Equivalent Optimization Problem Research Articles

Hybrid differential evolution algorithm for Falkner-Skan flow with rotation

A hybrid differential evolution algorithm is used in this work to study the rotating transport of Falkner-Skan flow. The problem is modeled as an equivalent optimization problem by using Padé rational approximation functions. The primary model of the governing partial differential equations is imposed to subjugate the error between profiles. A hybrid evolutionary algorithm based on differential evolution and a convergent version of Nelder-Mead direct search algorithm is employed to perform global exploratory search along with an enhanced exploitation to improve the accuracy of the proposed solution scheme. The resulting scheme is named as evolutionary Padé approximation (EPA) scheme. The performance of the proposed EPA scheme on the Falkner-Skan boundary value problem is investigated by considering various values of the rotation parameters. The developed optimizer in EPA scheme was able to minimize the residuals up to10−10. Results are displayed graphically in order to study the effect of various types of parameters. EPA scheme determined that angular velocity increases or decreases accordingly as fluid parameter (β) and the rotation parameter (λ) but shows inverse behavior with respect to power law index(n). Similarly, the response of velocity profile along y−axis was decreasing function of β as well as n but increasing function of λ. The performance of the proposed EPA scheme has been demonstrated by comparing results with a hybrid neural network scheme and found in excellent agreement.

A new class of high-order supplementary variable methods for the Klein–Gordon–Zakharov system

In this paper, based on the paradigm of supplementary variable method (SVM), we reformulate the Klein–Gordon–Zakharov system into equivalent optimization problem subject to PDE constraints, and then present a novel class of high-order energy-preserving numerical algorithms to solve it numerically. The optimization model is discretized by applying the Gauss collocation method as well as the prediction–correction technique in time and the sine pseudo-spectral method in space. Experimental results and some comparisons between this method and other reported ones are provided to favor the suggested method in the overall performance.

Neural network-based adaptive optimal tracking control for hypersonic morphing aircraft with appointed-time prescribed performance

In this paper, an adaptive optimal attitude tracking controller for hypersonic morphing aircraft with appointed-time prescribed performance is presented. A proper error transformation methodology is developed to ensure the performance constraints are met without knowing initial tracking conditions. Subsequently, an adaptive composite control scheme is devised for the transformed system, comprising a feedforward neuro-backstepping controller and a feedback optimal controller. The former is employed to convert the optimal tracking problem into an equivalent optimal regulation problem, while the latter is constructed using adaptive dynamic programming (ADP) with a single critic network to achieve optimality. A novel weight-tuning law for the critic network is introduced, incorporating a stability indicator and the concurrent learning technique. This approach relaxes the underlying restrictive conditions in ADP, improving the robustness and availability of the system. The stability of the closed-loop system is analyzed using the Lyapunov direct method. Finally, the effectiveness and improved performance of the proposed control scheme are validated through numerical simulations.

Iterative Chebyshev approximation method for optimal control problems

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production.

The method of determining optimal control of the thermoelastic state of piece-homogeneous body using a stationary temperature field

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequences. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demonstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries.

Energy-efficient UAV communication: A NOMA scheme with resource allocation and trajectory optimization.

This work investigates a downlink nonorthogonal multiple access (NOMA) scheme with unmanned aerial vehicle (UAV) aided wireless communication, where a single UAV was regarded as an air base station (ABS) to communicate with multiple ground users. Considering the constraints of velocity and maneuverability, a UAV energy efficiency (EE) model was proposed via collaborative design resource allocation and trajectory optimization. Based on this, an EE maximization problem was formulated to jointly optimize the transmit power of ground users and the trajectory of the UAV. To obtain the optimal solutions, this nonconvex problem was transformed into an equivalent convex optimization problem on the basis of three user clustering algorithms. After several alternating iterations, our proposed algorithms converged quickly. The simulation results show an enhancement in EE with NOMA because our proposed algorithm is nearly 99.6% superior to other OMA schemes.

Adaptive fault-tolerant optimal control for hypersonic vehicles with state constrains based on adaptive dynamic programming

This paper addresses a novel adaptive fault-tolerant control (FTC) design based on adaptive dynamic programming (ADP) technique for hypersonic vehicle (HSV) subject to actuator fault and state constrains. The total control input is constructed by the combination of backstepping-based adaptive FTC and the ADP-based adaptive optimal control to improve the tracking performance and fault-tolerant capacity. By introducing a barrier Lyapunov function (BLF) to deal with the state constraints, a backstepping-based FTC scheme is designed to transform the tracking control problem into an equivalent optimal control problem. Subsequently, an adaptive optimal control strategy is developed by using the ADP technique to provide a supplementary control action. A critic network is constructed to solve the Hamilton–Jacobi–Bellman (HJB) equation online. The convergence properties of the closed-loop system are developed by utilizing the Lyapunov stability theory. Finally, simulation results are carried out to illustrate the effectiveness and efficiency of the proposed adaptive ADP-based FTC strategy.

Fully Distributed Economic Dispatch with Random Wind Power Using Parallel and Finite-Step Consensus-Based ADMM

In this paper, a fully distributed strategy for the economic dispatch problem (EDP) in the smart grid is proposed. The economic dispatch model considers both traditional thermal generators and wind turbines (WTs), integrating generation costs, carbon trading expenses, and the expected costs associated with the unpredictability of wind power. The EDP is transformed into an equivalent optimization problem with only an equality constraint and thus can be solved by an alternating-direction method of multipliers (ADMM). Then, to tackle this problem in a distributed manner, the outer-layer framework of the proposed strategy adopts a parallel ADMM, where different variables can be calculated simultaneously. And the inner-layer framework adopts a finite-step consensus algorithm. Convergence to the optimal solution is achieved within a finite number of communication iterations, which depends on the scale of the communication network. In addition, leveraging local and neighbor information, a distributed algorithm is designed to compute the eigenvalues of the Laplacian matrix essential for the finite-step algorithm. Finally, several numerical examples are presented to verify the correctness and effectiveness of the proposed strategy.

Economic and low-carbon dispatch of industrial integrated energy system with EV load based on Stackelberg game framework

Industrial integrated energy systems (IESs) and electric vehicles (EVs) provide new solutions for addressing the increasing challenges of the energy crisis and environmental pollution. With the increasing number of EVs and smart charging stations in industrial IES, the uncoordinated charging load of EVs imposes significant pressure on IES. Therefore, a well-designed dispatch scheme is crucial for reducing the economic cost for both parties, alleviating the energy supply pressure on IES, and promoting the development of a low-carbon society. To this end, given the load characteristics of EVs in industrial IES, we propose a dispatch framework based on the Stackelberg game theory, which includes the leader and the follower. The leader IES is responsible for formulating both unit dispatch and demand response plans, as well as determining the charging pricing for the smart charging station. The follower smart charging station optimizes EVs charging power by minimizing the charging cost in order to protect the interest of EV owners. Additionally, we introduce the carbon emission flow model into charging station pricing to shift the responsibility for carbon emissions from the generation side to the EV load side. Considering that the two-layer game model is difficult to solve, we use the Karush–Kuhn–Tucker condition and duality theorem to transform it into an equivalent single-layer optimization problem, which is easily solved. Simulation results demonstrate that the proposed game framework effectively reduces the economic cost of IES and the charging cost of EVs, alleviates the pressure from charging load, and reduces the carbon emissions of industrial IES.

Robust receive beamforming and reflection coefficients optimization in an IRS‐aided decode‐and‐forward relay system

AbstractConsider a decode‐and‐forward wireless relay system assisted by an intelligent reflection surface (IRS), where a robust design on receive beamforming at the relay and reflection coefficients at the IRS is studied. The worst‐case signal‐to‐noise ratio maximization problem is formulated, subject to the reflection coefficients (with either continuous or discrete phases) constraints, under the assumption of imperfect channel state information for all channels. To cope with the hard problem, an equivalent nonconvex quadratic optimization problem with a simpler form is derived, and then the Cauchy‐Schwarz inequality is applied to update the beamforming and a cyclic process is proposed to update the reflection coefficients, where a closed‐form optimal solution is computed in each step. It turns out that the proposed algorithm achieves a locally optimal solution for the robust design problem. Simulation results show that the proposed robust design outperforms an existing non‐robust design and two robust designs via semidefinite relaxation technique.

Trajectory tracking control for underactuated autonomous vehicles via adaptive dynamic programming

In this paper, the trajectory tracking control problem for underactuated autonomous vehicles moving in 2-D space is studied. Firstly, the trajectory tracking error dynamics with the equivalent point being the origin are given. Since both model and input uncertainties of vehicles are considered, a robust control problem is formulated by giving the extended trajectory tracking error dynamics. Through showing that a robust control law is equivalent to an optimal control law for a nominal system under some conditions, the robust control problem is transformed into an equivalent optimal control problem and a cost function composed of uncertainty bounds and discount factor is constructed. In order to solve the optimal control problem, a policy iteration-based ADP algorithm is proposed, in which two neural networks, called critic and actor networks, are applied respectively to describe the unknown value function and control law. The weights of the two networks are tuned through the gradient descent method at each iteration and the optimality and convergence properties are provided in the sequel. At last, simulation results are given to illustrate the validity of the proposed algorithm.

Optimal management of DB pension fund under both underfunded and overfunded cases

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, a zero-coupon bond and a stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyzes to illustrate the manager's economic behaviors.

Cutting-Edge Trajectory Optimization through Quantum Annealing

This paper introduces an innovative approach to explore the capabilities of Quantum Annealing (QA) for trajectory optimization in dynamic systems. The proposed method involves transforming trajectory optimization problems into equivalent binary optimization problems using Quadratic Unconstrained Binary Optimization (QUBO) representation. The procedure is general and adaptable, making it applicable to a wide range of optimal control problems that entail the satisfaction of dynamic, boundary, and path constraints. Specifically, the trajectory is discretized and approximated using polynomials. In contrast to the conventional approach of determining the polynomial degree (n) solely based on the number of boundary conditions, a specific factor is introduced in our method to augment the polynomial degree. As a result, the ultimate polynomial degree is calculated as a composite of two components: n = l + (m−1), where m denotes the count of boundary conditions and l signifies the number of independent variables. By leveraging inverse dynamics, the control required to follow the approximated trajectory can be determined as a linear function of independent variables l. As a result, the optimization function, which is represented by the integral of the square of the control, can be formulated as a QUBO problem and the QA is employed to find the optimal binary solutions.

Parameter Space Optimization for Robust Controller Synthesis With Structured Feedback Gain.

Various optimal control and system designs involve searching for a feedback gain matrix with structural constraints. As an alternative solution, the parameter space methods map the constraints from the state space to another extended state-input space, in which an equivalent optimization problem is solved. However, to further extend its applications, there are still some issues need to be addressed, such as the limited type of structural constraints, the marginally stable solutions, and the low computation efficiency. In this article, we aim to make this method applicable to a class of structural constraints for some elements in the gain matrix being zero or with intrarow and intracolumn constraints. To address such structured control problem, we propose a procedure to transform the original system to an extended system with the decentralized feedback matrix. From here, the mapping rules to the parameter space are given for the decentralized feedback matrix with both intrarow and intracolumn constraints. To avoid oscillatory closed-loop dynamics, we include the closed-loop dominant pole constraints during optimization. In addition, to improve the computation efficiency during optimization, we revise the cutting plane logic, which allows adding multiple linear constraints within a single iteration. Simulation examples demonstrate the effectiveness of the proposed method.

Quasi-Newton algorithms for solving interval-valued multiobjective optimization problems by using their certain equivalence

In this paper, we investigate a class of interval-valued multiobjective optimization problems (IVMOP), and formulate a certain equivalent multiobjective optimization problem (MOP). We establish that any Pareto optimal solution of the equivalent MOP is also an effective solution of IVMOP. We introduce two variants of the quasi-Newton method to solve IVMOP when every component of the objective function has second-order partial generalized Hukuhara derivatives. In the first variant, we assume that the objective function of IVMOP is strongly convex, and approximate Hessian matrix of each component of the objective function of MOP by using the well-known BFGS method. The algorithm generated by this variant exhibits superlinear convergence to a locally effective solution of IVMOP. In the second variant, we approximate the Hessian matrix of each component of MOP by applying the Levenberg–Marquardt method. Moreover, this variant can be applied to non-convex functions, and the sequence generated by this variant converges to a locally effective solution of IVMOP as well. Finally, some numerical examples are furnished to illustrate the effectiveness of our developed methodology.

Full-Duplex Communication for ISAC: Joint Beamforming and Power Optimization

Beamforming design has been widely investigated for integrated sensing and communication (ISAC) systems with full-duplex (FD) sensing and half-duplex (HD) communication, where the base station (BS) transmits and receives radar sensing signals simultaneously while the integrated communication operates in either downlink or uplink. To achieve higher spectral efficiency, in this paper, we extend existing ISAC beamforming design to a general case by considering the FD capability for both radar and communication. Specifically, we consider an FD ISAC system, where the BS performs target detection and communicates with multiple downlink users and uplink users reusing the same time and frequency resources. We jointly optimize the downlink dual-functional transmit signal and the uplink receive beamformers at the BS and the transmit power at the uplink users. The problems are formulated under two criteria: power consumption minimization and sum rate maximization. The downlink and uplink transmissions are tightly coupled due to both the desired target echo and the undesired interference received at the BS, making the problems challenging. To handle these issues in both cases, we first determine the optimal receive beamformers in closed forms with respect to the BS transmit beamforming and the user transmit power. Subsequently, we invoke these results to obtain equivalent optimization problems and propose iterative algorithms to solve them. In addition, we consider a special case under the power minimization criterion and propose an alternative low complexity design. Numerical results demonstrate that the optimized FD communication-based ISAC brings tremendous improvements in terms of both power efficiency and spectral efficiency compared to the conventional ISAC with HD communication.

A Distributed Unscented Predictive Cooperation Approach for Networked Mobile Manipulators

Cooperation control of networked mobile manipulators is a challenging task to be met. This paper presents a distributed unscented model predictive control method to address the problem of cooperation control for networked mobile manipulators with uncertain parameters. Based on the Euler-Lagrange approach, an auxiliary model with uncertain parameters and disturbances is obtained via the nonlinear feedback technique. Under the framework of distributed predictive control, a local optimization problem with chance constraints is established. An equivalent optimization problem based on the probability distribution predictive model is derived using the unscented transform to approximate the probability distributions for nonlinear models. Moreover, a sufficient condition guaranteeing the overall system's stability is developed. The performance of the proposed distributed unscented predictive control strategy is validated by simulation examples.

A categorized and decomposed algorithm for thermal system simulation based on generalized benders decomposition

Accurate and efficient analysis of thermal systems is increasingly difficult with the increasing system scale and complexity. Many studies have to sacrifice the accuracy for calculation efficiency by simplifications. This work proposes a categorized and decomposed (C&D) algorithm to simulate thermal systems efficiently and robustly without introducing any simplification. It converts the system simulation into an equivalent optimization problem, which categorizes system constraints according to whether they are linear, and decomposes the optimization problem into a subproblem and a master problem to handle nonlinearity. Besides, variables’ gradient information is used to accelerate the convergence and enhance the stability. The proposed algorithm is applied to a supercritical carbon dioxide (sCO2) recompression Brayton cycle, and results show that it consumes about 8.84% calculation time compared with sequential modular method and requires much fewer initial values (about two orders of magnitude) compared with simultaneous equations method. Compared with the hierarchical and categorized algorithm, the proposed algorithm has a 48% larger convergence range regarding the deviation of initial values, and is more efficient when the initial value deviation is small. The proposed C&D algorithm owns a much higher efficiency and robustness and is a promising tool for complex thermal system simulation.

Elastoplastic and limit analysis of reinforced concrete with an equilibrium-based finite element formulation

This paper introduces a 3D equilibrium-based finite element formulation for reinforced concrete. The stress unknowns in the proposed formulation strongly satisfy the equilibrium equations throughout the entire volume, with the exception of the mesh interfaces where the rebars intersect. The concrete stress tensor is interpolated using tetrahedral piece-wise linear elements with statically admissible discontinuities, while the rebars are considered as 1D elements embedded into the concrete, intersecting the triangular mesh. Thus, the global equilibrium is ensured by writing for each face of the mesh the traction continuity equation, while including the rebar stress contribution for the intersected triangular faces. The elastic equilibrium formulation is then written into an equivalent optimization problem, extended to the elastoplastic case by simply adding semi-definite matrix constraints on the concrete stress tensor, corresponding to a Rankine or a truncated Mohr–Coulomb criterion. As for the rebars, they are considered to obey a 1D perfectly elastoplastic behavior. The present formulation is also developed for limit analysis, to directly obtain an estimate of the lower bound of the collapse load. The resulting semi-definite programming optimization problems are solved using an in–house interior point algorithm.

Optimal control of the treatment and the vaccination in an epidemic switched system using polynomial approach

We study the problem of optimal switching on a treatment-vaccination model with four switches. These switches provide a more realistic approach when dealing with the control of an epidemic. We have used a relatively new approach to investigate optimal switching. The main idea behind this approach is to convert the switched system, a nonlinear, nonconvex optimal control problem, into an equivalent optimal control problem with a linear and convex structure. This allows us to obtain an equivalent convex formulation better suited to being solved by high-performance numerical computing. Toward the end, we perform some numerical simulations and draw the conclusion that the presence of treatment is essential, whereas vaccination control can be used at intervals with progressively larger gaps between the intervals.