scholarly journals On mixed-integer optimal control with constrained total variation of the integer control

2020 ◽  
Author(s):  
Sebastian Sager ◽  
Clemens Zeile
Keyword(s):  
Optimal Control ◽  
Total Variation ◽  
Optimal Control Problems ◽  
Feasible Solution ◽  
Mixed Integer ◽  
Continuous Control ◽  
Mixed Integer Linear Program ◽  
Solution Quality ◽  
Frequent Switching ◽  
Control Trajectory

AbstractThe 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.

scholarly journals Optimality Condition-Based Sensitivity Analysis of Optimal Control for Hybrid Systems and Its Application

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Chunyue Song
Keyword(s):  
Optimal Control ◽  
Sensitivity Analysis ◽  
Hybrid Systems ◽  
Optimality Condition ◽  
Optimal Control Problems ◽  
Numerical Solutions ◽  
Equivalent Transformation ◽  
Control Problems ◽  
Continuous Control ◽  
Mode Transition

Gradient-based algorithms are efficient to compute numerical solutions of optimal control problems for hybrid systems (OCPHS), and the key point is how to get the sensitivity analysis of the optimal control problems. In this paper, optimality condition-based sensitivity analysis of optimal control for hybrid systems with mode invariants and control constraints is addressed under a priori fixed mode transition order. The decision variables are the mode transition instant sequence and admissible continuous control functions. After equivalent transformation of the original problem, the derivatives of the objective functional with respect to control variables are established based on optimal necessary conditions. By using the obtained derivatives, a control vector parametrization method is implemented to obtain the numerical solution to the OCPHS. Examples are given to illustrate the results.

scholarly journals Mixed-integer optimal control problems with switching costs: a shortest path approach

2020 ◽  
Author(s):  
Felix Bestehorn ◽  
Christoph Hansknecht ◽  
Christian Kirches ◽  
Paul Manns
Keyword(s):  
Optimal Control ◽  
Shortest Path ◽  
Switching Costs ◽  
Optimal Control Problems ◽  
Directed Acyclic Graphs ◽  
Shortest Path Problem ◽  
Mixed Integer ◽  
Programming Approach ◽  
Control Problems ◽  
Decomposition Approach

Abstract We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter $$\theta $$ θ . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in $$\theta $$ θ . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.

scholarly journals A solution framework for linear PDE-constrained mixed-integer problems

2021 ◽  
Author(s):  
Fabian Gnegel ◽  
Armin Fügenschuh ◽  
Michael Hagel ◽  
Sven Leyffer ◽  
Marcus Stiemer
Keyword(s):  
Computation Time ◽  
Mixed Integer ◽  
Continuous Control ◽  
State Variables ◽  
Mixed Integer Linear Program ◽  
Numerical Solution Method ◽  
Linear Pde ◽  
Finite Set ◽  
Continuous Constraints ◽  
Naive Approach

AbstractWe present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

Computational Approaches for Mixed Integer Optimal Control Problems with Indicator Constraints

2018 ◽  
Vol 46 (4) ◽  
pp. 1023-1051 ◽  
Author(s):  
Michael N. Jung ◽  
Christian Kirches ◽  
Sebastian Sager ◽  
Susanne Sass
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Mixed Integer ◽  
Control Problems ◽  
Computational Approaches ◽  
Indicator Constraints
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