For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three …
For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is a convex approximation of the original MINLP and relies on the Gauss–Newton approximation. We analyze the algorithm in terms of approximation properties and establish a first-order consistency result. Then, we investigate the proposed approach considering a numerical case study of Mixed-Integer Optimal Control (MIOC) of a renewable energy system. The investigation shows that the proposed formulation can yield an improved integer solution regarding the objective of the original MINLP compared with the established Combinatorial Integral Approximation (CIA) algorithm.
This work presents a whole-year simulation study on nonlinear mixed-integer Model Predictive Control (MPC) for a complex thermal energy supply system which consists of a heat pump, stratified water storages, …
This work presents a whole-year simulation study on nonlinear mixed-integer Model Predictive Control (MPC) for a complex thermal energy supply system which consists of a heat pump, stratified water storages, free cooling facilities, and a large underground thermal storage. For solution of the arising Mixed-Integer Non-Linear Programs (MINLPs) we apply an existing general and optimal-control-suitable decomposition approach. To compensate deviation of forecast inputs from measured disturbances, we introduce a moving horizon estimation step within the MPC strategy. The MPC performance for this study, which consists of more than 50,000 real time suitable MINLP solutions, is compared to an elaborate conventional control strategy for the system. It is shown that MPC can significantly reduce the yearly energy consumption while providing a similar degree of constraint satisfaction, and autonomously identify previously unknown, beneficial operation modes.
This work presents a whole-year simulation study on nonlinear mixed-integer Model Predictive Control (MPC) for a complex thermal energy supply system which consists of a heat pump, stratified water storages, …
This work presents a whole-year simulation study on nonlinear mixed-integer Model Predictive Control (MPC) for a complex thermal energy supply system which consists of a heat pump, stratified water storages, free cooling facilities, and a large underground thermal storage. For solution of the arising Mixed-Integer Non-Linear Programs (MINLPs) we apply an existing general and optimal-control-suitable decomposition approach. To compensate deviation of forecast inputs from measured disturbances, we introduce a moving horizon estimation step within the MPC strategy. The MPC performance for this study, which consists of more than 50,000 real time suitable MINLP solutions, is compared to an elaborate conventional control strategy for the system. It is shown that MPC can significantly reduce the yearly energy consumption while providing a similar degree of constraint satisfaction, and autonomously identify previously unknown, beneficial operation modes.
Abstract The collocation method meshed with non-linear programming techniques provides an efficient strategy for the numerical solution of optimal control problems. Good accuracy can be obtained for the state and …
Abstract The collocation method meshed with non-linear programming techniques provides an efficient strategy for the numerical solution of optimal control problems. Good accuracy can be obtained for the state and the control trajectories as well as for the value of the objective function. In addition, the control strategy can be quite flexible in form. However, it is necessary to select the appropriate number of collocation points and number of parameters in the approximating functions with care.
SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential …
SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.
This letter concerns the optimal control of a continuous-time dynamical system via continuous and discrete-valued control variables, where the objective functional also accounts for state-independent switching costs. The class of …
This letter concerns the optimal control of a continuous-time dynamical system via continuous and discrete-valued control variables, where the objective functional also accounts for state-independent switching costs. The class of mixed-integer optimal control problems (OCPs) is interpreted as a bilevel problem, involving both switching times optimization, for a given sequence of modes, and purely continuous optimal control. Additionally, an original nonconvex formulation for the switching costs is introduced, in terms of cardinality, inspired by sparse optimization and compressed sensing techniques. We then adopt proximal algorithms to solve the resulting bilevel OCP with composite objective function. An efficient routine for evaluating the proximal operator is developed. Two examples are numerically solved via a proximal gradient method, discussed and compared with the literature. Although this letter focuses on switched linear time-varying dynamics and quadratic cost functionals with a specific formulation of the switching costs, the proposed approach may also apply to more general mixed-integer OCPs.
Abstract The combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can …
Abstract The combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can be avoided by adding a constraint on the total variation to the MILP. Within this work, we present a fast heuristic way to solve this CIA problem and investigate in which situations optimality of the constructed feasible solution is guaranteed. In the second part of this article, we show tight bounds on the integrality gap between a relaxed continuous control trajectory and an integer feasible one in the case of two controls. Finally, we present numerical experiments to highlight the proposed algorithm’s advantages in terms of run time and solution quality.
We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty implies …
We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty implies that the considered integer control problems admit minimizers. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example.
Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. We consider a class of polyhedral functions that arise particularly as convex lower envelopes …
Multibang regularization and combinatorial integral approximation decompositions are two actively researched techniques for integer optimal control. We consider a class of polyhedral functions that arise particularly as convex lower envelopes of multibang regularizers and show that they have beneficial properties with respect to regularization of relaxations of integer optimal control problems. We extend the algorithmic framework of the combinatorial integral approximation such that a subsequence of the computed discrete-valued controls converges to the infimum of the regularized integer control problem.
We are interested in methods to solve mixed-integer nonlinear optimal control problems constrained by ordinary differential equations and combinatorial constraints on some of the control functions. To solve these problems …
We are interested in methods to solve mixed-integer nonlinear optimal control problems constrained by ordinary differential equations and combinatorial constraints on some of the control functions. To solve these problems we use a first discretize, then optimize approach to get a specially structured mixed-integer nonlinear program (MINLP). We decompose this MINLP into a nonlinear program (NLP) and a mixed-integer linear program (MILP), which is called the combinatorial integral approximation problem (CIAP). Previous results guarantee an integer gap for the MINLP depending on the objective function value of the CIAP. The focus of this study is the analysis of the CIAP and of a tailored branch-and-bound method. We link the huge computational gains compared to state-of-the-art MILP solvers to an analysis of subproblems on the branching tree. To this end we study properties of the Lagrangian relaxation of the CIAP. Special focus is given to special ordered set constraints that are present due to an outer convexification of the control problem. Also subproblems that arise by the application of branch-and-bound schemes are of interest. We prove polynomial runtime of the algorithm for special cases and give numerical evidence for efficiency by means of a numerical benchmark problem.