A duality approach for solving control-constrained linear-quadratic optimal control problems ∗
01 May 2014-Siam Journal on Control and Optimization (Society for Industrial and Applied Mathematics)-Vol. 52, Iss: 3, pp 1423-1456
TL;DR: By solving the dual of the optimal control problem, instead of the primal one, significant computational savings can be achieved, and it is proved that strong duality and saddle point properties hold.
Abstract: We use a Fenchel duality scheme for solving control-constrained linear-quadratic optimal control problems We derive the dual of the optimal control problem explicitly, where the control constraints are embedded in the dual objective functional, which turns out to be continuously differentiable We specifically prove that strong duality and saddle point properties hold We carry out numerical experiments with the discretized primal and dual formulations of the problem, for which we implement powerful existing finite-dimensional optimization techniques and associated software We illustrate that by solving the dual of the optimal control problem, instead of the primal one, significant computational savings can be achieved Other numerical advantages are also discussed
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TL;DR: Hestenes as mentioned in this paper has been a leading researcher on optimization theory since the early 1930's and this book contains much of his original work in the field, including the classical, calculus of variations, approach of first and second variations.
Abstract: M. R. Hestenes London: John Wiley. 1967. Pp. xii + 405. Price £5. Professor Hestenes has been a leading researcher on optimization theory since the early 1930's and this book contains much of his original work in the field. The first three chapters are of an introductory nature and contain the classical, calculus of variations, approach of first and second variations.
129 citations
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TL;DR: This study presents how matrix-valued reproducing kernels allow for an alternative viewpoint in the linear quadratic regulator problem, and introduces a strengthened continuous-time convex optimization problem which can be tackled exactly with finite dimensional solvers, and which solution is interior to the constraints.
Abstract: The linear quadratic regulator problem is central in optimal control and was investigated since the very beginning of control theory. Nevertheless, when it includes affine state constraints, it remains very challenging from the classical ``maximum principle`` perspective. In this study we present how matrix-valued reproducing kernels allow for an alternative viewpoint. We show that the quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories. Drawing upon kernel formalism, we introduce a strengthened continuous-time convex optimization problem which can be tackled exactly with finite dimensional solvers, and which solution is interior to the constraints. When refining a time-discretization grid, this solution can be made arbitrarily close to the solution of the state-constrained Linear Quadratic Regulator. We illustrate the implementation of this method on a path-planning problem.
11 citations
Cites background from "A duality approach for solving cont..."
...(2014) and references within), but also for the theoretical aspects, even without constraints (Bourdin and Trélat, 2017) or just control constraints (Burachik et al., 2014)....
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...…stand at the origin of control theory, research is still active in the field, not only for its numerous applications (see e.g. the examples of Burachik et al. (2014) and references within), but also for its theoretical aspects, even without constraints (Bourdin and Trélat, 2017) or just…...
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...…research is still active in the field, not only for its numerous applications (see e.g. the examples of Burachik et al. (2014) and references within), but also for its theoretical aspects, even without constraints (Bourdin and Trélat, 2017) or just control constraints (Burachik et al., 2014)....
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TL;DR: The collaborative optimization is improved to tackle multidisciplinary problems with coupling design variables within disciplines to achieve the better optimal results as compared with other methods.
11 citations
TL;DR: In this article, the authors consider the minimum energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator.
Abstract: We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i) Dykstra’s algorithm, (ii) the Douglas–Rachford (DR) method and (iii) the Aragon Artacho–Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.
10 citations
Cites background from "A duality approach for solving cont..."
...This problem should be considered within the framework of control-constrained linear-quadratic optimal control problems for which new numerical methods are constantly being developed—see for example [1, 12] and the references therein....
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Posted Content•
TL;DR: In this note the authors correct and improve a zero duality gap result in extended monotropic programming given by Bertsekas (J Optim. Theory Appl. 139:209–225, 2008).
Abstract: In this note we correct and improve a zero duality gap result in extended monotropic programming given by Bertsekas in [1].
9 citations
References
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"A duality approach for solving cont..." refers methods in this paper
...We use here the classical Fenchel conjugate of a functional f : X → R, which is defined as f∗(v) := sup{〈v, x〉 − f(x) |x ∈ X} [5]....
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Abstract: The fourth and final volume in this comprehensive set presents the maximum principle as a wide ranging solution to nonclassical, variational problems. This one mathematical method can be applied in a variety of situations, including linear equations with variable coefficients, optimal processes with delay, and the jump condition. As with the three preceding volumes, all the material contained with the 42 sections of this volume is made easily accessible by way of numerous examples, both concrete and abstract in nature.
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Abstract: Preface to the classics edition Preface Part I. Fundamentals of Convex Analysis. I. Convex functions 2. Minimization of convex functions and variational inequalities 3. Duality in convex optimization Part II. Duality and Convex Variational Problems. 4. Applications of duality to the calculus of variations (I) 5. Applications of duality to the calculus of variations (II) 6. Duality by the minimax theorem 7. Other applications of duality Part III. Relaxation and Non-Convex Variational Problems. 8. Existence of solutions for variational problems 9. Relaxation of non-convex variational problems (I) 10. Relaxation of non-convex variational problems (II) Appendix I. An a priori estimate in non-convex programming Appendix II. Non-convex optimization problems depending on a parameter Comments Bibliography Index.
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01 Jan 1993TL;DR: An efficient translator is implemented that takes as input a linear AMPL model and associated data, and produces output suitable for standard linear programming optimizers.
Abstract: Practical large-scale mathematical programming involves more than just the application of an algorithm to minimize or maximize an objective function. Before any optimizing routine can be invoked, considerable effort must be expended to formulate the underlying model and to generate the requisite computational data structures. AMPL is a new language designed to make these steps easier and less error-prone. AMPL closely resembles the symbolic algebraic notation that many modelers use to describe mathematical programs, yet it is regular and formal enough to be processed by a computer system; it is particularly notable for the generality of its syntax and for the variety of its indexing operations. We have implemented an efficient translator that takes as input a linear AMPL model and associated data, and produces output suitable for standard linear programming optimizers. Both the language and the translator admit straightforward extensions to more general mathematical programs that incorporate nonlinear expressions or discrete variables.
3,176 citations
"A duality approach for solving cont..." refers methods in this paper
...Table 5 Example 4—numerical performance of the AMPL-Ipopt suite with problems (P2) and (D2)....
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...Example 2—optimal state variables for problems (P1) and (D1), obtained with large N . obtain a solution to problem (P1) in any of the 300 runs of the AMPL-Ipopt suite, starting with randomly generated initial guesses as in (6.1)....
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...Table 2 Example 2—numerical performance of the AMPL-Ipopt suite with problems (P1) and (D1)....
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...Within AMPL, we employ the interior-point optimization software Ipopt, version 3.2.4s [49]....
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...On the other hand, we were able to find a solution to the discretized (D2) in every single one of the 300 runs of the AMPL-Ipopt suite for each value of N listed in the table, with randomly generated initial guesses as prescribed in (6.1)....
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