Approximate Optimal Value Research Articles

A note on the satisficing policy of the secretary problem

Wu and Li (2022) presented a satisficing policy for the secretary problem (SP). Moreover, Wu and Li (2022) derived asymptotic formulas of the maximum probability of choosing one of the k satisfactory applicants with special cases k = 2 and 3, respectively, in the satisficing policy. “Based on the work of Wu and Li (2022), this paper derives the asymptotic formulas for the probability of selecting one of the k satisfactory applicants and provides the equation for the approximate optimal cutoff value to achieve the asymptotic maximum probability in the general case of k satisfactory applicants under the satisficing policy.” Moreover, a series of computer simulations are conducted to test the theoretical results for k = 4 and 5.

A collocation method using generalized Laguerre polynomials for solving nonlinear optimal control problems governed by integro-differential equations

The Laguerre polynomials represent a new set of basic functions adopted to solve a class of optimal control problems (OCP) specified by integro-differential equation namely IOCP. The corresponding operational matrices of derivatives are computed to extend the solution of the problem in terms of Laguerre polynomials. Using the results of the considered basis functions with a joint application of collocation method, the IOCP is reduced to solve a system of nonlinear algebraic system. Come acrossing an error bound and convergence analysis for the approximate optimal value of the performance index have been proved by the proposed method. Finally, some examples are given to bring about the validity and applicability of this technique.

Further Optimization of Maxwell-Type Dynamic Vibration Absorber with Inerter and Negative Stiffness Spring Using Particle Swarm Algorithm

Dynamic vibration absorbers (DVAs) are widely used in engineering practice because of their good vibration control performance. Structural design or parameter optimization could improve its control efficiency. In this paper, the viscoelastic Maxwell-type DVA model with an inerter and multiple stiffness springs is investigated with the combination of the traditional theory and an intelligent algorithm. Firstly, the expressions and approximate optimal values of the system parameters are obtained using the fixed-point theory to deal with the H∞ optimization problem, which can provide help with the range of parameters in the algorithm. Secondly, we innovatively introduce the particle swarm optimization (PSO) algorithm to prove that the algorithm could adjust the value of the approximate solution to minimize the maximum amplitude by analyzing and optimizing the single variable and four variables. Furthermore, the validity of the parameters is further verified by simulation between the numerical solution and the analytical solution using the fourth-order Runge–Kutta method. Finally, the DVA demonstrated in this paper is compared with typical DVAs under random excitation. The timing sequence and variances, as well as the decreased ratios of the displacements, show that the presented DVA has a more satisfactory control performance. The inerter and negative stiffness spring can indeed bring beneficial effects to the vibration absorber. Remarkably, the intelligent algorithm can make the resonance peaks equal in the parameter optimization of the vibration absorber, which is quite difficult to achieve with theoretical methods at present. The results may provide a theoretical and computational basis for the optimization design of DVA.

How to grow an oscillators' network with enhanced synchronization.

We study a way to set the natural frequency of a newly added oscillator in a growing network to enhance synchronization. Population growth is one of the typical features of many oscillator systems for which synchronization is required to perform their functions properly. Despite this significance, little has been known about synchronization in growing systems. We suggest effective growing schemes to enhance synchronization as the network grows under a predetermined rule. Specifically, we find that a method based on a link-wise order parameter outperforms that based on the conventional global order parameter. With simple solvable examples, we verify that the results coincide with intuitive expectations. The numerical results demonstrate that the approximate optimal values from the suggested method show a larger synchronization enhancement in comparison with other naïve strategies. The results also show that our proposed approach outperforms others over a wide range of coupling strengths.

On an Approximation of Average Cost per Unit Time Impulse Control of Markov Processes

In this paper we consider impulse control of continuous time Markov processes with average cost per unit time functional. This problem is approximated using impulse control problems stopped at the first exit time from an increasing sequence of open sets. We find a solution to the Bellman equation corresponding to the original problem and show that stopped impulse control problems approximate optimal value of the cost functional.

Convexification with Bounded Gap for Randomly Projected Quadratic Optimization

Random projection techniques based on the Johnson--Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, which leads to smaller-scale optimization problems. D'Ambrosio et al. [Math. Program., 183 (2020), pp. 619--647] have applied random projection to a quadratic optimization problem so as to decrease the number of decision variables. Although the problem size becomes smaller, the projected problem will also almost surely be nonconvex if the original problem is nonconvex and hence will be hard to solve. In this paper, by focusing on the fact that the level of the nonconvexity of a nonconvex quadratic optimization problem can be alleviated by random projection, we find an approximate global optimal value of the problem by attributing it to a convex problem with smaller size. To the best of our knowledge, our paper is the first to use random projection for convexification of nonconvex optimization problems. We evaluate the approximation error between optimum values of a nonconvex optimization problem and its convexified randomly projected problem.

Impact of Energy Storage on Economic Dispatch of Distribution Systems: A Multi-Parametric Linear Programming Approach and its Implications

As a main flexible resource, energy storage helps smooth the volatility of renewable generation and reshape the load profile. This paper aims to characterize the impact of energy storage unit on the economic operation of distribution systems in a geometric manner that is convenient for visualization. Posed as a multi-parametric linear programming problem, the optimal operation cost is explicitly expressed as a convex piecewise linear function in the MW/MWh parameter of the energy storage unit. Based on duality theory, a dual linear programming based algorithm is proposed to calculate an approximate optimal value function (OVF) and critical regions, circumventing the difficulty of degeneracy, a common challenge in the existing multi-parametric linear programming solvers. When the uncertainty of renewable generation is considered, the expected OVF can be readily established based on OVFs in the individual scenarios, which is scalable in the number of scenarios. The OVF delivers abundant sensitivity information that is useful in energy storage sizing. Leveraging the OVFs, a robust stochastic optimization model is proposed to determine the optimal MW-MWh size of the storage unit subject to a given budget, which gives rise to a simple linear program. Case study provides a clear sketch of the outcome of the proposed method, and suggests that the optimal energy-power ratio of an energy storage unit is between 5 and 6 from the economical perspective.

Optimal trajectory tracking control based on reinforcement learning for the deployment process of space tether system

Space tether system has a wide application prospect in space mission. Due to the characteristics of strong non-linearity and under-actuation, as well as the interference of complex space environment, it is difficult to model the tethered system accurately. Hence, the controller based on the parameters of the system model will cause large errors in the process of control. In this paper, an adaptive dynamic programming algorithm based on reinforcement learning theory is adopted. By training two Back Propagation (BP) neural networks, namely critic neural network (NN) and actor NN, the performance index function and control law of the system approach approximate optimal values respectively. The controller design is independent of the system model, so model-free control of the system is realized by implementing this control method. First, assuming that the out-of-plane motion of the system is stable, the optimal deployment trajectory of the tethered system is obtained by parameter optimization based on Nelder-Mead method. The optimal trajectory is taken as the nominal trajectory and the trajectory tracking is carried out by reinforcement learning controller. The simulation results show that the reinforcement learning algorithm has a good control effect on the in-plane trajectory tracking of the tethered system, which proves the feasibility and robustness of the control method.

A numerical approach for solving fractional optimal control problems using modified hat functions

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann–Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann–Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

Distributed optimal control for nonholonomic systems with input constraints and uncertain interconnections

This paper studies a distributed optimal tracking control method for nonholonomic mobile mechanical multi-agent systems under complex conditions such as input constraints, the presence of both kinematic and dynamic disturbances, and uncertain interconnections. Initially, novel feed-forward control inputs are proposed to transform the inherently separate systems of kinematics and dynamics into an equivalent integrated system. Successively, an online distributed $$\mathcal {L}_2$$ -bounded optimal control algorithm is designed by utilizing adaptive dynamic programming and the theory of cooperative differential graphical games. In the algorithm, a single neural network instead of three for each agent is chosen, and the online weight-tuning laws for which are designed without identifying uncertain parameters directly or indirectly. Additionally, the optimal control and worst disturbance policies are synchronously updated in only one iterative loop. It is shown that during the convergence of the value functions to the approximate optimal values when the agents perform the algorithm, overall tracking and function approximation errors are uniformly ultimately bounded. Finally, as a successful application of the study, control of the wheeled mobile multi-robot system is discussed through simulations.

A multi-criteria approach to approximate solution of multiple-choice knapsack problem

We propose a method for finding approximate solutions to multiple-choice knapsack problems. To this aim we transform the multiple-choice knapsack problem into a bi-objective optimization problem whose solution set contains solutions of the original multiple-choice knapsack problem. The method relies on solving a series of suitably defined linearly scalarized bi-objective problems. The novelty which makes the method attractive from the computational point of view is that we are able to solve explicitly those linearly scalarized bi-objective problems with the help of the closed-form formulae. The method is computationally analyzed on a set of large-scale problem instances (test problems) of two categories: uncorrelated and weakly correlated. Computational results show that after solving, in average 10 scalarized bi-objective problems, the optimal value of the original knapsack problem is approximated with the accuracy comparable to the accuracies obtained by the greedy algorithm and an exact algorithm. More importantly, the respective approximate solution to the original knapsack problem (for which the approximate optimal value is attained) can be found without resorting to the dynamic programming. In the test problems, the number of multiple-choice constraints ranges up to hundreds with hundreds variables in each constraint.

Convergence Conditions for the Observed Mean Method in Stochastic Programming

The paper analyzes convergence conditions of the method of observed mean under nonstandard conditions, where dependent observations of random parameters are used and probabilistic optimization functions may be discontinuous indicators. For the case of dependent observations, large deviation type theorems for approximate optimal values and solutions are established.

Estimation of a noisy subordinated Brownian motion via two-scales power variations

High frequency based estimation methods for a semiparametric pure-jump subordinated Brownian motion exposed to a small additive microstructure noise are developed building on the two-scales realized variations approach originally developed by Zhang et al. (2005) for the estimation of the integrated variance of a continuous Itô process. The proposed estimators are shown to be robust against the noise and, surprisingly, to attain better rates of convergence than their precursors, method of moment estimators, even in the absence of microstructure noise. Our main results give approximate optimal values for the number K of regular sparse subsamples to be used, which is an important tune-up parameter of the method. Finally, a data-driven plug-in procedure is devised to implement the proposed estimators with the optimal K-value. The developed estimators exhibit superior performance as illustrated by Monte Carlo simulations and a real high-frequency data application.

Markov approximation for multilayered selection in satellite network

A satellite network system comprises three layers of satellites: LEO (Low Earth Orbit), MEO (Middle Earth Orbit) and GEO (Geostationary Orbit). In the system, users can choose a layer according to their demands, including QoS (Quality of Service), congestion, energy cost, etc. The utility that users gain will change when they access satellites in different layers. The mobility of satellites in the LEO and MEO leads to frequent handover among satellites in the same layer. These characteristics of multi-layered satellite networks make it possible for us to exploit the optimal distribution of users, which will maximize the utility of the entire satellite network. While the proposed problem is an NP-hard problem, we analyze the system based on the Markov chain and use the Markov approximation to approach the maximum utility. In addition, we use the count down and select algorithm to implement the process of Markov chain. The simulation results validate the convergence of the Markov approximation. In addition, the gap between the approximate value and optimal values decreases with an increase in ß, which is a positive constant in Markov formulation, according to the simulation results.

Numerical hybrid Taguchi-random coordinate search algorithm for path synthesis

We present a new algorithm for application to path synthesis of a four-bar linkage with continuous variables called the Hybrid Taguchi random coordinate search algorithm (HTRCA). The HTRCA combines two algorithms of the Taguchi method (TM) and the random coordinate search algorithm (RCA). The RCA adjusts one variable simultaneously with two directions and various step sizes in random order to escape a local minimum. Since the RCA modifies one variable at a time according to the random order, the result of the RCA is sensitive to an initial condition of variables. The TM was adopted to generate nearly optimal initial conditions. Using the TM, the approximate optimal value can be found in even multi-modal functions. Also, the TM is used with different setting values to escape a local minimum point by changing two or more variables in one step. By combining the two methods, a global optimal value can be found efficiently. Seven test functions were optimized and the robust and efficient performance was verified by comparison with other hybrid optimization algorithms. Finally, path synthesis of four-bar linkage is done by the developed optimization algorithm, and the result is compared to previous works on the same problem with evolutionary algorithm.

Optimization of Cellulose Enzyme in the Simultaneous Saccharification and Fermentation of Sugarcane Bagasse on the Second-generation Bioethanol Production Technology

Sugarcane bagasse as raw material were treated on chemical delignification with NaOH and used for Simultaneous Saccharification and Fermentation (SSF). SSF was incubated during 5 days by adding Sacharomyches cereviceae, cellulose enzymes (Trichoderma reesei) and nutrients. The variable of this research was the cellulose enzyme variation of 10-60 fpu with 10 fpu on intervals. Optimal results were obtained on 60 fpu which producing ethanol content of 1.3544% or dry weight conversion of 21.3724g/L with potential prediction shaped on logarithmic equation. In the advanced prediction can be obtained approximate optimal value of enzyme contents on 96.6802 fpu.

Dynamic multi-objective optimization of industrial radial-flow fixed-bed reactor of heavy paraffin dehydrogenation in LAB plant using NSGA-II method

Due to the increasing global demand for olefin production, the development of catalytic dehydrogenation of paraffins to olefins has attracted considerable attention in recent years. Thus, a dynamic multi-objective optimization of heavy paraffin dehydrogenation over Pt–Sn–Al2O3 catalyst for production of olefin in an industrial radial-flow fixed-bed reactor (RF-FBR) using non-dominated sorting genetic algorithm-II (NSGA-II) is the subject of this study. The optimization problem involves two objective functions namely, maximization of olefin production rate and selectivity. Two points, A′ and B′, are considered on the obtained Pareto optimal fronts. The result reveals that the midpoints of A′B′ have the approximate optimal values of both objective functions. Moreover, the 3D distributions of the optimized decision variables chromosomes are studied during 29 operating days. Concentrations of the chromosomes are increasing toward the lower limit of temperature and pressure, as well as higher limit of total molar flow rate. Optimization results give the optimum values of operating conditions, under which the maximum olefin production rate and selectivity can be obtained.

Robust Adaptive Optimal Control for Unknown Dynamical Systems without Re-Learning

AbstractIn this paper, we propose an algorithm which directly derive an approximate optimal control law for a new criterion function when some approximate optimal value functions are obtained. The proposed algorithm requires to solve HJB equation for a nested cost function which is defined by the optimal cost function, and a method called Robust Extended Policy Iteration can solve the problem. Furthermore, the proposed algorithm doesn't require the drift terms of the system. The proposed algorithm can be used in order to tune the response. The validity of the proposed algorithm is shown by numerical simulations.

A RECURRENCE METHOD FOR SIMPLE CONTINUOUS LINEAR PROGRAMMING PROBLEMS

In this paper, we discuss a special class of continuous linear programming problems which can be called simple continuous linear programming problems (SP). A practical and efficient method for finding an approximate optimal value and optimal solution of (SP) is presented. The main work of computing an approximate optimal value in the provided method is only to solve finite linear programming problems by using recurrence relations. Furthermore, a simple algorithm can be employed not only to easily solve the (SP) problem but also to provide an error bound of optimal value as well. Some numerical examples are given to implement the algorithm.

A RECURRENCE METHOD FOR SIMPLE CONTINUOUS LINEAR PROGRAMMING PROBLEMS

ABSTRACT In this paper, we discuss a special class of continuous linear programming problems which can be called simple continuous linear programming problems (SP). A practical and efficient method for finding an approximate optimal value and optimal solution of (SP) is presented. The main work of computing an approximate optimal value in the provided method is only to solve finite linear programming problems by using recurrence relations. Furthermore, a simple algorithm can be employed not only to easily solve the (SP) problem but also to provide an error bound of optimal value as well. Some numerical examples are given to implement the algorithm.