Optimal Approximation Problem Research Articles

Efficient mini-batch stochastic gradient descent with Centroidal Voronoi Tessellation for PDE-constrained optimization under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is frequently utilized in engineering, biology and finance. Although stochastic gradient descent is a well-known stochastic optimization technique for solving the approximate optimal control problem, it often exhibits a slow convergence rate due to the inherent variance in gradient approximation. To address this issue, we propose a more efficient stochastic optimization algorithm based on Centroidal Voronoi Tessellation sampling. This strategy reduces the error in gradient estimation compared to the conventional mini-batch stochastic gradient descent method. Our approach involves dividing the whole snapshot set into several Voronoi cells with low variance and extracting samples with good uniformity from each region to construct an unbiased estimation of the full gradient. Our method can economically and stably approximate numerical optimal control functions even with a fixed step size. Numerical results demonstrate that the proposed method is a reliable algorithm with great potential for applications in the field of optimization.

Approximate balancing weights for clustered observational study designs.

In a clustered observational study, a treatment is assigned to groups and all units within the group are exposed to the treatment. We develop a new method for statistical adjustment in clustered observational studies using approximate balancing weights, a generalization of inverse propensity score weights that solve a convex optimization problem to find a set of weights that directly minimize a measure of covariate imbalance, subject to an additional penalty on the variance of the weights. We tailor the approximate balancing weights optimization problem to the clustered observational study setting by deriving an upper bound on the mean square error and finding weights that minimize this upper bound, linking the level of covariate balance to a bound on the bias. We implement the procedure by specializing the bound to a random cluster-level effects model, leading to a variance penalty that incorporates the signal-to-noise ratio and penalizes the weight on individuals and the total weight on groups differently according to the the intra-class correlation.

Approximation of optimization problems with constraints through kernel Sum-of-Squares

Handling an infinite number of inequality constraints in infinite-dimensional spaces occurs in many fields, from global optimization to optimal transport. These problems have been tackled individually in several previous articles through kernel Sum-of-Squares (kSoS) approximations. We propose here a unified theorem to prove convergence guarantees for these schemes. Pointwise inequalities are turned into equalities within a class of nonnegative kSoS functions. Assuming further that the functions appearing in the problem are smooth, focusing on pointwise equality constraints enables the use of scattering inequalities to mitigate the curse of dimensionality in sampling the constraints. Our approach is illustrated in learning vector fields with side information, here the invariance of a set.

Optimal control of an elastic–rigid obstacle problem

In this paper, we consider the optimal control problem of an elastic–rigid obstacle problem. The force is considered as the control, and the corresponding solution to the elastic–rigid obstacle problem is taken to be the state. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem, and the connection between the two formulations is established through some convergence results. Existence of optimal pairs is proved and necessary optimality conditions are derived.

An optimal approximation problem for free polynomials

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial f f in d d freely noncommuting arguments, find a free polynomial p n p_n , of degree at most n n , to minimize c n ≔ ‖ p n f − 1 ‖ 2 c_n ≔\|p_nf-1\|^2 . (Here the norm is the ℓ 2 \ell ^2 norm on coefficients.) We show that c n → 0 c_n\to 0 if and only if f f is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the d d -shift.

Adaptive critic learning for approximate optimal event-triggered tracking control of nonlinear systems with prescribed performances

This paper studies the approximate optimal event-triggered tracking control problem for a class of second-order nonlinear systems with prescribed performances. By employing a prescribed performances function, restrictions on tracking errors are removed for prescribed performances including convergence rate, maximum overshoot, and steady-state error. On this basis, a complex optimal tracking control problem can be transformed into an optimal stabilising control problem by constructing an augmented system. Subsequently, an event-triggered mechanism is introduced to reduce communication resources. Then, an adaptive critic learning algorithm is used to solve the Hamilton-Jacobi-Bellman equation, where the weights in the critic networks are tuned through the gradient descent approach and experience replay technology. Through the developed learning method, the persistence of excitation condition is released, and the data efficiency is improved. The Lyapunov stability theory is introduced to verify the stability of the close-loop system and the tracking errors are uniformly ultimately bounded. Finally, simulation results are presented to show the effectiveness of the proposed approach.

Power utility maximization problem under rough stochastic local volatility models with continuous-time Markov chain approximation

This paper studies the portfolio optimization problem based on rough stochastic local volatility models. Due to the challenge that this model is neither a semimartingale nor a Markov process, we first introduce the semimartingale approximate optimization problem and present the convergence analysis. We use the classic stochastic control method and continuous-time Markov chain approximation approach to solve the approximate optimization problem respectively. Then the optimal investment strategy and the HJB equation satisfied by the value function are obtained respectively. Furthermore, for the classic stochastic control method, we demonstrate the existence of solutions for the associated partial differential equation via the nonlinear Feynman–Kac formula. For continuous-time Markov chain approximation method, we prove the convergence of the value function. Finally, we provide a numerical analysis to illustrate the above results by using the power utility function. The numerical analysis shows that continuous-time Markov chain approximation method is effective in solving the portfolio optimization problem under rough stochastic local volatility models.

Approximate Optimal Control for Nonlinear Systems With Periodic Event-Triggered Mechanism.

This article investigates the approximate optimal control problem for nonlinear affine systems under the periodic event triggered control (PETC) strategy. In terms of optimal control, a theoretical comparison of continuous control, traditional event-based control (ETC), and PETC from the perspective of stability convergence, concluding that PETC does not significantly affect the convergence rate than ETC. It is the first time to present PETC for optimal control target of nonlinear systems. A critic network is introduced to approximate the optimal value function based on the idea of reinforcement learning (RL). It is proven that the discrete updating time series from PETC can also be utilized to determine the updating time of the learning network. In this way, the gradient-based weight estimation for continuous systems is developed in discrete form. Then, the uniformly ultimately bounded (UUB) condition of controlled systems is analyzed to ensure the stability of the designed method. Finally, two illustrative examples are given to show the effectiveness of the method.

A dynamical method for optimal control of the obstacle problem

Abstract In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method.

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

Data-driven optimal control via linear transfer operators: A convex approach

This paper is concerned with the data-driven optimal control of nonlinear systems. We present a convex formulation of the optimal control problem with a discounted cost function. We consider optimal control problems with both positive and negative discount factors. The convex approach relies on lifting nonlinear system dynamics in the space of densities using the linear Perron–Frobenius operator. This lifting leads to an infinite-dimensional convex optimization formulation of the optimal control problem. The data-driven approximation of the optimization problem relies on the approximation of the Koopman operator and its dual: the Perron–Frobenius operator, using a polynomial basis function. We write the approximate finite-dimensional optimization problem as a polynomial optimization which is then solved efficiently using a sum-of-squares-based optimization framework. Simulation results demonstrate the efficacy of the developed data-driven optimal control framework.

Robust approximate optimal control for air‐breathing hypersonic vehicle

AbstractThis paper investigates the robust approximate optimal control problem for air‐breathing hypersonic vehicle (AHV). By input‐output linearzation and introducing auxiliary variables, the high‐order nonlinear AHV dynamics are transformed to a second‐order feedback decoupling model. Then a virtual command is designed by back‐stepping strategy to guarantee the practical finite‐time convergence of tracking error. In order to drive the AHV with uncertainties to follow the virtual command, a robust approximate optimal controller based on adaptive dynamic programming (ADP) is proposed. Different from the conventional robust method, both the robustness and the input cost are taken into consideration in the proposed controller, which can optimize the control actions and hence avoid the chattering. Besides, the proposed controller adopts a single critic network design instead of actor‐critic structure, which ensures the stability of the system autonomously and simplifies the implementation procedure. The simulation results show the effectiveness of the proposed scheme.

Concurrent Transmission and Multiuser Detection of LoRa Signals

This article investigates a new model to improve the scalability of low-power long-range (LoRa) networks by allowing a group of multiple end devices (EDs) to communicate with multiple multi-antenna gateways simultaneously (i.e., in the same time slot) on the same frequency band and using the same spreading factor. The maximum-likelihood (ML) decision rule is first derived for noncoherent detection of information bits transmitted by multiple devices in a group. To overcome the high complexity of the ML detection, we propose a suboptimal two-stage detection algorithm to balance the computational complexity and error performance. In the first stage, we identify transmitted chirps (without knowing which EDs transmit them). In the second stage, we determine the EDs that transmit the specific chirps identified from the first stage. To improve the detection performance in the second stage, we also optimize the transmit powers of EDs to minimize the similarity, measured by the Jaccard coefficient, between the received powers of any pair of EDs in the same group. As the power control optimization problem is nonconvex, we use concepts from successive convex approximation to transform it to an approximate convex optimization problem that can be solved iteratively and guaranteed to reach a suboptimal solution. Simulation results demonstrate and justify the tradeoff between transmit power penalties and network scalability of the proposed LoRa network model. In particular, by grouping two or three EDs in each group for concurrent transmission, the uplink capacity of the proposed network can be doubled or tripled over that of a conventional LoRa network, albeit at the expense of additional 3.0 or 4.7 dB transmit power.

Hierarchical Multi-Agent Deep Reinforcement Learning for Energy-Efficient Hybrid Computation Offloading

Mobile edge computing (MEC) provides an economical way for the resource-constrained edge users to offload computational workload to MEC servers co-located with the access point (AP). In this article, we consider a hybrid computation offloading scheme that allows edge users to offload workloads by using active RF communications and backscatter communications. We aim to maximize the overall energy efficiency subject to the completion of all workload by jointly optimizing the AP's beamforming and the users' offloading decisions. Considering a dynamic environment, we propose a hierarchical multi-agent deep reinforcement learning (H-MADRL) framework to solve this problem. The high-level agent resides in the AP and optimizes the beamforming strategy, while the low-level user agents learn and adapt individuals' offloading strategies. To further improve the learning efficiency, we propose a novel optimization-driven learning algorithm that allows the AP to estimate the low-level users' actions by solving approximate optimization problem efficiently. Then, the action estimation can be shared with all users and drive them to update individuals' actions independently. Simulation results reveal that our algorithm can improve the system performance by 50%. The learning efficiency and reliability are also improved significantly comparing to the model-free learning methods.

Finite-Time Distributed Approximate Optimization Algorithms of Higher Order Multiagent Systems via Penalty-Function-Based Method

This article investigates the finite-time distributed approximate optimization problem of higher order multiagent systems, where the local cost functions are considered to be quadratic functions. This problem is solved via penalty-function-based method. First, by the penalty-function method, a global approximate cost function is constructed. Second, nonlinear distributed optimization algorithms are proposed for higher order multiagent systems by the tool of adding a power integrator technique. In the optimization algorithms design, the gradients of the approximate cost function are utilized. Under the proposed optimization algorithms, the agents approach the approximate optimal solution in finite time. Although there exist errors (they may be called approximation errors) between the approximate minimizers and the global accurate minimizer, the approximation errors can be regulated by penalty parameter and the relationship between the bound of the approximation errors and the penalty parameter is given explicitly. Furthermore, the proposed distributed approximate optimization algorithms are applied to the optimal rendezvous problem of wheeled multimobile robots, making the mobile robots achieve approximate optimization rendezvous in finite time. The effectiveness of the proposed distributed optimization algorithms and their applications to optimal rendezvous problem are validated by simulations.

Optimal Control for Parabolic Uncertain System Based on Wavelet Transformation

In this paper, we study a new type of optimal control problem subject to a parabolic uncertain partial differential equation where the expected value criterion is adopted in the objective function. The basic idea of Haar wavelet transformation is to transform the proposed problem into an approximate uncertain optimal control problem with arbitrary accuracy because the dimension of Haar basis tends to infinity. The relative convergence theorem is proved. An application to an optimal control problem with an uncertain heat equation is dealt with to illustrate the efficiency of the proposed method.

Training neural networks for solving 1-D optimal piecewise linear approximation

Recently, the interpretability of deep learning has attracted a lot of attention. A plethora of methods have attempted to explain neural networks by feature visualization, saliency maps, model distillation, and so on. However, it is hard for these methods to reveal the intrinsic properties of neural networks. In this work, we studied the 1-D optimal piecewise linear approximation (PWLA) problem and associated it with a designed neural network, named lattice neural network (LNN). We asked four essential questions as following: (1) What are the characters of the optimal solution of the PWLA problem? (2) Can an LNN converge to the global optimum? (3) Can an LNN converge to the local optimum? (4) Can an LNN solve the PWLA problem? Our main contributions are that we propose the theorems to characterize the optimal solution of the PWLA problem and present the LNN method for solving it. We evaluated the proposed LNNs on approximation tasks, forged an empirical method to improve the performance of LNNs. The experiments verified that our LNN method is competitive with the start-of-the-art method.

Intelligent Reflecting Surface Aided Millimeter Wave Communication Using Subarray-Connected Structure

In this paper, we investigate an intelligent reflecting surface (IRS) aided millimeter wave (mmWave) communication system using hybrid precoding of subarray-connected structure at the base station. To maximize the received signal power of the user, we jointly optimize the active beamforming at the BS and passive beamforming at the IRS, respectively. Exploiting the rank-one property of mmWave channels, we transform the original problem into an approximate optimization problem and give the closed-form solution of the approximated problem. Simulation results show that our proposed scheme can effectively improve the received signal quality of the user with a high energy efficiency.

A gradient-based algorithm for non-smooth constrained optimization problems governed by discrete-time nonlinear equations with application to long-term hydrothermal optimal scheduling control

This paper considers a class of non-smooth constrained optimization problems governed by discrete-time nonlinear equations. In these problems, the non-smooth feature comes from the objective function. Our main contributions are as follows. Firstly, by using a smoothing method, the objective function can be approximated by a smooth function, which leads to an approximate optimization problem. Then, based on the idea of l1 penalty function, the state and control constraints are appended to the objective function to form an augmented objective function, which leads to a smooth unconstrained optimization problem. Next, a gradient-based algorithm with a novel line search is proposed for solving this smooth unconstrained optimization problem, and this line search is proved to have a good property similar to the Armijo line search. This is helpful to easily establish convergence results of the proposed algorithm under several mild assumptions. Finally, the proposed algorithm is adopted to solve a long-term hydrothermal optimal scheduling problem, the numerical results show that the proposed algorithm is effective, and the parameter sensitivity analysis results show that the proposed algorithm is also robust.

A computational study of optimal control of Markov jump systems

This paper considers a class of optimal control problems governed by Markov jump systems. Our focus is to develop a computational method, based on the control parametrization approach, for solving this class of optimal control problems. Due to the existence of the continuous-time Markov chain, the optimal control problem under consideration is a stochastic optimal control problem, and hence the control parametrization technique cannot be applied directly. For this, a derandomization approach is introduced to obtain a representative deterministic optimal control problem. Then, the control parametrization method is applied to obtain an approximate finite dimensional optimization problem which can be computed numerically using the gradient-based optimization method. For this, the gradient formulas of the cost function and the constraint functions with respect to control variables are derived. Finally, a practical application involving a RLC circuit model is solved using the method proposed.