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The Inexact Restoration method for Euler discretization of state and control constrained optimal control problems is studied. Convergence of the discretized (finite-dimensional optimization) problem to an approximate solution using the Inexact Restoration method and convergence of the approximate solution to a continuous-time solution of the origin...
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... relationship between the costates of the direct and indirect adjoining approaches are furnished by (19) and Remark 3.1. The relationship is then illustrated in Figures 6 and 10 for the crane control example. ...
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... problem studied in the aforementioned references is briefly described as follows. The container crane is equipped with a trolley drive motor and a hoist motor, which independently generate torque to actuate the crane; see Figure 1. The aim of the control process is to keep the swing angle as small as possible since a large swing of the container load during the transfer may become dangerous. ...
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... in the case of c = 0.01, we provide the graphs of the multipliers μ 5 , ν 5 and ψ 5 , in Figure 10. Now, ψ 5 for c = 0 also agrees with that presented in [24]. ...
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... robot can be controlled by the thrust of two jets. The planar motion of the FFR is depicted in Figure 11, where we use x 1 and x 2 for the co-ordinates of the FFR, x 4 and x 5 for the corresponding velocities, x 3 for the direction of thrust, x 6 for the angular velocity, and u 1 and u 2 for the thrusts of the two jets. ...
Citations
... The current technique for solving these control-and state-constrained LQ problems is a direct discretization approach whereby the infinite-dimensional problem is reduced to a large-scale finitedimensional problem by using a discretization scheme, e.g., a Runge-Kutta method [5,22]. This discretized problem is then solved through the use of an optimization modelling language such as AMPL [19], paired with (finite-dimensional) large-scale optimization software such as Ipopt [33]. ...
... In the numerical experiments, we used Matlab version 2023b with the DR algorithm and compared its performance with that of the AMPL-Ipopt optimization computational suite [19,33] with Ipopt version 3.12.13. We chose to make comparisons to Ipopt since it is a free and easily obtainable solver that is used for problems such as those presented in this paper (also see [5]). All computations applied = 10 −8 from Algorithm 1 or, in the case of Ipopt, tol = 10 −8 , and were run on a PC with an i5-10500T 2.30GHz processor with 8GB RAM. ...
We consider the application of the Douglas–Rachford (DR) algorithm to solve linear-quadratic (LQ) control problems with box constraints on the state and control variables. We have split the constraints of the optimal control problem into two sets: one involving the ordinary differential equation with boundary conditions, which is affine, and the other, a box. We have rewritten the LQ control problems as the minimization of the sum of two convex functions. We have found the proximal mappings of these functions, which we then employ for the projections in the DR iterations. We propose a numerical algorithm for computing the projection onto the affine set. We present a conjecture for finding the costates and the state constraint multipliers of the optimal control problem, which can, in turn, be used to verify the optimality conditions. We conducted numerical experiments with two constrained optimal control problems to illustrate the performance and the efficiency of the DR algorithm in comparison with the traditional approach of direct discretization.
... We achieve this aim by addressing two optimal control problems that do not have an analytical solution available. These problems are the control-constrained double integrator and the free flying robot [2,3,27,28]. While the first one of these problems is convex, the second one is highly nonconvex. ...
... That is to say, the increment of c k+1 − c k is a quantity in the range of [2,6]. ...
... We use x 1 and x 2 for the coordinates of the FFR, x 4 and x 5 for the corresponding velocities, x 3 for the direction of thrust, x 6 for the angular velocity, and u 1 and u 2 for the thrusts of the two jets. The model was formulated initially in [27] and further studied in [2,3,28]. We use the control constraints as in [2,3,28]. ...
We propose a primal--dual technique that applies to infinite dimensional equality constrained problems, in particular those arising from optimal control. As an application of our general framework, we solve a control-constrained double integrator optimal control problem and the challenging control-constrained free flying robot optimal control problem by means of our primal--dual scheme. The algorithm we use is an epsilon-subgradient method that can also be interpreted as a penalty function method. We provide extensive comparisons of our approach with a traditional numerical approach.
... One of the most successful applications of the IR algorithm is electronic structure calculation, as shown in [8]. Moreover, the IR algorithm has also been successful applied to optimization problems with the box constraint in [9] and problems with multiobjective constraints under weighted-sum scalarization in [10]. For more applications, please see [11,12]. ...
For a better understanding of the bilevel programming on Riemannian manifolds, a semivectorial bilevel programming scheme is proposed in this paper. The semivectorial bilevel programming is firstly transformed into a single-level programming problem by using the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem, which is convex and satisfies the Slater constraint qualification. Then, the single-level programming is divided into two stages: restoration and minimization, based on which an Inexact Restoration algorithm is developed. Under certain conditions, the stability and convergence of the algorithm are analyzed.
... To solve the discretized problem (which is a finite-dimensional optimization problem with around 7,000 variables) we employ the optimization modelling language AMPL [4], which is paired up with the optimization software Ipopt [23] that uses an interior point method. Theory and applications of discretized solutions of constrained optimal control problems can be found in [1,6,7,13,14]. .75 ...
... Problem (P3) with state constraints is much more challenging both analytically and numerically, resulting in boundary arcs when a state constraint becomes active [9]. This extension would typically result in bang-singular-boundary arcs in a similar fashion to the type of solutions obtained for another example (a container crane) in [1]. ...
This paper is concerned with finding an optimal path for an observer, or sensor, moving at a constant speed, which is to estimate the position of a stationary target, using only bear- ing angle measurements. The generated path is optimal in the sense that, along the path, information, and thus the effi- ciency of a potential estimator employed, is maximized. In other words, an observer path is deemed optimal if it max- imizes information so that the location of the target is esti- mated with smallest uncertainty, in some sense. We formulate this problem as an optimal control problem maximizing the determinant of the Fisher information matrix, which is one of the possible measures of information. We derive analytical results for optimality using the Maximum Principle. We carry out numerical experiments and discuss the multiple (locally) optimal solutions obtained. We verify graphically that the nec- essary conditions of optimality are verified by the numerical solutions. Finally we provide a comprehensive list of possible extensions for future work.
... Subsequently, there are many effective numerical methods available in the literature for solving various optimal control problems. Examples include iterative dynamic programming [25], control parametrisation [26][27][28][29][30][31], collocation methods [32,33], and full parametrisation [34][35][36][37][38]. ...
Abstract 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.
... Theoretical papers concerning Inexact Restoration methods for constrained optimization include [41,10,24]. Algorithmic variations are discused in [41,27,4,22,25,26,28], and applications may be found in [43,1,23,38,15,37,5,31,30,6,29,14,47]. ...
... For every iteration of BIRA, by Lemma 5.5, the Restoration Phase finishes after at most N R evaluations of h and ∇ x h. At Step 2 of BIRA, there are no evaluations of ∇ x h and additional evaluations of h are not necessary, since h(x k , y k R ) and h(x k R , y k R ) have been already computed in RESTA or to check (5). ...
In a recent paper an Inexact Restoration method for solving continuous constrained optimization problems was analyzed from the point of view of worst-case functional complexity and convergence. On the other hand, the Inexact Restoration methodology was employed, in a different research,to handle minimization problems with inexact evaluation and simple constraints. These two methodologies are combined in the present report, for constrained minimization problems in which both the objective function and the constraints, as well as their derivatives, are subject to evaluation errors. Together with a complete description of the method, complexity and convergence results will be proved.
... We have discretized the 1 -model, using the trapezoidal rule, in a fashion similar to the way optimal control problems are discretized; see, for example, [21,22]. Challenging optimal control problems have successfully been solved using direct discretization previously [1,2,6,23]. ...
We propose new mathematical optimization models for generating sparse dynamical graphs, or networks, that can achieve synchronization. The synchronization phenomenon is studied using the Kuramoto model, defined in terms of the adjacency matrix of the graph and the coupling strength of the network, modelling the so-called coupled oscillators. Besides sparsity, we aim to obtain graphs which have good connectivity properties, resulting in small coupling strength for synchronization. We formulate three mathematical optimization models for this purpose. Our first model is a mixed integer optimization problem, subject to ODE constraints, reminiscent of an optimal control problem. As expected, this problem is computationally very challenging, if not impossible, to solve, not only because it involves binary variables but also some of its variables are functions. The second model is a continuous relaxation of the first one, and the third is a discretization of the second, which is computationally tractable by employing standard optimization software. We design dynamical graphs that synchronize, by solving the relaxed problem and applying a practical algorithm for various graph sizes, with randomly generated intrinsic natural frequencies and initial phase variables. We test robustness of these graphs by carrying out numerical simulations with random data and constructing the expected value of the network’s order parameter and its variance under this random data, as a guide for assessment.
... To solve the discretized problem (which is a finite-dimensional optimization problem with around 7,000 variables) we employ the optimization modelling language AMPL [4], which is paired up with the optimization software Ipopt [23] that uses an interior point method. Theory and applications of discretized solutions of constrained optimal control problems can be found in [1,6,7,13,14]. .75 ...
... Problem (P3) with state constraints is much more challenging both analytically and numerically, resulting in boundary arcs when a state constraint becomes active [9]. This extension would typically result in bang-singular-boundary arcs in a similar fashion to the type of solutions obtained for another example (a container crane) in [1]. ...
This paper is concerned with finding an optimal path for an observer, or sensor, moving at a constant speed, which is to estimate the position of a stationary target, using only bearing angle measurements. The generated path is optimal in the sense that, along the path, information, and thus the efficiency of a potential estimator employed, is maximized. In other words, an observer path is deemed optimal if it maximizes information so that the location of the target is estimated with smallest uncertainty, in some sense. We formulate this problem as an optimal control problem maximizing the determinant of the Fisher information matrix, which is one of the possible measures of information. We derive analytical results for optimality using the Maximum Principle. We carry out numerical experiments and discuss the multiple (locally) optimal solutions obtained. We verify graphically that the necessary conditions of optimality are verified by the numerical solutions. Finally we provide a comprehensive list of possible extensions for future work.
... The employment of filters associated with Inexact Restoration was exploited in [35,24]. Applications to control problems were given in [37,12,36,3]. In [25], Inexact Restoration was used to obtain global convergence of a sequential programming method. ...
The Inexact Restoration approach has proved to be an adequate tool for handling the problem of minimizing an expensive function within an arbitrary feasible set by using different degrees of precision in the objective function. The Inexact Restoration framework allows one to obtain suitable convergence and complexity results for an approach that rationally combines low- and high-precision evaluations. In the present research, it is recognized that many problems with expensive objective functions are nonsmooth and, sometimes, even discontinuous. Having this in mind, the Inexact Restoration approach is extended to the nonsmooth or discontinuous case. Although optimization phases that rely on smoothness cannot be used in this case, basic convergence and complexity results are recovered. A derivative-free optimization phase is defined and the subproblems that arise at this phase are solved using a regularization approach that take advantage of different notions of stationarity. The new methodology is applied to the problem of reproducing a controlled experiment that mimics the failure of a dam.
... We have discretized the ℓ 1 -model, using the Trapezoidal rule, in a fashion similar to the way optimal control problems are discretized; see, for example, [25,26]. Challenging optimal control problems have successfully been solved using direct discretization previously [27][28][29][30]. ...
We propose new mathematical optimization models for generating sparse graphs/networks that can achieve synchronization. The synchronization phenomenon is studied using the Kuramoto model, defined in terms of the adjacency matrix of the graph and the coupling strength of the network, modelling the so-called coupled oscillators. Besides sparsity, we aim to obtain graphs which have good connectivity properties, resulting in small coupling strength for synchronization. These properties are in practice known to be related to a small spectral ratio, namely the ratio of the largest and the second smallest eigenvalues of the Laplacian of the graph. We formulate three mathematical optimization models for this purpose. The new objective function and constraints that we propose promote a narrow range of small degrees. Our first model is a mixed integer optimization problem, which, computationally, is very challenging, if not impossible, to solve, not only because it involves binary variables but also some of its variables are functions. The second model is a continuous relaxation of the first one, and the third is a discretization of the second, which is computationally tractable by employing standard optimization software. We design graphs by solving the relaxed problem and applying a practical algorithm for various graph sizes, with randomly gen- erated intrinsic natural frequencies and initial phase variables. We test robustness of these graphs by carrying out numerical simulations with random data and constructing the expected value of the network’s order parameter and its variance under this random data, as a guide for assessment.
