Smooth Penalty Function Research Articles

Distributed predefined-time optimal economic dispatch for microgrids

With the massive popularization of distributed generators, optimal economic dispatch has been a key optimization problem to maintain stable and efficient work of the whole system. In this paper, a new smooth reconstruction penalty function with continuous and piecewise linear differential is designed to deal with generation power constraints, which promotes to obtain a better suboptimal solution compared with the existing smooth penalty methods. A distributed predefined-time optimal economic dispatch strategy is presented by utilizing a time-based function. By employing the proposed strategy, the minimization of the generation cost with transmission loss under the power balance constraint and generation minimum/maximum constraints can be realized within a predefined settling time. The performance of the proposed optimization strategy is evaluated by simulations and hardware-in-the-loop experiments in terms of validity verification, robustness to load change and topology reconfiguration, plug-and-play functionality, and comparison with the existing results to illustrate the advantages of fast convergence and near optimal results.

Fast generation of collision-free low-thrust trajectories for asteroid landing using deep neural networks

This study presents a fast generation method of time-optimal low-thrust trajectories based on deep neural networks (DNN) for the collision-free asteroid landings. An equivalent unconstrained differentiable problem is transformed via the introduction of smooth penalty function, so that it can be addressed with the optimal control framework. To efficiently solve the equivalent problem to generate sufficient dataset for training DNNs, a double homotopy method is developed. Specifically, the equivalent problem is connected to the time-optimal control problem without anti-collision through two consecutive simplifications, and a solving process with double backward continuation is also formulated. Besides, two DNN models are constructed to provide high-quality predictions for the minimum glide-slope constraint angle of an initial state, and the initial costates and landing time for time-optimal low-thrust trajectories with anti-collision constraints, respectively. As such, the efficiency of collision-free low-thrust landing trajectory generation is improved significantly. For the cases within the range of training dataset or within a certain extension, the low-thrust landing trajectories with a very small terminal landing error can be generated using the predicted initial costates. Meanwhile, with the high-quality initial costates, the high-precision landing trajectories can be obtained by the double homotopy method in only a few iterative steps. Finally, the promising results in a Bennu collision-free asteroid landing scenario validate the efficiency of the proposed method.

A Smooth Penalty Function Algorithm for Computing Nonlinear Inequality Constraint Optimization Problem

A Smooth Penalty Function Algorithm for Computing Nonlinear Inequality Constraint Optimization Problem

An exact penalty function optimization method and its application in stress constrained topology optimization and scenario based reliability design problems

A smooth penalty function method which does not involve dual and slack implicit variables to formulate constrained optimization conditions is proposed. New active and loss functions are devised to independently and adaptively handle the violation of each constraint function. A single unconstrained minimization of the proposed penalty function produces a solution to the original optimization problem. The constrained optimization conditions depending only on the primal variable are derived and reduce to the canonical Karush-Kuhn-Tucker(KKT) conditions in a much more general framework. The derivative of the proposed loss functions with respect to constraint function can be interpreted as the Lagrange multiplier of the traditional Lagrangian function. The appearance of the first-order Hessian information is unveiled. This is in contrast to the existing works where the classical Lagrange Hessian contains only second order derivatives. Finally, some numerical examples including medium-scale stress constrained topology optimization and scenario based reliability design problems are presented to demonstrate the effectiveness of the proposed methodology.

Hierarchical Nash equilibrium seeking strategies of quadratic time‐varying games with Euler–Lagrange players

AbstractThis article investigates the distributed Nash equilibrium seeking problem of quadratic time‐varying games with Euler–Lagrange (EL) players, where external disturbances and parametric uncertainties are involved. A gradient‐based hierarchical algorithm consisting of a game layer and a control layer is proposed. Specifically, in the game layer, EL players communicate with neighbors through a graph to reach the consensus on potential aggregate values, which will be employed to calculate the gradient of each player's objective function, and then, a gradient‐based sliding mode controller is developed to track time‐varying gradient in the control layer. Thus, the convergence results are hierarchically obtained through the Lyapunov stability method. In addition, the hierarchical control strategy is extended to address the constrained problems through the utilization of a smooth penalty function. By appropriately choosing control parameters, the Nash equilibrium seeking errors can be arbitrarily small. The relation between the optimal solutions of the original problem and the dual one is further discussed. Finally, the proposed methods are numerically verified.

Data-driven strategies for extractive distillation unit optimization

We provide insights for the development of a fast, dynamic, and adaptive way of modeling and optimizing chemical processes. We investigate two data-driven methodologies to optimize the energy cost of an extractive distillation process using an Aspen simulator. The first method uses surrogate-based optimization to explore two model-building techniques: Automated Learning of Algebraic Models’s (ALAMO) generalized linear models and neuron network models with rectifier (ReLU) activation function. We compare the accuracy and performance of these models when embedded into mathematical programming models. The second approach uses black-box optimization (BBO) to optimize problems directly using simulation results. We compare four BBO solvers and three different penalty functions to address constraints. We find that ALAMO performs well for a less complex system, whereas the ReLU network performs better for a more complex one. BBO with a smooth penalty function for constraints is the more effective approach for problems considered in this study.

Autonomous obstacle avoidance control method based on safe adaptive reinforcement learning

Obstacle avoidance is an important issue in the motion planning of autonomous unmanned systems. Therefore, designing an effective avoidance control method is crucial. For further improving the decision-making process, this paper presents a novel autonomous obstacle avoidance control method based on reinforcement learning that generates a safe motion trajectory in an adaptive manner. First, the barrier function is utilized to design a smooth penalty function in the cost function, thereby transforming the avoidance problem into an unconstrained optimal control problem. Then, adaptive reinforcement learning is implemented by using an actor-critic neural network architecture and policy iteration, in which the critic network uses the state-following kernel function to approximate the cost function while the actor network provides an approximate optimal control policy. During this learning process, the simulated experience is obtained through state extrapolation such that the critic network can use experience replay for reliable local exploration. Finally, simulation experiments on simplified drone systems and a nonlinear numerical system are provided. The proposed method can generate a safe motion trajectory in real time with comparable performance.

Sparsity-based method for ring artifact elimination in computed tomography.

Ring artifact elimination is one of the popular problems in computed tomography (CT). It appears in the reconstructed image in the form of bright or dark patterns of concentric circles. In this paper, based on the compressed sensing theory, we propose a method for eliminating the ring artifact during the image reconstruction. The proposed method is based on representing the projection data by a sum of two components. The first component contains ideal correct values, while the latter contains imperfect error values causing the ring artifact. We propose to minimize some sparsity-induced norms corresponding to the imperfect error components to effectively eliminate the ring artifact. In particular, we investigate the effect of using different sparse models, i.e. different sparsity-induced norms, on the accuracy of the ring artifact correction. The proposed cost function is optimized using an iterative algorithm derived from the alternative direction method of multipliers. Moreover, we propose improved versions of the proposed algorithms by incorporating a smoothing penalty function into the cost function. We also introduce angular constrained forms of the proposed algorithms by considering a special case as follows. The imperfect error values are constant over all the projection angles, as in the case where the source of ring artifact is the non-uniform sensitivity of the detector. Real data and simulation studies were performed to evaluate the proposed algorithms. Results demonstrate that the proposed algorithms with incorporating smoothing penalty and their angular constrained forms are effective in ring artifact elimination.

Inequality Constrained Optimal Control for Rope-Assisted ASV Docking Maneuvers

In difficult whether conditions the complexity of docking maneuvers of autonomous surface vessels (ASVs) increases. Docking maneuvers take place in proximity to piers and other vessels. The slow speed at which the maneuvers take place decreases the maneuverability of the vessel and environmental disturbances such as wind and waves increase in effect. In these scenarios ropes can be utilized to support the maneuver and compensate for the compromised actuation capabilities. The modeling of the modified mechanics plays a center role. It has been shown in previous work that this setup, connecting the vessel to the pier with a constant distance (e.g. with a rod) can be modeled using the concept of differential-algebraic equations (DAEs). With the adapted vessel dynamics a trajectory generation problem can be solved using optimal control theory. Here, a direct numerical method, in this case a full discretization of the optimal control problem (OCP), is used. The contribution in this paper allows for slackness in the rope. This is done by using a smooth penalty function, as well as solving a complementary problem. Both methods lead to feasible solutions. The penalty function shows are more smooth behavior but suffers from numerical robustness in comparison to the linear complementary solution.

Progressive coherence and spectral norm minimization scheme for measurement matrices in compressed sensing

Compressed sensing is a novel signal sampling theory based on the sparsity of signals. To recover a sparse signal from fewer measurements, it is desired that measurement matrices have low coherence and low spectral norm. For this purpose, a progressive coherence and spectral norm minimization (PCSNM) scheme is proposed by dividing the optimization problem into several ℓ∞-minimization problems with orthogonality constraints. Utilizing the smoothing technique and penalty function method, these non-smooth constrained problems are transformed into unconstrained approximate optimization problems and solved by a gradient-type based alternating minimization approach. Simulation results demonstrate that the proposed scheme outperforms some previous methods, especially in decreasing the spectral norm.

Smooth penalty function method for rigorous optimization of distillation processes

AbstractInteger decisions on stage numbers and feed locations, and global optimality are still challenging for rigorous optimization of distillation processes. In the present article, we propose a smooth penalty function method to address both these problems. The proposed method is based on the relaxation of the integer decision problem into continuous nonlinear programming (NLP) problem by adopting the bypass efficiency model developed by Dowling and Biegler. A smooth penalty term (SPT) is proposed and added to the total annual cost (TAC) function to form a new objective function, namely, the smooth penalty function. Using the new objective function, the problem is initially solved with negative weight coefficients for the SPTs regarding each column section to get an optimum near the global optimum of the SPT. Then, starting from this solution, the problem is solved again iteratively by increasing the values of the weight coefficients until all the stage numbers become integers. The performance of the method is validated by an illustrating problem and in three case studies, including a reactive distillation optimization problem.

The smoothing objective penalty function method for two-cardinality sparse constrained optimization problems

The two-cardinality sparse constrained optimization problems include sparse optimization problems and constrained sparse optimization problems in many fields, such as signal processing, image processing, securities investment etc. A smoothing penalty function method and a smoothing objective penalty function method are studied for two-cardinality sparse constrained optimization problems respectively. Some error estimations are proved for the smoothing penalty function and the smoothing objective penalty function. Based on the smoothing penalty function and smoothing objective penalty function, two algorithms are designed to solve two-cardinality sparse constrained optimization problems and their convergence is proved respectively. Numerical results show that the two algorithms have the similar effectiveness in finding out an approximate solution for two-cardinality sparse constrained optimization problems.

Smoothing Approximation to the New Exact Penalty Function with Two Parameters

In this paper, we propose a new non-smooth penalty function with two parameters for nonlinear inequality constrained optimization problems. And we propose a twice continuously differentiable function which is smoothing approximation to the non-smooth penalty function and define the corresponding smoothed penalty problem. A global solution of the smoothed penalty problem is proved to be an approximation global solution of the non-smooth penalty problem. Based on the smoothed penalty function, we develop an algorithm and prove that the sequence generated by the algorithm can converge to the optimal solution of the original problem.

Network Optimization via Smooth Exact Penalty Functions Enabled by Distributed Gradient Computation

This article proposes a distributed algorithm for a network of agents to solve an optimization problem with separable objective function and locally coupled constraints. Our strategy is based on reformulating the original constrained problem as the unconstrained optimization of a smooth (continuously differentiable) exact penalty function. Computing the gradient of this penalty function in a distributed way is challenging even under the separability assumptions on the original optimization problem. Our technical approach shows that the distributed computation problem for the gradient can be formulated as a system of linear algebraic equations defined by separable problem data. To solve it, we design an exponentially fast, input-to-state stable distributed algorithm that does not require the individual agent matrices to be invertible. We employ this strategy to compute the gradient of the penalty function at the current network state. Our distributed algorithmic solver for the original constrained optimization problem interconnects this estimation with the prescription of having the agents follow the resulting direction. Numerical simulations illustrate the convergence and robustness properties of the proposed algorithm.

A New Unified Path to Smoothing Nonsmooth Exact Penalty Function for the Constrained Optimization

We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.

A class of smooth exact penalty function methods for optimization problems with orthogonality constraints

ABSTRACT Updating the augmented Lagrangian multiplier by closed-form expression yields efficient first-order infeasible approach for optimization problems with orthogonality constraints. Hence, parallelization becomes tractable in solving this type of problems. Inspired by this closed-form updating scheme, we propose a novel penalty function with compact convex constraints (PenC). We show that PenC can act as an exact penalty model which shares the same global minimizers as the original problem with orthogonality constraints. Based on PenC, we first propose a first-order algorithm called PenCF and establish its global convergence and local linear convergence rate under some mild assumptions. For the case that the computation and storage of Hessian is achievable, and we pursue high precision solution and fast local convergence rate, a second-order approach called PenCS is proposed for solving PenC. To avoid expensive calculation or solving a hard subproblem in computing the Newton step, we propose a new strategy to do it approximately which still leads to quadratic convergence locally. Moreover, the main iterations of both PenCF and PenCS are orthonormalization-free and hence parallelizable. Numerical experiments illustrate that PenCF is comparable with the existing first-order methods. Furthermore, PenCS shows its stability and high efficiency in obtaining high precision solution comparing with the existing second-order methods.

Optimal Control Problem Solution with Phase Constraints for Group of Robots by Pontryagin Maximum Principle and Evolutionary Algorithm

A numerical method based on the Pontryagin maximum principle for solving an optimal control problem with static and dynamic phase constraints for a group of objects is considered. Dynamic phase constraints are introduced to avoid collisions between objects. Phase constraints are included in the functional in the form of smooth penalty functions. Additional parameters for special control modes and the terminal time of the control process were introduced. The search for additional parameters and the initial conditions for the conjugate variables was performed by the modified self-organizing migrating algorithm. An example of using this approach to solve the optimal control problem for the oncoming movement of two mobile robots is given. Simulation and comparison with direct approach showed that the problem is multimodal, and it approves application of the evolutionary algorithm for its solution.

A new perturbed smooth penalty function for inequality constrained optimization

A new perturbed smooth penalty function for inequality constrained optimization

Engineering-oriented dynamic optimal control of a greenhouse environment using an improved genetic algorithm with engineering constraint rules

To effectively and practically solve the dynamic economic optimal control problem in greenhouse environments, an improved genetic algorithm with engineering constraint rules (R-GA) is proposed. Based on a dynamic greenhouse-crop model and control vector parameterization (CVP) method to discretize the control variables, the economic optimal control problem is transformed into a nonlinear programming (NLP) problem with finite-dimension parameters, and then R-GA is used to effectively solve the NLP problem. Three components were investigated, such as a smooth penalty function to deal with state variable path constraints, engineering constraint rules to improve the optimization performance and the algorithm feasibility, and the number of collocation points (Nc) to meet the actual control laws. The simulation results demonstrate that the proposed approach greatly improves the effectiveness and feasibility to solve the dynamic optimal control problem in greenhouse environments.

Smoothing Partially Exact Penalty Function of Biconvex Programming

In this paper, a smoothing partial exact penalty function of biconvex programming is studied. First, concepts of partial KKT point, partial optimum point, partial KKT condition, partial Slater constraint qualification and partial exactness are defined for biconvex programming. It is proved that the partial KKT point is equal to the partial optimum point under the condition of partial Slater constraint qualification and the penalty function of biconvex programming is partially exact if partial KKT condition holds. We prove the error bounds properties between smoothing penalty function and penalty function of biconvex programming when the partial KKT condition holds, as well as the error bounds between objective value of a partial optimum point of smoothing penalty function problem and its [Formula: see text]-feasible solution. So, a partial optimum point of the smoothing penalty function optimization problem is an approximately partial optimum point of biconvex programming. Second, based on the smoothing penalty function, two algorithms are presented for finding a partial optimum or approximate [Formula: see text]-feasible solution to an inequality constrained biconvex optimization and their convergence is proved under some conditions. Finally, numerical experiments show that a satisfactory approximate solution can be obtained by the proposed algorithm.