Optimal Parameter Selection Problem Research Articles

최신의 하이퍼파라미터 최적화 프레임워크들의 상대적 성능 평가

Machine learning (ML) has proven to be highly effective in solving complex problems in various domains, thanks to its ability to identify specific data tasks, perform feature engineering, and learn quickly. However, designing and training ML models is a complicated task and requires optimization. The effectiveness of ML models is highly dependent on the selection of hyperparameters that determines their performance. Hyperparameter optimization (HPO) is a systematic search process to find the optimal combinations of hyperparameters to achieve robust performance. Traditional HPO methods such as grid and random search take a lot of computing time when used in large-scale applications. Recently, various automated search strategies, such as Bayesian optimization (BO) and evolutionary algorithms, have been developed to significantly reduce the computing time. In this paper, we use state-of-the-art HPO frameworks, namely BO, Optuna, HyperOpt, and Keras Tuner, for optimizing the ML and deep learning models for the classification tasks and evaluate their comparative performance using two different sets of experiments. The first one uses different ML classifiers to solve the optimal parameter selection problem with HPO. The second one attempts to optimize the convolutional neural network (CNN) architecture using HPO frameworks to improve its performance in the image classification task. We use four publicly available real-world datasets including one image dataset. The experimental results show that HyperOpt - TPE outperforms the other HPO frameworks for the ML classifiers and achieves up to 94.12% of accuracy with 30 minutes for performing the optimization. Similarly, for the CNN model, HyperOpt-TPE outperforms the other HPO frameworks by improving 34% of the classification accuracy, while taking 2 hours and 24 minutes of computing time.

Control parameterization approach to time-delay optimal control problems: A survey

<p style='text-indent:20px;'>Control parameterization technique is an effective method to solve optimal control problems. It works by approximating the control function by piece-wise constant (or linear) function. In this way, the optimal control problems are approximated by optimal parameter selection problems, which can be regarded as finite-dimensional optimization problems, where the control heights and switching times of the piece-wise constant function are taken as the decision variables. They can be solved by using gradient-based optimization methods. For this, it requires the gradient formulas of the objective and constraint functions with respect to the decision variables. There are two methods for deriving the required gradient formulas. The aim of this paper is to survey various numerical methods developed based on control parameterization for solving optimal control problems governed by time-delay systems. Particular focus is on those numerical methods for optimizing switching time points. This is because the process of optimizing the switching time points is much more complicated for optimal control problems with time delays. In addition, we shall also review the computational methods developed based on control parameterization technique for solving two optimal control problems involving more complicated time delays.</p>

Learning-PDE-Based Approximate Optimal Control for an MHD System With Uncertainty Quantification

Handling uncertainty is one of the most important challenges in real physical systems. In this article, we study an approximate optimal magnetic control strategy for a one-dimensional (1-D) magnetohydrodynamic (MHD) system with uncertainty quantification within the learning framework of the underlying MHD model. First, the MHD flow system is modeled by coupled partial differential equations (PDEs) wherein the Reynolds number is not deterministic but random. Then, the optimal magnetic control problem is formulated and reduced to a parameter selection problem with stochastic PDE constraints by means of the control parameterization method. Significantly different from the conventional sensitivity analysis and adjoint methods, a polynomial chaos expansion (PCE) based on a multifidelity model is developed to construct the underlying PDEs model by polynomial functions. Thus, the relationship between the objective function and the control sequence is derived explicitly, which, in turn, transforms the optimal parameter selection problem with stochastic PDE constraints into a typical algebraic optimization problem that can be easily solved by the existing nonlinear programming algorithm. Numerical simulations are illustrated to demonstrate the high performance of our proposed method.

Sequential adaptive switching time optimization technique for optimal control problems

The control parameterization method used together with time-scaling transformation is an effective approach to transform optimal control problems into optimal parameter selection problems. The approximate problems can then be solved by gradient-based optimization methods. However, the conventional time-scaling transformation requires that all the control component functions switch simultaneously, which can be undesirable in practice. In this paper, we introduce a novel technique where the switching times for each of the control component functions can be adaptively selected. To demonstrate the effectiveness and flexibility of the proposed approach, two example problems with control input and terminal state constraints are solved using the method proposed. Numerical results show that the new method achieves much better objective values with some increase in computation time compared to the conventional time-scaling transformation technique.

Optimal control of nonlinear Markov jump systems by control parametrisation technique

This paper considers an optimal control problem of nonlinear Markov jump systems with continuous state inequality constraints. Due to the presence of continuous-time Markov chain, no existing computation method is available to solve such an optimal control problem. In this paper, a derandomisation technique is introduced to transform the nonlinear Markov jump system into a deterministic system, which simultaneously gives rise to an equivalent deterministic dynamic optimisation problem. The control parametrisation technique is then used to partition the time horizon into a sequence of subintervals such that the control function is approximated by a piecewise constant function consistent with the partition. The heights of the piecewise constant function on the corresponding subintervals are taken as decision variables to be optimised. In this way, the approximate dynamic optimisation problem is an optimal parameter selection problem, which can be viewed as a finite dimensional optimisation problem. To solve it using a gradient-based optimisation method, the gradient formulas of the cost function and the constraint functions are derived. Finally, a real-world practical problem involving a bioreactor tank model is solved using the method proposed.

Reconfiguration Optimization of Relative Motion Between Elliptical Orbits Using Lyapunov-Floquet Transformation

This paper investigates the reconfiguration optimization of the formation flying in elliptical orbits. Based on the Tschauner-Hempel equations, the time-varying system has been reduced to a time-independent one by Lyapunov-Floquet transformation without losing its relative motion characteristics. A geometric configuration invariant of the formation is introduced and general relative configurations are considered as linear combinations of essential components in proportion to this invariant. Benefitting from the evolution of the configuration invariant, the reconfiguration optimization is converted into an optimal parameter selection problem and the transfer trajectory is parameterized by a functional integral. In particular, the indirect method of optimal control based on the reduced dynamics avoids the time-varying derivative of costate variables, which simplifies the optimization problem. The numerical results of both low-thrust and impulsive reconfigurations verify the effectiveness of the reconfiguration optimization based on the reduced relative dynamics.

A Computational Approach to Optimal Control Problems with Almost Smooth Controls

In this paper, we consider a class of optimal control problems involving continuous control and state inequality constraints where the control is almost smooth. We first employ the control parametrization technique via approximating the control signal by a piecewise linear function. Then, we develop a time scaling transformation procedure for transforming the approximate problem into an equivalent problem that can be solved readily using conventional methods. On this basis, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. The gradient formulas and convergent properties ensure that the transformed unconstrained optimal parameter selection problems can be solved by existing optimization algorithms or software packages. Finally, an example is solved showing the effectiveness and applicability of the approach proposed.

Optimal control for controllable stochastic linear systems

This paper is concerned with a constrained stochastic linear-quadratic optimal control problem, in which the terminal state is fixed and the initial state is constrained to lie in a stochastic linear manifold. The controllability of stochastic linear systems is studied. Then the optimal control is explicitly obtained by considering a parameterized unconstrained backward LQ problem and an optimal parameter selection problem. A notable feature of our results is that, instead of solving an equation involving derivatives with respect to the parameter, the optimal parameter is characterized by a matrix equation.

A linear optimal feedback control for producing 1, 3-propanediol via microbial fermentation

In this paper, we consider a multistage feedback control strategy for the production of 1, 3-propanediol(1, 3-PD) in microbial fermentation. The feedback control strategy is widely used in industry, and to the best of our knowledge, this is the first time it is applied to 1, 3-PD. The feedback control law is assumed to be linear of the concentrations of biomass and glycerol, and the coefficients in the controller are continuous. A multistage feedback control law is obtained by using the control parameterization method on the coefficient functions. Then, the optimal control problem can be transformed into an optimal parameter selection problem. The time horizon is partitioned adaptively. The corresponding gradients are derived, and finally, our numerical results indicate that the strategy is flexible and efficient.

An optimal pid tuning method for a single-link manipulator based on the control parametrization technique

A control parametrization based optimal PID tuning scheme for a single-link manipulator is developed in this paper. The performance specifications of the control system are formulated as continuous state inequality constraints. Then, the PID optimal tuning problem of the single-link manipulator can be formulated as an optimal parameter selection problem subject to continuous inequality constraints. These continuous inequality constraints are handled by the constraint transcription method together with a local smoothing technique. In such a way, the transformed problem becomes an optimal parameter selection problem in a canonical form, which can be solved efficiently by control parametrization method. Since approach is using the gradient-based method, the corresponding gradient formulas for the cost function and the constraints are derived, respectively. The effectiveness of the proposed method is demonstrated by numerical simulations.

Optimal parameter selection problem of the state dependent impulsive differential equations

The optimal parameter selection problem for the state dependent impulsive equations is a challenging problem since the impulse number and impulse moments are dependent on the control parameters implicitly. In the simple continuous case, the state variables are continuous with respect to the control parameter when the impulse number is fixed. We prove the existence of the optimal solution and present the gradient of the cost functional such that the optimal problem can be solved by some gradient based optimization methods. For the difficult discontinuous case, i.e., the unfixed impulse number, we construct an approximation problem, and prove that its optimal value converges to that of the original optimal problem by the definition of epi-convergence. Then the corresponding gradient and solving algorithm are given. Finally, two numerical examples are presented to show the efficiency of the algorithm.

Analysis and optimization based on a sex pheromone and pesticide pest model with gestation delay

Sex pheromone, aiming at mating disruption (MD), being species specific and leaving no toxic residues in the produce grown, offers an attractive alternative to conventional pesticides. In this paper, by incorporating the gestation delay and sex pheromone, we explore the impact of MD control on the dynamic behaviors of pest system. Firstly, the boundness, stability and bifurcation of system are deliberated. Secondly, an optimal control problem based on sex pheromone and pesticide is transformed into an equivalent optimal parameter selection problem by introducing the constrain violation function. Additionally, the gradients of the cost function with respect to the dose of sex pheromone and the killing rate are given. Furthermore, simulations are executed to validate the validity of our method. Meanwhile, our results indicate that gestation delay increases the extinction risk of the population and liberating sex pheromone destroys the stability of equilibrium states.

Optimal parameter selection problem with application to the synchronization of delivery cycles in a supply chain with Wiener process

SummaryIn this paper, we use optimal parameter selection technique to develop two models involving single‐vendor–multiple‐buyer supply chain, which are called the dynamic independent optimization (DIO) model and the dynamic synchronized cycles (DSC) model, respectively. These models are, respectively, similar to the traditional static independent policy model and the traditional static synchronized cycle model, except that the deterministic demands of the buyers in the above two static models are now being replaced by the stochastic demands satisfying a Wiener process, which have more real‐life applications. Similar to the above static synchronized cycles model, the synchronization of the supply chain in our DSC model is also achieved by scheduling the delivery days of the buyers and coordinating them with the vendor's production cycle. Finding the optimal expected system costs of the DIO model and the DSC model involves solving optimal parameter selection problems governed by ordinary differential equations, whose final times are continuous decision variables and discrete decision variables, respectively. Computational methods have been developed for solving these problems. Numerical results show that the coordinated policy is better than the independent optimization policy, in terms of minimizing the expected system cost of the entire supply chain. Sensitivity analysis is performed to test the effect of changing the cost coefficients and the value on the performances of these models, where is the ratio of the total mean demand rate of all the buyers over the vendor's production rate.

Minimizing almost smooth control variation in nonlinear optimal control problems

In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.

A new computational approach for optimal control problems with multiple time-delay

The control parameterization method used together with the time-scaling transformation is an effective approach to approximating optimal control problems into optimal parameter selection problems when no time delays are involved. The approximate problems can then be solved by gradient-based optimization algorithms. However, the time-scaling transformation, which works well for optimizing variable switching times of the approximate piecewise constant/linear control functions obtained after the application of the control parameterization method, is not applicable to optimal control problems with time delays. In this paper, we consider a class of nonlinear optimal control problems with multiple time delays subject to canonical equality and inequality constraints. Our aim is to develop a novel transformation procedure that converts a given time-delay system into an equivalent system – defined on a new time horizon – in which the control switching times are fixed, but the dynamic system contains multiple variable time delays expressed in terms of the durations between the switching times for each of the approximate control functions in the original time horizon. On this basis, we show that an optimal control policy for the equivalent system can be obtained efficiently using gradient-based optimization techniques. This optimal control policy can then be used to determine the optimal switching times and optimal control variables for the original system. We conclude the paper by solving two example problems.

The growth and productivity of potato (Solanum tuberosum L.) in photoculture

Potato ( Solanum tuberosum L.) is one of the most valuable agricultural plants, which explains high interest in its optimal cultivating conditions study. Potato productivity in the conditions of northern latitudes can be greatly enhanced due to the use of additional light sources. The review focuses on the problem of optimal parameters selection of supplementary lighting: the intensity of the luminous flux, length of photoperiod, pulse and continuous regimes of illumination and spectral structure.The article observes the influence of light intensity and the length of photoperiod on the growth and development of potato plantlets. It also reveals the problem of selection of optimal conditions in connection with cultivar-specific reactions of plants. The question of usefulness of applying pulse and combined regimes of illumination in photoculture is discussed in terms of productivity enhancing and electricity saving. The study gives experimental data from different authors on the influence of various light spectrums on the growth and development of plants and the pigment synthesis. The choice of illumination with wide spectral characteristics and a certain ratio of separate spectrums intensity is justified. The possibility of applying short-time UV radiation for potato plantlets was revealed as a way to stimulate the growth and biosynthetic processes. Infrared radiation can be useful for thermoregulation of optimal conditions for photosynthesis and other physiological processes.

Computational bilinear optimal control for a class of one-dimensional MHD flow systems

In this paper, we consider a bilinear optimal control problem arising in a one-dimensional (1-D) MHD flow modeled by an array of coupled partial differential equations (PDEs). The external control input (external induction of magnetic field) in the model takes the multiplicative effect on both state variables (i.e., momentum and magnetic components). Our aim is to drive the flow velocity to within close proximity of a desired target flow velocity at the pre-indicated terminal time. We first use the Galerkin method combined with a set of basis quadratic B-spline functions to obtain a semi-discrete approximation problem. Next, the convergence of the semi-discrete approximation problem is proved. Then the control parameterization method combined with the time-scaling transformation technique is utilized to obtain an approximate optimal parameter selection problem, in which the exact gradients of the cost function with respect to the decision parameters are computed based on our analytical equations. The approximate problem are then solved by using the gradient-based optimization techniques such as the sequential quadratic programming (SQP). Finally, the numerical results validate the effectiveness of our method.

Linear Quadratic Optimal Control Problems with Fixed Terminal States and Integral Quadratic Constraints

This paper is concerned with a linear quadratic (LQ, for short) optimal control problem with fixed terminal states and integral quadratic constraints. A Riccati equation with infinite terminal value is introduced, which is uniquely solvable and whose solution can be approximated by the solution for a suitable unconstrained LQ problem with penalized terminal state. Using results from duality theory, the optimal control is explicitly derived by solving the Riccati equation together with an optimal parameter selection problem. It turns out that the optimal control is a target-dependent feedback of the current state. Some examples are presented to illustrate the theory developed.

Optimal-tuning of proportional-integral-derivative-like controller for constrained nonlinear systems and application to ship steering control

In this paper, we consider an optimal-tuning problem of proportional-integral-derivative-like (PID-like) controller for constrained nonlinear systems with continuous-time inequality constraints and a terminal state constraint. Due to the complexity of such constraints, it is difficult to solve this problem by using conventional optimization methods. To overcome this difficulty, a constraint transcription approach and a smoothing function are applied to these continuous–time inequality constraints. An augmented cost functional is obtained by appending these continuous-time inequality constraints to the cost functional. Then, this optimal–tuning problem can be converted into an approximate optimal parameter selection problem. The gradient formulae of the augmented cost function and the terminal constraint function are derived, and a gradient-based numerical algorithm is developed for solving this approximate problem. Convergence results show that there only exist a finite number of iterates such that the sequence obtained by using the algorithm is infeasible, and any optimal solution of the approximate problem is also an optimal solution of the original problem. Finally, a ship steering control problem is solved to illustrate the effectiveness of the approach proposed by us.

Optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow

ABSTRACTThis article considers a dynamic optimization problem arising in a one-dimensional magnetohydrodynamic flow, which can be modelled by coupled partial differential equations, where the external control input (external induction of magnetic field) takes a multiplicative effect on the system states (momentum and magnetic components). The aim is to drive the flow velocity to within close proximity of a desired target of flow velocity at the pre-indicated terminal time. First, the Galerkin method is utilized to reduce the original dynamic optimization problem to a lower finite-dimensional dynamic optimization problem governed by a set of ordinary differential equations. Then the control parameterization method is employed to parameterize the finite-dimensional optimization problem and thus obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques, such as sequential quadratic programming. The exact gradients of the cost functional with respect to the decision parameters, which are the key advantages of this approach, are computed using the analytical equations. Finally, numerical examples are illustrated to verify the effectiveness of the proposed method. The novel scheme of this article is to develop an effective computational optimal control method for realizing the optimal tracking control of flow velocity in a one-dimensional magnetohydrodynamic flow.