Given the set of discrete solution points or nodes, called the skeleton, generated by an ODE solver, we study the problem of fitting a curve passing through the nodes in the skeleton minimizing a norm of the residual vector of the ODE. We reformulate this interpolation problem as a multi-stage optimal control problem and, for the minimization of two different norms, we apply the associated maximum principle to obtain the necessary conditions of optimality. We solve the problem analytically for the Dahlquist test problem and a variant of the leaky bucket problem, in terms of the given skeleton. We also consider the Van der Pol equation, for which we obtain interpolating curves with minimal residual norms by numerically solving a direct discretization of the problem through optimization software. With the skeletons obtained by various ODE solvers of MATLAB, we make comparisons between the residuals obtained by our approach and those obtained by the MATLAB function deval.

We investigate the behaviour of a primal–dual penalty scheme applied to all possible instances, i.e., boundary conditions, of an optimal control problem involving the double integrator. The proposed duality-based framework enables us to derive closed-form expressions for the associated dual functions as well as for the optimal control variables. Furthermore, we provide a complete characterization of the least exact penalty parameter for this problem. The iteration dynamics of the algorithm are examined by constructing and analyzing the trajectories of the constraint functions for every admissible instance of the problem. Finally, we compare the two step-size rules employed in our primal–dual algorithm with two alternative Polyak-type step-size strategies.

We consider the problem of finding (in some sense) the best approximation control for an infeasible double integrator. The control function is constrained by upper and lower bounds that are too tight and thus cause infeasibility. The infeasibility is characterized by a gap function (representing the separation between two constraint sets) whose squared L^2-norm is to be minimized to find the best approximation control solution. First, we review the existing results for problems involving a general linear control system. Then, for the infeasible double integrator problem, we present an analytical solution for the bang–bang control with at most one switching. The infinite-dimensional optimization problem is reduced to the problem of solving two algebraic equations in two variables, to compute the switching time and gap function. We discuss numerical approaches to solving the system of equations. Finally, we describe the (relaxed) Douglas–Rachford algorithm for the double integrator problem and carry out numerical experiments to illustrate the implementation of the algorithm and test performance.

We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We consider the case when simple bounds (constraints) are also imposed on the curvature along the curve. The curvature at the endpoints may or may not be specified. We prove via optimal control theory that the optimal curve is some concatenation of Euler spiral arcs, circular arcs, and straight line segments. When the curvature is not constrained (or when the curvature constraint does not become active), an optimal curve is only made up of a concatenation of Euler spiral arcs, unless the oriented endpoints lie in a line segment or a circular arc of the prescribed length, in which case the whole curve is either a straight line segment or a circular arc segment, respectively. We propose numerical methods and illustrate these methods and the results by means of three example problems of finding such curves.

We employ optimal control theory to study the problem of estimating the probability density function from a data set originating from an unknown probability distribution. The original variational problem is reformulated as a multi-stage optimal control problem and the associated maximum principle, or conditions of optimality, is reduced to a two-point boundary-value problem with interior conditions. A numerical scheme is proposed to solve the discretization of this problem. Estimates of density functions for synthetic and real data are computed using the proposed approach. The real data come from the Old Faithful geyser and the speeds of a group of galaxies. Comparisons are made with the popular statistics software R.

Splitting and projection-type algorithms have been applied to many optimization problems due to their simplicity and efficiency, but the application of these algorithms to optimal control is less common. In this paper we utilize the Douglas–Rachford (DR) algorithm to solve control-constrained minimum-energy optimal control problems. Instead of the traditional approach where one discretizes the problem and solves it using large-scale finite-dimensional numerical optimization techniques we split the problem in two subproblems and use the DR algorithm to find an optimal point in the intersection of the solution sets of these two subproblems hence giving a solution to the original problem. We derive general expressions for the projections and propose a numerical approach. We obtain analytic closed-form expressions for the projectors of pure, under-, critically- and over-damped harmonic oscillators. We illustrate the working of our approach to solving not only these example problems but also a challenging machine tool manipulator problem. Through numerical case studies, we explore and propose desirable ranges of values of an algorithmic parameter which yield smaller number of iterations.

We explore the relationship between the dual of a weighted minimum-energy control problem, a special case of linear-quadratic optimal control problems, and the Douglas-Rachford (DR) algorithm. We obtain an expression for the fixed point of the DR operator as applied to solving the optimal control problem, which in turn devises a certificate of optimality that can be employed for numerical verification. The fixed point and the optimality check are illustrated in an example optimal control problem.

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.

We prove that if the total factor productivity A of an aggregative economy is right at the barrier s + ?, with s being the growth rate of labor force and ? the real interest rate, then the unique policy to optimally control the economy is the same as the one for optimally controlling weak economies, where A < s + ?. This result gives a complete answer for the interesting open question raised by Vu Thi Huong in her recent paper [Optimal eco-nomic growth problems with high values of total factor productivity. Appl Anal. 2022;101:1315-1329].

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 consider the problem of finding curves of minimum pointwise-maximum curvature, i.e., curves of minimax curvature, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We reformulate the problem in terms of optimal control and use the maximum principle, as well as some geometrical arguments, to produce a classification of the types of solutions. Using the classification, we devise a numerical method which reduces the infinite-dimensional optimization problem to a finite-dimensional problem with just six variables. The solution types, together with some further observations on optimality, are illustrated via numerical examples.

We consider optimal control problems involving two constraint sets: one comprised of linear ordinary differential equations with the initial and terminal states specified and the other defined by the control variables constrained by simple bounds. When the intersection of these two sets is empty, typically because the bounds on the control variables are too tight, the problem becomes infeasible. In this paper, we prove that, under a controllability assumption, the “best approximation” optimal control minimizing the distance (and thus finding the “gap”) between the two sets is of bang–bang type, with the “gap function” playing the role of a switching function. The critically feasible control solution (the case when one has the smallest control bound for which the problem is feasible) is also shown to be of bang–bang type. We present the full analytical solution for the critically feasible problem involving the (simple but rich enough) double integrator. We illustrate the overall results numerically on various challenging example problems.

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.

We present a framework for the optimization of networks across three perspectives of complexity. Two of these refer to structural properties of a system, termed objective and subjective complexity: the former sees an advantage in connectivity; the latter views complexity as creating opaqueness from within a subjective perspective. Here, the distance from a reference for simplicity network—, a tree, the quintessential management structure—models subjective complexity. The third property relates to the dynamics of a network for which we utilize the proxy of Kuramoto synchronization. Within this we propose a two-stage optimization process for effective synchronization within a constraint of load balance and minimum subjective and objective complexities. Separate objective functions for each of these views of complexity are presented. This builds on previous work on the optimization of Kuramoto synchronization. We demonstrate the computational approach within an illustrative framework for stakeholder engagement in the management of complexity in organizational decision making. We show how the different perspectives may be balanced to arrive at choices of an optimal network design.

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Multi-objective integer or mixed-integer programming problems typically have disconnected feasible domains, making the task of constructing an approximation of the Pareto front challenging. The present article shows that certain algorithms that were originally devised for continuous problems can be successfully adapted to approximate the Pareto front for integer, and mixed-integer, multi-objective problems. Relationships amongst various scalarization techniques are established to motivate the choice of a particular scalarization in these algorithms. The proposed algorithms are tested by means of two-, three- and four-objective integer and mixed-integer problems, and comparisons are made. In particular, a new four-objective algorithm is used to solve a rocket injector design problem with a discrete variable, which is a challenging mixed-integer programming problem.

The Douglas-Rachford and Peaceman-Rachford algorithms have been successfully employed to solve convex optimization problems, or more generally find zeros of monotone inclusions. Recently, the behaviour of these methods in the inconsistent case, i.e., in the absence of solutions has triggered significant consideration. It has been shown that under mild assumptions the shadow sequence of the Douglas-Rachford algorithm converges weakly to a generalized solution when the underlying operators are subdifferentials of proper lower semicontinuous convex functions. However, no convergence behaviour has been proved in the case of Peaceman-Rachford algorithm. In this paper, we prove the convergence of the shadow sequences associated with the Douglas-Rachford algorithm and Peaceman-Rachford algorithm when one of the operators is uniformly monotone and 3^* monotone but not necessarily a subdifferential. Several examples illustrate and strengthen our conclusion. We carry out numerical experiments using example instances of optimization problems.

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.

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.

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.