Min-max Optimization Problem Research Articles

Predictive control for visual servoing control of cyber physical systems with packet loss

Networked robotic visual servoing system is a representative cyber-physical system (CPS). This paper studies the robust predictive control problem of networked robotic visual servoing system with packet losses and uncertainty. Firstly, according to eye-to-hand framework, the system is modeled as a nonlinear discrete model with packet loss based on image-based visual servoing (IBVS) approach, where the packet losses obey a Bernoulli distribution. Since the discretization error of the system is a norm-bounded parameter, a nonlinear system with bounded uncertainty is derived. With regard to the principle of the MPC and the stochastic system method, the upper bound of the MPC performance index and the min-max optimization problem with uncertainty are presented. Based on robust least-square approach, the parameter-dependent predictive controller is obtained and an iterative algorithm is presented to solve the horizon optimization problem. Finally, numerical simulations and experiments are proposed to verify the effectiveness of the proposed algorithm.

Robust model predictive control of HVAC systems with uncertainty in building parameters using linear matrix inequalities

ABSTRACT In this work, a new robust controller is proposed for building climate control in presence of parametric uncertainties. The design of the controller is based on the Model Predictive Control (MPC) framework and it includes time-varying constraints. The robust design is implemented by explicitly considering parametric uncertainty in the synthesis of the control law. Variations of the parameters of the buildings are represented in the form of polytopic uncertainty. The robust control action is obtained by minimizing an appropriate ‘worst-case’ cost function, which leads to the definition of a min–max optimization problem. This optimization problem is formulated using Linear Matrix Inequalities (LMIs) that allow for efficient numerical computation of the control command. Simulation results show that the proposed approach is successful in keeping the indoor temperature of the building in the desired range even in presence of large model uncertainties. The proposed controller is also compared with a nominal controller synthesized without accounting for parametric uncertainty. Numerical results confirm 24% better performance of the robust design in comparison with the nominal controller with same conditions. Further, simulation results also demonstrate that the robust control system achieves 17% better performance in the case of severe conditions of uncertainty.

Resource Allocation for Wireless Power Transmission Over Full-Duplex OFDMA/NOMA Mobile Wireless Networks

The current wireless networks have been designed solely for data communication purposes, imposing many new challenges when supporting both power and information transmissions simultaneously. One of these main challenges lies in resource allocations. To overcome these difficulties, we propose the resource allocation schemes to efficiently support wireless power transmission over a relay-assisted full-duplex (FD) wireless network, where the access point (AP) conducts wireless information and power transfer (WIPT) to multiple mobile users in the downlink (DL), and in the meantime, mobile users transmit information to the AP via relays' assistances in the uplink (UL). Under DL WIPT, each mobile user uses the power splitting technique to harvest energy and receive data simultaneously. Aiming at maximizing the minimum (max-min) sum of DL and UL transmit rates among all mobile users, our proposed schemes jointly optimize the allocation of subcarriers and powers, and the selection of relays and power splitting ratios. Also, our proposed schemes consider two scenarios, where the network employs either orthogonal frequency-division multiple access (OFDMA) or non-orthogonal multiple access (NOMA) taking into account both perfect channel state information (CSI) estimation and imperfect CSI estimation cases. We first approximate the formulated non-convex max-min optimization problems as convex optimization problems. Then, we develop an asymptotically optimal algorithm and a suboptimal algorithm for OFDMA case and NOMA case, respectively. Finally, we validate and evaluate the performances of our proposed schemes through numerical analyses.

Infinite Horizon Average Cost Dynamic Programming Subject to Total Variation Distance Ambiguity

We analyze the per unit-time infinite horizon average cost Markov control model, subject to a total variation distance ambiguity on the controlled process conditional distribution. This stochastic optimal control problem is formulated as a minimax optimization problem in which the minimization is over the admissible set of control strategies, while the maximization is over the set of conditional distributions which are in a ball, with respect to the total variation distance, centered at a nominal distribution. We derive two new equivalent dynamic programming equations, and a new policy iteration algorithm. The main feature of the new dynamic programming equations is that the optimal control strategies are insensitive to inaccuracies or ambiguities in the controlled process conditional distribution. The main feature of the new policy iteration algorithm is that the policy evaluation and policy improvement steps are performed using the maximizing conditional distribution, which is obtained via a water filling solution of aggregating states together to form new states. Throughout the paper, we illustrate the new dynamic programming equations and the corresponding policy iteration algorithm to various examples.

A Task-Driven Updated Discrete Graph Assisted Minimum Delivery Delay Routing for Remote Sensing Disruption-Tolerant Networks

Benefiting from the extensive coverage, remote sensing satellites have been playing important roles in surveillance for regions on the earth surface. In order to accommodate with intermittent connected and partitioned characteristics of satellite topology, disruption-tolerant network (DTN) develops a feasible solution for networking among such satellites. However, utilized widely and frequently in the future, numerous image covering hundreds of spectral bands and kilometers width is supposed to be delivered simultaneously. Due to the timeliness constrain, efficient routing design with minimum delivery delay for bulk and concurrent image data has become the bottleneck of remote sensing application. Considering data transmission initiated as different tasks, therefore, the Task-driven Updated Discrete Graph (TUDG) is designed to depict the topology evolution, with task data and edge capacity model for BP/LTP incorporated Remote Sensing DTN Network (RS-DTNet). In particular, the multitask-based delivery delay analytical framework is proposed based on the TUDG graph model, by solving a mixed integer Max-Min optimization problem. The Multi-Task Minimum Delay Routing (MTMDR) is designed with the delay-optimal flow distribution, dispatching appropriate bundle data to edges in the TUDG. This flow-based routing avoids the path selection procedure, which may degenerate the delivery delay performance. Through numerical simulation based on the representative RS-DTNet scene, the proposed MTMDR routing strategy shows to advantage on delivery delay, compared with typical path-based routing.

A Novel Collaborate Neural Dynamic System Model for Solving a Class of Min–Max Optimization Problems with an Application in Portfolio Management

AbstractIn this paper, we describe a new neural network model for solving a class of non-smooth optimization problems with min–max objective function. The basic idea is to replace the min–max function by a smooth one using an entropy function. With this smoothing technique, the non-smooth problem is converted into an equivalent differentiable convex programming problem. A neural network model is then constructed based on Karush–Kuhn–Tucker optimality conditions. It is investigated that the proposed neural network is stable in the sense of Lyapunov and can converge to an exact optimal solution of the original problem. As an application in economics, we use the proposed scheme to a min–max portfolio optimization problems. The effectiveness of the method is demonstrated by several numerical simulations.

A Uniform-Forcing Transceiver Design for Over-the-Air Function Computation

The future Internet-of-Things network is expected to connect billions of sensors, which incurs high latency for data aggregation. To overcome this challenge, a new technique called over-the-air function computation was recently developed to enable fusion center to receive a desired function directly. It utilizes the superposition property of wireless channel to realize the uniform summation of the desired function. In order to compensate the non-uniform fading of different sensors, we propose a novel uniform-forcing transceiver design for over-the-air function computation. A corresponding min-max optimization problem is formulated to minimize the distortion of the computation which is measured by mean squared error. Due to the non-convexity of the problem, it is relaxed to semidefinite programming first. Then, the performance of the initial solution is improved through successive convex approximation. Simulation results show that the proposed design is able to achieve significant performance gain with low complexity.

Multidirectional Grey Wolf Optimizer Algorithm for Solving Global Optimization Problems

In this paper, we propose a new hybrid population-based meta-heuristics algorithm inspired by grey wolves in order to solve integer programming and minimax problems. The proposed algorithm is called Multidirectional Grey Wolf Optimizer (MDGWO) algorithm. In the proposed algorithm, we try to accelerate the standard grey wolf optimizer algorithm (GWO) by invoking the multidirectional search method with it in order to accelerate the search instead of letting the standard GWO run for more iterations without significant improvement in the results. MDGWO starts the search by applying the standard GWO search for a number of iterations, and then the best-obtained solution is passed to the multidirectional search method as an intensification process in order to accelerate the search and overcome the slow convergence of the standard GWO algorithm. We test MDGWO algorithm on seven integer programming problems and 10 minimax problems. Moreover, we compare against 11 algorithms for solving integer programming problems and 10 algorithms for solving minimax problems. Furthermore, we show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time by giving several results of the experiments.

Robust Differential Received Signal Strength Based Localization With Model Parameter Errors

In this letter, we address the differential received signal strength based problem with model parameter errors. To deal with the model parameter errors, we adopt the robust weighted least squares criterion, leading to a minimax optimization problem. By assuming the model parameter errors lie in a ball, the minimax problem is transformed into a tractable reformulation via the S-Lemma. Confronted with the nonconvexity of the reformulated problem, we approximately solve it by applying the semidefinite relaxation. The proposed approach only requires the knowledge of the upper bounds of the model parameter errors, which are practically easy to acquire. Simulation results show that the proposed method is robust to the model parameter errors and outperforms the existing methods.

Robust deterministic least-squares filtering for uncertain time-varying nonlinear systems with unknown inputs

The augmented state robust regularized least-squares filter (ASRRLSF) and two-stage robust regularized least-squares filter (TSRRLSF) are proposed for discrete time-varying nonlinear systems with unknown inputs and norm-bounded uncertainties. Unknown inputs affect both state-space model and measurements equation of the system. Combining system states and unknown inputs as an augmented state, the ASRRLSF is developed by converting a deterministic min–max optimization problem to a robust regularized least-squares problem. If dimension of the augmented state increases, the performance of the proposed ASRRLSF will reduce and the computational cost will increase rapidly. Therefore, in the following, the TSRRLSF is proposed by decoupling the ASRRLSF to lower order filters as system states filter and unknown inputs filter using T transformation. Finally, two numerical examples are given in order to illustrate the performance of the proposed filtering approaches.

A new gradient-based neural dynamic framework for solving constrained min-max optimization problems with an application in portfolio selection models

A neural network model based on a nonlinear dynamic model for solving a class of min-max problems, which is motivated as a non-differentiable optimization problem, is proposed. The main idea is to convert the non-differentiable problem into an equivalent differentiable convex optimization problem using a smoothing scheme called an entropy procedure. A neural network model is then constructed for solving the obtained convex problem. The stability of the equilibrium point and the convergence of the optimal solution are discussed. As an application in economics, we use the proposed scheme to a min-max portfolio optimization problem. Several clarifying examples and simulation results are provided to demonstrate the correctness of the results and the good performance of the presented model.

Hadamard Directional Differentiability of the Optimal Value Function of a Quadratic Programming Problem

In this paper, we consider the stability analysis of a convex quadratic programming (QP) problem and its restricted Wolfe dual when all parameters in the problem are perturbed. Based on the continuity of the feasible set mapping, we establish the upper semi-continuity of the optimal solution mappings of the convex QP problem and the restricted Wolfe dual problem. Furthermore, by characterizing the optimal value function as a min–max optimization problem over two compact convex sets, we demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.

Efficient Convex Optimization for Energy-Based Acoustic Sensor Self-Localization and Source Localization in Sensor Networks.

The energy reading has been an efficient and attractive measure for collaborative acoustic source localization in practical application due to its cost saving in both energy and computation capability. The maximum likelihood problems by fusing received acoustic energy readings transmitted from local sensors are derived. Aiming to efficiently solve the nonconvex objective of the optimization problem, we present an approximate estimator of the original problem. Then, a direct norm relaxation and semidefinite relaxation, respectively, are utilized to derive the second-order cone programming, semidefinite programming or mixture of them for both cases of sensor self-location and source localization. Furthermore, by taking the colored energy reading noise into account, several minimax optimization problems are formulated, which are also relaxed via the direct norm relaxation and semidefinite relaxation respectively into convex optimization problems. Performance comparison with the existing acoustic energy-based source localization methods is given, where the results show the validity of our proposed methods.

A dynamic game approach to distributionally robust safety specifications for stochastic systems

This paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances. Our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times, subject to the worst possible disturbance distribution in an ambiguity set. We propose a dynamic game formulation of constructing such policies and identify conditions under which a non-randomized Markov policy is optimal. Based on this existence result, we develop a practical design approach to safety-oriented stochastic controllers with limited information about disturbance distributions. However, an associated Bellman equation involves infinite-dimensional minimax optimization problems since the disturbance distribution may have a continuous density. To alleviate computational issues, we propose a duality-based reformulation method that converts the infinite-dimensional minimax problem into a semi-infinite program that can be solved using existing convergent algorithms. We prove that there is no duality gap, and that this approach thus preserves optimality. The results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances, while a standard stochastic safety verification tool is not.

Robust model predictive control for greenhouse temperature based on particle swarm optimization

Application of model predictive control (MPC) in horticultural practice requires detailed models. However, even highly sophisticated greenhouse climate models are often known to have unknown dynamics affected by bounded uncertainties. To enforce robustness during the controller design stage, this paper proposes a particle swarm optimization (PSO)-based robust MPC strategy for greenhouse temperature systems. The strategy is based on a nonlinear physical temperature affine model. The robust MPC technique requires online solution of a minimax optimal control problem, which optimizes the tradeoff between set point tracking and cost requirements reduction. The minimax optimization problem is reformulated to a nonlinear programming problem with constraints. PSO is used to solve the reformulated problem and priority ranking of constraint fitness is proposed to guarantee that the constraints are satisfied. The results of simulations performed using the proposed control system show that the controller can effectively achieve the set point in the presence of disturbances and that it offers more suitable control variables, higher control precision, and stronger robustness than the conventional MPC.

Global stabilization of the linearized three-axis axisymmetric spacecraft attitude control system by bounded linear feedback

In this paper, the three-axis attitude stabilization of the axisymmetric spacecraft with bounded inputs is studied. By constructing some novel state transformations, saturated linear state feedback controllers are constructed for the considered attitude control system. By constructing suitable quadratic plus integral Lyapunov functions, globally asymptotic stability of the closed-loop systems is proved if the feedback gain parameters satisfy some explicit conditions. By solving some min–max optimization problems, a global optimal feedback gain for the underactuated attitude stabilization system is proposed such that the convergence rate of the linearized closed-loop system is maximized. Numerical simulations show the effectiveness of the proposed approaches.

Adaptive Robust Expansion Planning for a Distribution Network With DERs

Distribution expansion planning (DEP) is used to determine the best expansion plan of a distribution system. Considering the uncertainties of loads and of power productions of wind distributed energy resources (DERs) and incorporating AC power flow equations make the DEP problem increasingly challenging. In this context, this paper presents a new adaptive robust distribution expansion planning model to identify the timing of feeder reinforcements in addition to the location, capacity, and installation time of dispatchable and wind DERs. The proposed approach characterizes the uncertain nature of loads and power productions of wind DERs through polyhedral uncertainty sets and the robustness of the solution can be controlled by means of a budget of uncertainty. Since AC power flow equations rather than DC ones are considered, the proposed model is a nonlinear and non-convex min–max–min optimization problem, which cannot be solved directly by commercial optimization packages. Hence, a tri-level decomposition algorithm using both primal and dual cutting planes is introduced to solve the problem. Additionally, to attain a smoother optimization problem, the non-convex AC power flow equations are convexified. The working of the proposed model is illustrated using 33-bus and 123-bus distribution networks.

An efficient algorithm for the solution of min-max problems in multiaxial fatigue

Numerous fatigue criteria require the evaluation of critical quantities to estimate fatigue damage. The approach based on the construction of the minimum circle or hypersphere is often employed in multiaxial high-cycle fatigue criteria to compute critical quantities as the amplitude and mean value of the shear stress. Such a construction can be mathematically formulated as a min-max optimization problem for which efficient numerical strategies are of utmost importance. The present paper proposes a novel algorithm for solving such an optimization problem. The algorithm is based on the alternating direction method of multipliers (ADMM) and is very simple to implement. A wide range of fatigue loading paths, formulated in the two-dimensional and deviatoric spaces, are considered for numerical testing. Comparisons with reference solutions and with two alternative optimization approaches demonstrate the accuracy as well as the high efficiency of the proposed ADMM-based algorithm. A discussion on the adopted stopping criterion and on algorithm parameters is also addressed.

Enhanced Max-Min SINR for Uplink Cell-Free Massive MIMO Systems

In this paper, we consider the max-min signal-to-interference plus noise ratio (SINR) problem for the uplink transmission of a cell-free Massive multiple-input multiple-output (MIMO) system. Assuming that the central processing unit (CPU) and the users exploit only the knowledge of the channel statistics, we first derive a closed-form expression for uplink rate. In particular, we enhance (or maximize) user fairness by solving the max-min optimization problem for user rate, by power allocation and choice of receiver coefficients, where the minimum uplink rate of the users is maximized with available transmit power at the particular user. Based on the derived closed-form expression for the uplink rate, we formulate the original user max-min problem to design the optimal receiver coefficients and user power allocations. However, this max-min SINR problem is not jointly convex in terms of design variables and therefore we decompose this original problem into two sub-problems, namely, receiver coefficient design and user power allocation. By iteratively solving these sub-problems, we develop an iterative algorithm to obtain the optimal receiver coefficient and user power allocations. In particular, the receiver coefficients design for a fixed user power allocation is formulated as generalized eigenvalue problem whereas a geometric programming (GP) approach is utilized to solve the power allocation problem for a given set of receiver coefficients. Numerical results confirm a three-fold increase in system rate over existing schemes in the literature.

Linear Optimal Signal Designs for Multi-Color MISO-VLC Systems Adapted to CCT Requirement

In visible light communications (VLC), multi-color light emitting diodes (LEDs) have higher modulation bandwidth and better chromatic property compared with traditional phosphor-converted LEDs. For multi-color multi-input single-output (MISO) VLC systems, multiple colors of one LED are not independent with each other since they are limited jointly with chromaticity constraint. In this paper, we develop a novel linear optimal signal constellation (LOSC) for three-color LEDs and four-color LEDs to maximize the minimum Euclidean distance of received signal constellation under lighting constraints. The main signal design problem is formulated as a max–min optimization problem with continuous variables and discrete variables. To solve this problem, we disassemble it into two parts. In the inner part, we develop a linear optimal signal structure under maximum likelihood detection criterion for the fixed average optical power of each color. In the outer part, we transform the non-convex optimization problem into several convex optimization problems with polar coordinate transformations and slack variables. Combining these two parts, our proposed LOSC design scheme shows a lower computation complexity than the exhaustive search scheme. For different given correlated color temperature (CCT) values, the LOSC would always provide an efficient signal design method adaptively. Simulation results demonstrate that our proposed LOSC achieves better bit error rate performances than the conventional schemes.