established and a game-theoretic interpretation of the computed solutions is provided. Numerical experiments show that the computed solutions of the nonconvex robust optimization problems are less sensitive to the data perturbation compared with the unrobustfied ones.

Stability and robustness analysis of minmax solutions for differential graphical games

Recent studies have shown that, in general, Nash equilibrium cannot be achieved by the players of a differential graphical game by using distributed control policies. Alternative solution concepts that do not necessarily lead to Nash equilibrium can be proposed to allow the players in the game determine distributed optimal strategies. This paper analyzes the performance properties of the solution concept regarded as minmax strategies. The minmax formulation is shown to provide distributed control policies for linear systems under mild assumptions. The stability and robustness characteristics of the proposed solution are studied in terms of gain and phase margins, and related to the robustness properties of the single-agent LQR controller. The results of our analysis are finally tested by means of a simulation example.

Delay-Constrained Beamforming and Resource Allocation in Full Duplex Systems

This paper studies the beamforming and resource allocation problem in a multiuser full duplex (FD) system with delay-awareness. We design a power-efficient algorithm to minimize the long-term sum transmit power under delay constraints. We do this by jointly optimizing the uplink transmit power and the downlink beamforming vectors while satisfying the long-term stability constraints on the queue buffers for the downlink and uplink users as well as quality of service constraints. Due to the stochastic nature of the problem, we exploit the classic drift-plus-penalty function and subsequently simplify the problem into a difference of convex functions. Building upon the transformed problem, we propose two algorithm designs, one, that exploits the users with good channel conditions for efficient resource allocation, and the other that ensures delay-fairness among all users based on the max-min formulation. Simulation results show significant gains achieved by the proposed FD schemes compared with the baseline HD schemes.

Stochastic AUC optimization with general loss

Recently, there is considerable work on developing efficient stochastic optimization algorithms for AUC maximization. However, most of them focus on the least square loss which may be not the best option in practice. The main difficulty for dealing with the general convex loss is the pairwise nonlinearity w.r.t. the sampling distribution generating the data. In this paper, we use Bernstein polynomials to uniformly approximate the general losses which are able to decouple the pairwise nonlinearity. In particular, we show that this reduction for AUC maximization with a general loss is equivalent to a weakly convex (nonconvex) min-max formulation. Then, we develop a novel SGD algorithm for AUC maximization with per-iteration cost linearly w.r.t. the data dimension, making it amenable for streaming data analysis. Despite its non-convexity, we prove its global convergence by exploring the appealing convexity-preserving property of Bernstein polynomials and the intrinsic structure of the min-max formulation. Experiments are performed to validate the effectiveness of the proposed approach.

Improving Energy Efficiency Fairness of Wireless Networks: A Deep Learning Approach

Achieving energy efficiency (EE) fairness among heterogeneous mobile devices will become a crucial issue in future wireless networks. This paper investigates a deep learning (DL) approach for improving EE fairness performance in interference channels (IFCs) where multiple transmitters simultaneously convey data to their corresponding receivers. To improve the EE fairness, we aim to maximize the minimum EE among multiple transmitter–receiver pairs by optimizing the transmit power levels. Due to fractional and max-min formulation, the problem is shown to be non-convex, and, thus, it is difficult to identify the optimal power control policy. Although the EE fairness maximization problem has been recently addressed by the successive convex approximation framework, it requires intensive computations for iterative optimizations and suffers from the sub-optimality incurred by the non-convexity. To tackle these issues, we propose a deep neural network (DNN) where the procedure of optimal solution calculation, which is unknown in general, is accurately approximated by well-designed DNNs. The target of the DNN is to yield an efficient power control solution for the EE fairness maximization problem by accepting the channel state information as an input feature. An unsupervised training algorithm is presented where the DNN learns an effective mapping from the channel to the EE maximizing power control strategy by itself. Numerical results demonstrate that the proposed DNN-based power control method performs better than a conventional optimization approach with much-reduced execution time. This work opens a new possibility of using DL as an alternative optimization tool for the EE maximizing design of the next-generation wireless networks.

Worst-case Satisfaction of STL Specifications Using Feedforward Neural Network Controllers

In this paper, a reinforcement learning approach for designing feedback neural network controllers for nonlinear systems is proposed. Given a Signal Temporal Logic (STL) specification which needs to be satisfied by the system over a set of initial conditions, the neural network parameters are tuned in order to maximize the satisfaction of the STL formula. The framework is based on a max-min formulation of the robustness of the STL formula. The maximization is solved through a Lagrange multipliers method, while the minimization corresponds to a falsification problem. We present our results on a vehicle and a quadrotor model and demonstrate that our approach reduces the training time more than 50 percent compared to the baseline approach.

Why Botnets Work: Distributed Brute-Force Attacks Need No Synchronization

In September 2017, the McAfee Labs quarterly report estimated that brute-force attacks represent 20% of total network attacks, making them the most prevalent type of attack ex-aequo with browser-based vulnerabilities. These attacks have sometimes catastrophic consequences, and understanding their fundamental limits may play an important role in the risk assessment of password-secured systems and in the design of better security protocols. While some solutions exist to prevent online brute-force attacks that arise from one single IP address, attacks performed by botnets are more challenging. In this paper, we analyze these distributed attacks by using a simplified model. Our aim is to understand the impact of distribution and asynchronization on the overall computational effort necessary to breach a system. Our result is based on guesswork, a measure of the number of queries (guesses) before a correct sequence, such as a password, is found in an optimal attack. Guesswork is a direct surrogate for time and computational effort of guessing a sequence from a set of sequences with the associated likelihoods. We model the lack of synchronization by a worst-case optimization in which the queries made by multiple adversarial agents are received in the worst possible order for the adversary, resulting in a min–max formulation. We show that, even without synchronization, and for sequences of growing length, the asymptotic optimal performance is achievable by using randomized guesses drawn from an appropriate distribution. Therefore, randomization is key for the distributed asynchronous attacks. In other words, asynchronous guessers can asymptotically perform brute-force attacks as efficiently as synchronized guessers.

An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme involves the knowledge of the shape gradient which is established through the minimax formulation. We illustrate the feasibility of the proposed method and highlight its advantage over the classical setting of tracking the Neumann data through several numerical examples.

A certified model reduction approach for robust parameter optimization with PDE constraints

We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting non-linear optimization problem has a bi-level structure due to the min-max formulation. To approximate the worst-case in the optimization problem we propose linear and quadratic approximations. However, this approach still turns out to be very expensive, therefore we propose an adaptive model order reduction technique which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which is then utilized in the numerical optimization. Numerical results are presented to validate the presented approach.

Developing Bidding and Offering Curves of a Price-Maker Energy Storage Facility Based on Robust Optimization

This paper presents an algorithm to construct hourly bidding and offering curves to purchase and sell electricity for a price-maker merchant energy storage facility participating in a day-ahead electricity market. Hourly generation and demand price quota curves (GPQCs and DPQCs) are used to model the price impact of storage operation in the participation strategy problem of a price-maker storage facility in the market. This paper introduces a max-min mixed-integer linear programming model to present a participation strategy to manage the risk of uncertainty associated with forecasted GPQCs and DPQCs by robust optimization. The max-min formulation is converted to its equivalent linear maximization formulation based on the worst case scenario during charging and discharging hours. Then, an algorithm is proposed to build hourly bidding and offering curves. To do so, the confidence intervals for GPQCs and DPQCs are divided into subintervals and the robust mixed-integer linear formulation is solved sequentially for each subinterval of GPQCs and DPQCs. Sequential constraints are applied to ensure the decreasing and increasing nature of the bidding and offering curves, respectively. Then, hourly bidding and offering curves are built based on the obtained scheduling and corresponding price results. The proposed bidding and offering strategy is presented and validated using numerical simulations.

Shape optimization approach to defect-shape identification with convective boundary condition via partial boundary measurement

We aim to identify the geometry (i.e., the shape and location) of a cavity inside an object through the concept of thermal imaging. More precisely, we present an identification procedure to determine the geometric shape of a cavity with convective boundary condition in a heat-conducting medium using the measured temperature on a part of the surface of the object. The inverse problem of identifying the cavity is resolved by shape optimization techniques, specifically by minimizing a least-squares type cost functional over a set of admissible geometries. The computation of the first-order shape derivative or shape gradient of the cost is carried out through minimax formulation, which is then justified by the Correa–Seeger theorem coupled with function space parametrization technique. We further characterize its boundary integral form using some identities from tangential calculus. Then, we utilize the computed expression for the shape gradient to implement an effective boundary variation algorithm for the numerical resolution of the shape optimization problem. To avoid boundary oscillations or irregular shapes in our approximations, we execute the gradient-based scheme using the $$H^1$$ gradient method with perimeter regularization. Also, we propose a novel application of the said method in computing the mean curvature of the free boundary appearing in the shape gradient of the cost functional. We illustrate the feasibility of the proposed method by testing the numerical scheme to several cavity identification problems. Additionally, we also give some numerical examples for the case of corrosion detection since its inverse problem interpreted in the framework of electrostatic imaging is closely related to the focused problem.

Beamforming Techniques for Nonorthogonal Multiple Access in 5G Cellular Networks

In this paper, we develop various beamforming techniques for downlink transmission for multiple-input single-output nonorthogonal multiple access (NOMA) systems. First, a beamforming approach with perfect channel state information is investigated to provide the required quality of service for all users. Taylor series approximation and semidefinite relaxation (SDR) techniques are employed to reformulate the original nonconvex power minimization problem to a tractable one. Furthermore, a fairness-based beamforming approach is proposed through a max–min formulation to maintain fairness between users. Next, we consider a robust scheme by incorporating channel uncertainties, where the transmit power is minimized while satisfying the outage probability requirement at each user. Through exploiting the SDR approach, the original nonconvex problem is reformulated in a linear matrix inequality form to obtain the optimal solution. Numerical results demonstrate that the robust scheme can achieve better performance compared to the nonrobust scheme in terms of the rate satisfaction ratio. Furthermore, simulation results confirm that NOMA consumes a little over half transmit power needed by orthogonal multiple access for the same data rate requirements. Hence, NOMA has the potential to significantly improve the system performance in terms of transmit power consumption in future 5G networks and beyond.

Distributionally robust scheduling on parallel machines under moment uncertainty

This paper investigates a distributionally robust scheduling problem on identical parallel machines, where job processing times are stochastic without any exact distributional form. Based on a distributional set specified by the support and estimated moments information, we present a min-max distributionally robust model, which minimizes the worst-case expected total flow time out of all probability distributions in this set. Our model doesn’t require exact probability distributions which are the basis for many stochastic programming models, and utilizes more information compared to the interval-based robust optimization models. Although this problem originates from the manufacturing environment, it can be applied to many other fields when the machines and jobs are endowed with different meanings. By optimizing the inner maximization subproblem, the min-max formulation is reduced to an integer second-order cone program. We propose an exact algorithm to solve this problem via exploring all the solutions that satisfy the necessary optimality conditions. Computational experiments demonstrate the high efficiency of this algorithm since problem instances with 100 jobs are optimized in a few seconds. In addition, simulation results convincingly show that the proposed distributionally robust model can hedge against the bias of estimated moments and enhance the robustness of production systems.

Quickest Change Detection in Adaptive Censoring Sensor Networks

The problem of quickest change detection with communication rate constraints is studied. A network of wireless sensors with limited computation capability monitors the environment and sends observations to a fusion center via wireless channels. At an unknown time instant, the distributions of observations at all the sensor nodes change simultaneously. Due to limited energy, the sensors cannot transmit at all the time instants. The objective is to detect the change at the fusion center as quickly as possible, subject to constraints on false detection and average communication rate between the sensors and the fusion center. A minimax formulation is proposed. The cumulative sum (CuSum) algorithm is used at the fusion center and censoring strategies are used at the sensor nodes. The censoring strategies, which are adaptive to the CuSum statistic, are fed back by the fusion center. The sensors only send observations that fall into prescribed sets to the fusion center. This CuSum adaptive censoring (CuSum-AC) algorithm is proved to be an equalizer rule and to be globally asymptotically optimal for any positive communication rate constraint, as the average run length to false alarm goes to infinity. It is also shown, by numerical examples, that the CuSum-AC algorithm provides a suitable trade-off between the detection performance and the communication rate.

Parallel Scenario Decomposition of Risk-Averse 0-1 Stochastic Programs

In this paper, we extend a recently proposed scenario decomposition algorithm for risk-neutral 0-1 stochastic programs to the risk-averse setting. Specifically, we consider two-stage risk-averse 0-1 stochastic programs with objective functions based on coherent risk measures. Using a dual representation of a coherent risk measure, we first derive an equivalent minimax reformulation of the considered problem. We then develop three variants of the scenario decomposition algorithm for this minimax formulation based on different relaxations of the nonanticipaticity constraints. The algorithms proceed by solving scenario subproblems to obtain candidate solutions and bounds and subsequently cutting off the candidate solutions from the search space to achieve convergence to an optimal solution. We design three parallelization schemes for implementing the algorithms with different tradeoffs between overhead time and computation time. Our computational results with risk-averse extensions of two standard stochastic 0-1 programming test instances demonstrate the scalability of the proposed decomposition and parallelization framework.

On the Discretization Schemes in Map-Aided Indoor Localization

Pedestrian dead reckoning (PDR) implemented on the discretized map is considered as an efficient solution for indoor localization. Finer map discretization can provide better localization accuracy at the cost of higher computation complexity. In this letter, we characterize the effect of map discretization in PDR-based indoor localization. In particular, the closed-form expressions for the localization errors are derived for two typical discretization schemes. We then optimize the quantization intervals by a minimax formulation, which can guarantee the localization accuracy for users with different step sizes. We show that the low-complexity scheme only allowing transitions between adjacent discretization points on the map can achieve a robust and accurate performance when the quantization interval is optimized. Numerical results validate the effectiveness of the proposed scheme.

A proximal point algorithm for generalized fractional programs

In this paper, the problem of solving generalized fractional programs will be addressed. This problem has been extensively studied and several algorithms have been proposed. In this work, we propose an algorithm that combines the proximal point method with a continuous min–max formulation of discrete generalized fractional programs. The proposed method can handle non-differentiable convex problems with possibly unbounded feasible constraints set, and solves at each iteration a convex program with unique dual solution. It generates two sequences that approximate the optimal value of the considered problem from below and from above at each step. For a class of functions, including the linear case, the convergence rate is at least linear.

Layout design modelling for a real world just-in-time warehouse

The retail industry sector is one of the most competitive and where each company has to improve their operations on a daily basis to remain competitive. The struggle to move to just-in-time delivery requires distribution centres to readapt to this reality. Most of the literature in warehouse layout design is focused in traditional warehouses, where the main focus is on product storage and picking. However, when operating in a cross-docking basis, new approaches are required to plan the internal layout of the warehouse. In this paper, it is proposed a mathematical programming approach, based on a min-max formulation that returns the optimized layout of a cross-docking warehouse that feeds a just-in-time distribution operation. In this case, the layout requires the allocation of floor spaces to stores’ demands. Products are picked up at the receiving dock, and are transported by the worker along the warehouse up to the location where the products for a given store are located. Then, the products are left in the required quantity and the worker moves to the next store location that requires that product. To this end, we perform clusters of floor locations that may be used as locations to visit in product distribution routes. Our approach was tested in a real world case study of a Portuguese retail chain in a fruits and vegetables warehouse, which supplies more than 200 stores per day. We show that the distance travelled in the warehouse can be reduced in more than 2000km/month by just reallocating stores to different floor locations.

Robust shortest path planning and semicontractive dynamic programming

AbstractIn this article, we consider shortest path problems in a directed graph where the transitions between nodes are subject to uncertainty. We use a minimax formulation, where the objective is to guarantee that a special destination state is reached with a minimum cost path under the worst possible instance of the uncertainty. Problems of this type arise, among others, in planning and pursuit‐evasion contexts, and in model predictive control. Our analysis makes use of the recently developed theory of abstract semicontractive dynamic programming models. We investigate questions of existence and uniqueness of solution of the optimality equation, existence of optimal paths, and the validity of various algorithms patterned after the classical methods of value and policy iteration, as well as a Dijkstra‐like algorithm for problems with nonnegative arc lengths.© 2016 Wiley Periodicals, Inc. Naval Research Logistics 66:15–37, 2019

Data-Efficient Minimax Quickest Change Detection With Composite Post-Change Distribution

The problem of quickest change detection is studied, where there is an additional constraint on the cost of observations used before the change point and where the post-change distribution is composite. Minimax formulations are proposed for this problem. It is assumed that the post-change family of distributions has a member which is least favorable in a well-defined sense. An algorithm is proposed in which ON–OFF observation control is employed using the least favorable distribution, and a generalized likelihood ratio-based approach is used for change detection. Under additional conditions on the post-change family of distributions, it is shown that the proposed algorithm is asymptotically optimal, uniformly for all possible post-change distributions.