Finite-dimensional Optimization Research Articles

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

Machine-Synthesized Control of Nonlinear Dynamic Object Based on Optimal Positioning of Equilibrium Points

When solving an optimal control problem with both direct and indirect approaches, the main technique is to transfer the optimal control problem from the class of infinite-dimensional optimization to a finite-dimensional one. However, with all these approaches, the result is an open-loop program control that is sensitive to uncertainties, and for the implementation of which in a real object it is necessary to build a stabilization system. The introduction of the stabilization system changes the dynamics of the object, which means that the optimal control and the optimal trajectory should be calculated for the object already taking into account the stabilization system. As a result, it turns out that the initial optimal control problem is complex, and often the possibility of solving it is extremely dependent on the type of object and functionality, and if the object becomes more complex due to the introduction of a stabilization system, the complexity of the problem increases significantly and the application of classical approaches to solving the optimal control problem turns out to be time-consuming or impossible. In this paper, a synthesized optimal control method is proposed that implements the designated logic for developing optimal control systems, overcoming the computational complexity of the problem posed through the use of modern machine learning methods based on symbolic regression and evolutionary optimization algorithms. According to the approach, the object stabilization system is first built relative to some point, and then the position of this equilibrium point becomes a control parameter. Thus, it is possible to translate the infinite-dimensional optimization problem into a finite-dimensional optimization problem, namely, the optimal location of equilibrium points. The effectiveness of the approach is demonstrated by solving the problem of optimal control of a mobile robot.

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.

Parameter optimization decomposition and synthesis algorithm for a bundle of rotation shells connected with a ring frame

The method of weight optimization of a shell structure consisting of a power ring frame connected to it on each side of non-homogeneous shells of rotation with variable wall thickness under the action of a spatially asymmetric load is presented. The construction decomposition algorithm is applied. The optimization of shells is carried out based on the necessary Pontryagin's optimality conditions with phase constraints. Finite-dimensional optimization methods are used to seek the optimal configuration of the ring frame. The synthesis of the construction is carried out by the method of successive approximations. Numerical optimization results are presented

Mesh Refinement with Early Termination for Dynamic Feasibility Problems

We propose a novel early-terminating mesh refinement strategy using an integrated residual method to solve dynamic feasibility problems. As a generalization of direct collocation, the integrated residual method is used to approximate an infinite-dimensional problem into a sequence of finite-dimensional optimization subproblems. Each subproblem in the sequence is a finer approximation of the previous. It is shown that these subproblems need not be solved to a high precision; instead, an early termination procedure can determine when mesh refinement should be performed. The new refinement strategy, applied to an inverted pendulum swing-up problem, outperforms a conventional refinement method by up to a factor of three in function evaluations.

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>

Distributionally robust reinsurance with expectile

AbstractWe study a distributionally robust reinsurance problem with the risk measure being an expectile and under expected value premium principle. The mean and variance of the ground-up loss are known, but the loss distribution is otherwise unspecified. A minimax problem is formulated with its inner problem being a maximization problem over all distributions with known mean and variance. We show that the inner problem is equivalent to maximizing the problem over three-point distributions, reducing the infinite-dimensional optimization problem to a finite-dimensional optimization problem. The finite-dimensional optimization problem can be solved numerically. Numerical examples are given to study the impacts of the parameters involved.

Particle dual averaging: optimization of mean field neural network with global convergence rate analysis* *This article is an updated version of: Nitanda A, Wu D and Suzuki T 2021 Particle dual averaging: optimization of mean field neural network with global convergence rate analysis Advances in Neural Information Processing Systems vol 34 ed M Ranzato, A Beygelzimer, Y Dauphin, P S

We propose the particle dual averaging (PDA) method, which generalizes the dual averaging method in convex optimization to the optimization over probability distributions with quantitative runtime guarantee. The algorithm consists of an inner loop and outer loop: the inner loop utilizes the Langevin algorithm to approximately solve for a stationary distribution, which is then optimized in the outer loop. The method can thus be interpreted as an extension of the Langevin algorithm to naturally handle nonlinear functional on the probability space. An important application of the proposed method is the optimization of neural network in the mean field regime, which is theoretically attractive due to the presence of nonlinear feature learning, but quantitative convergence rate can be challenging to obtain. By adapting finite-dimensional convex optimization theory into the space of measures, we analyze PDA in regularized empirical/expected risk minimization, and establish quantitative global convergence in learning two-layer mean field neural networks under more general settings. Our theoretical results are supported by numerical simulations on neural networks with reasonable size.

Multi-objective optimal control of bioconversion process considering system sensitivity and control variation

This paper aims to determine a time-varying dilution rate of the feed medium which maximizes mean productivity while minimizing both system sensitivity and control cost in glycerol continuous culture. A multi-objective optimal control problem subjected to a nonlinear control system and its auxiliary system is proposed and transformed into a finite-dimensional large-scale multi-objective optimization problem via discretization and time-scaling transformation. A multi-objective competitive swarm optimization algorithm is constructed to solve this transformed problem based on pairwise competition, mutation operation and environment selection. The Pareto front obtained shows the distribution of the optimal target value which indicates the rationality and feasibility of the control system under the approximate optimal strategy. Numerical simulations verify the robustness of the optimal control system and the effectiveness of our numerical solution method. The value of IGD shows the good accuracy and distribution of the proposed multi-objective optimization algorithm.

Optimal control of nonlinear systems with integer‐valued control inputs and stochastic constraints

AbstractPractical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of nonlinear systems with stochastic constraints can be available to address practical industrial process problems with integer‐valued control inputs. In general, obtaining an analytical solution of this optimal control problem is very difficult due to the discrete nature of the control inputs and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a finite dimensional static constrained optimization problem. First, the integer‐valued control input is relaxed into a continuous‐valued control input by imposing a penalty term on the objective function. Convergence results show that the relaxed solution obtained is an integer solution as long as the penalty parameter is sufficiently large. In addition, it should be noted that no any constraint is introduced in the novel relaxation technique, which can effectively avoid introducing any extra extreme point for the original optimal control problem. Next, a novel smooth approximation function is used to construct a subset of feasible region for this optimal control problem. It is proved that the smooth approximation can converge uniformly to the stochastic constraints as the adjusting parameter reduces. Following that, a numerical computation method is proposed for solving the original optimal control problem. Finally, in order to illustrate the effectiveness of the proposed method, an electric vehicle energy management problem is extended by considering some stochastic constraints. The numerical results show that the proposed method is less conservative compared with the existing typical approaches and can obtain a stable and robust performance when considering the initial condition small perturbations.

Design of impulsive asteroid flybys and scheduling of time-minimal optimal control arcs for the construction of a Dyson ring (GTOC 11)

This paper describes the approach used by the team named the Eccentric Anomalies to obtain the sixth best solution to the problem of the eleventh Global Trajectory Optimization Competition, whose futuristic scenario of Dyson sphere building remained mathematically relevant for current space mission design and engineering in general. As usual for this recurring challenge, it involved large-scale combinatorics at a high level and a multitude of optimal control problems at a lower level. Furthermore it proposed additional layers of complexity by adding a strong scheduling component to the usual flyby sequencing, and by featuring both impulsive and continuous-thrust trajectories. The authors took advantage of modern theoretical techniques and open-source tools to put together a sequential process including analysis based on analytical trajectory models, tree searches using efficient data structures, global and local finite-dimensional optimization and multi-objective trade-offs. The optimal control part was both tackled with direct transcription as well as indirect shooting methods, and the mixed-integer scheduling reformulated as a bi-level optimization. From a programming point of view, the main framework was set in an interpreted language whilst using as much as possible dependencies written in compiled ones for speed.

Warm-started semionline trajectory planner for ship’s automatic docking (berthing)

In the usual framework of control, a reference trajectory is needed as the set point for a feedback controller. This reference trajectory can be generated by solving a trajectory optimization problem. This problem is a continuous optimal control problem (OCP) that is transcribed into a finite-dimensional nonlinear optimization problem (NLP) and solved by SQP. For an underactuated conventional vessel, the mathematical model can be very intricate, hence the NLP itself. This causes significant computational time. This article demonstrates that the balance between the feasibility of the reference trajectory and the computational time can be achieved for an underactuated vessel in a disturbed and restricted environment. This is done by: (1) using an almost-globally optimal offline solution as a warm start in a semionline trajectory optimization to speed up the calculation, (2) including the prediction of wind dynamics, and (3) representing the ship as a rigid body and using a predefined boundary to generate the necessary spatial constraints via a point-in-polygon method that ensure a collision-free trajectory in a nonconvex region. Incorporation of these three things maintains a safe and dynamically feasible trajectory where the warm start gives a considerable computational speedup and better results than that without a warm start.

SYNTHESIS OF ZONAL CONTROLS USING INFORMATION ON THE OBJECT’S STATE AT THE CURRENT AND PREVIOUS MOMENTS OF TIME

It the work, it is assumed that only a part of the components of the phase state vector is controllable. To form the current values of the control actions, we use the measured values of the components both at the current and in some previous moments of time. As a result, the process under study is described by differential equations with timelagging arguments in the phase variable. Another feature of the approach to the feedback control is that the parameters of the dependence of the control actions on the measured values of the state are constant on subsets (zones) of the phase space, into which it is divided in advance. We call such feedback parameters zonal. Due to the fact that the phase space is divided into a finite number of zones, the number of optimizable feedback parameters in the control problem is also finite. Accordingly, the original feedback control problem is reduced to a finite-dimensional optimization problem with the values of the zonal feedback parameters as design variables.

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.

Parametric shape optimization using the support function

The optimization of shape functionals under convexity, diameter or constant width constraints shows numerical challenges. The support function can be used in order to approximate solutions to such problems by finite dimensional optimization problems under various constraints. We propose a numerical framework in dimensions two and three and we present applications from the field of convex geometry. We consider the optimization of functionals depending on the volume, perimeter and Dirichlet Laplace eigenvalues under the aforementioned constraints. In particular we confirm numerically Meissner’s conjecture, regarding three dimensional bodies of constant width with minimal volume.

A Fokker-Planck feedback control framework for optimal personalized therapies in colon cancer-induced angiogenesis.

In this paper, a new framework for obtaining personalized optimal treatment strategies in colon cancer-induced angiogenesis is presented. The dynamics of colon cancer is given by a Itó stochastic process, which helps in modeling the randomness present in the system. The stochastic dynamics is then represented by the Fokker-Planck (FP) partial differential equation that governs the evolution of the associated probability density function. The optimal therapies are obtained using a three step procedure. First, a finite dimensional FP-constrained optimization problem is formulated that takes input individual noisy patient data, and is solved to obtain the unknown parameters corresponding to the individual tumor characteristics. Next, a sensitivity analysis of the optimal parameter set is used to determine the parameters to be controlled, thus, helping in assessing the types of treatment therapies. Finally, a feedback FP control problem is solved to determine the optimal combination therapies. Numerical results with the combination drug, comprising of Bevacizumab and Capecitabine, demonstrate the efficiency of the proposed framework.

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.

Data to Controller for Nonlinear Systems: An Approximate Solution

This letter considers the problem of determining an optimal control action based on observed data. We formulate the problem assuming that the system can be modeled by a nonlinear state-space model, but where the model parameters, state and future disturbances are not known and are treated as random variables. Central to our formulation is that the joint distribution of these unknown objects is conditioned on the observed data. Crucially, as new measurements become available, this joint distribution continues to evolve so that control decisions are made accounting for uncertainty as evidenced in the data. The resulting problem is intractable which we obviate by providing approximations that result in finite dimensional deterministic optimization problems. The proposed approach is demonstrated in simulation on a nonlinear system.

Genetic Algorithms as Computational Methods for Finite-Dimensional Optimization

Introduction. As early as 1744, the great Leonhard Euler noted that nothing at all took place in the universe in which some rule of maximum or minimum did not appear [12]. Great many today’s scientific and engineering problems faced by humankind are of optimization nature. There exist many different methods developed to solve optimization problems, the number of these methods is estimated to be in the hundreds and continues to grow. A number of approaches to classify optimization methods based on various criteria (e.g. the type of optimization strategy or the type of solution obtained) are proposed, narrower classifications of methods solving specific types of optimization problems (e.g. combinatorial optimization problems or nonlinear programming problems) are also in use. Total number of known optimization method classes amounts to several hundreds. At the same time, methods falling into classes far from each other may often have many common properties and can be reduced to each other by rethinking certain characteristics. In view of the above, the pressing task of the modern science is to develop a general approach to classify optimization methods based on the disclosure of the involved search strategy basic principles, and to systematize existing optimization methods. The purpose is to show that genetic algorithms, usually classified as metaheuristic, population-based, simulation, etc., are inherently the stochastic numerical methods of direct search. Results. Alternative statements of optimization problem are given. An overview of existing classifications of optimization problems and basic methods to solve them is provided. The heart of optimization method classification into symbolic (analytical) and numerical ones is described. It is shown that a genetic algorithm scheme can be represented as a scheme of numerical method of direct search. A method to reduce a given optimization problem to a problem solvable by a genetic algorithm is described, and the class of problems that can be solved by genetic algorithms is outlined. Conclusions. Taking into account the existence of a great number of methods solving optimization problems and approaches to classify them it is necessary to work out a unified approach for optimization method classification and systematization. Reducing the class of genetic algorithms to numerical methods of direct search is the first step in this direction. Keywords: mathematical programming problem, unconstrained optimization problem, constrained optimization problem, multimodal optimization problem, numerical methods, genetic algorithms, metaheuristic algorithms.

Efficient demands in a multi-product monopoly

This paper characterizes the efficient market demands among those with a fixed surplus level in a multi-product monopoly, where the monopolist is able to produce a continuum of quality-differentiated products with a cost function that is convex in quality. We show that any efficient market demand must be affine-unit-elastic. This further reduces the problem of characterizing the efficient frontier to a finite dimensional constraint optimization problem. From this characterization, it follows that deadweight losses are positive even under efficient demands; that both consumer surplus and total welfare are nonmonotonic in cost; and that the monopolist sells at most two distinct quality levels under any efficient market demand.