On the time transformation of mixed integer optimal control problems using a consistent fixed integer control function

Nonlinear control systems with instantly changing dynamical behavior can be modeled by introducing an additional control function that is integer valued in contrast to a control function that is allowed to have continuous values. The discretization of a mixed integer optimal control problem (MIOCP) leads to a non differentiable optimization problem and the non differentiability is caused by the integer values. The paper is about a time transformation method that is used to transform a MIOCP with integer dependent constraints into an ordinary optimal control problem. Differentiability is achieved by replacing a variable integer control function with a fixed integer control function and a variable time allows to change the sequence of active integer values. In contrast to other contributions, so called control consistent fixed integer control functions are taken into account here. It is shown that these control consistent fixed integer control functions allow a better accuracy in the resulting trajectories, in particular in the computed switching times. The method is verified on analytical and numerical examples.     

Time transformed mixed integer optimal control problems with impacts

The solutions of mixed integer optimal control problems (MIOCPs) yield optimized trajectories for dynamical systems with instantly changing dynamical behavior. The instant change is caused by a changing value of the integer valued control function. For example, a changing integer value can cause a car to change the gear, or a mechanical system to close a contact. The direct discretization of a MIOCP leads to a mixed integer nonlinear program (MINLP) and can not be solved with gradient based optimization methods at once. We extend the work by Gerdts [1] and reformulate a MIOCP with integer dependent constraints as an ordinary optimal control problem (OCP). The discretized OCP can be solved using gradient based optimization methods. We show how the integer dependent constraints can be used to model systems with impact and present optimized trajectories of computational examples, namely of a lockable double pendulum and an acyclic telescope walker. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete mechanics and mixed integer optimal control of dynamical systems

Mixed integer optimal control problems are a generalization of ordinary optimal control problems that include additional integer valued control functions. The integer control functions are used to switch instantaneously from one system to another. We use a time transformation (similar as in [1]) to get rid of the integer valued functions. This allows to apply gradient based optimization methods to approximate the mixed integer optimal control problem. The time transformation from [1] is adapted such that problems with distinct state domains for each system can be solved and it is combined with the direct discretization method DMOC [2,3] (Discrete Mechanics and Optimal Control) to approximate trajectories of the underlying optimal control problems. Our approach is illustrated with the help of a first example, the hybrid mass oscillator. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A decomposition approach for optimal control problems with integer innerpoint state constraints

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed‐integer optimal control problems (MIOCPs). A decomposition approach is suggested to solve a special subclass of MIOCPs with mixed integer inner point state constraints. It is the intrinsic combinatorial complexity of the discrete variables in addition to the high nonlinearity of the continuous optimal control problem that forms the challenges in the theoretical and numerical solution of MIOCPs. During the solution procedure the problem is decomposed at the inner time points into a multiphase problem with mixed integer boundary constraints and phase transitions at unknown switching points. Due to a discretization of the state space at the switching points the problem can be decoupled into a family of continuous optimal control problems (OCPs) and a problem similar to the asymmetric group traveling salesman problem (AGTSP). The OCPs are transcribed by direct collocation to large‐scale nonlinear programming problems, which are solved efficiently by an advanced SQP method. The results are used as weights for the edges of the graph of the corresponding TSP‐like problem, which is solved by a Branch‐and‐Cut‐and‐Price (BCP) algorithm. The proposed approach is applied to a hybrid optimal control benchmark problem for a motorized traveling salesman. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a pair of nonlinear mixed integer programming problems

We consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense. Under weaker (pseudo-convex/pseudo-concave) assumptions, we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem. As a particular case, this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Further, this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.     

State constrained optimal control problems with time delays

In a recent, related, paper, necessary conditions in the form of a Maximum Principle were derived for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions covered fixed end-time problems and, under additional hypotheses, free end-time problems. These conditions improved on previous conditions in the following respects. They provided the first fully non-smooth Pontryagin Maximum Principle for problems involving delays in both state and control variables, only special cases of which were previously available. They provide a strong version of the Weierstrass condition for general problems with possibly non-commensurate control delays, whereas the earlier literature does so only under structural assumptions about the dynamic constraint. They also provided a new ‘two-sided’ generalized transversality condition, associated with the optimal end-time. This paper provides an extension of the Pontryagin Maximum Principle of the earlier paper for time delay systems, to allow for the presence of a unilateral state constraint. The new results fully recover the necessary conditions of the earlier paper when the state constraint is absent, and therefore retain all their advantages but in a setting of greater generality.     

Lagrangian methods for optimal control problems governed by a mixed quasi-variational inequality

This paper studies an optimal control problem where the state of the system is defined by a mixed quasi-variational inequality. Several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem are obtained by using nonlinear Lagrangian methods. Our results are applied to an example where the mixed quasi-variational inequality leads to a bilateral obstacle problem.     

On the mixed integer signomial programming problems

This paper proposes an approximate method to solve the mixed integer signomial programming problem, for which the objective function and the constraints may contain product terms with exponents and decision variables, which could be continuous or integral. A linear programming relaxation is derived for the problem based on piecewise linearization techniques, which first convert a signomial term into the sum of absolute terms; these absolute terms are then linearized by linearization strategies. In addition, a novel approach is included for solving integer and undefined problems in the logarithmic piecewise technique, which leads to more usefulness of the proposed method. The proposed method could reach a solution as close as possible to the global optimum.     

An integral transformation for integer programming problems

It is shown that every integer programming problem can be transformed into an equivalent integer program with free variables in polynomial time. The transformation is advantageous because the equivalent problem it generates can be solved very easily in some restricted cases.     

Superconvergence of mixed finite element methods for optimal control problems

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

     

A maximum principle for optimal control problems with mixed constraints

Necessary conditions in the form of maximum principles are derivedfor optimal control problems with mixed control and state constraints.Traditionally, necessary condtions for problems with mixed constraintshave been proved under hypothesis which include the requirementthat the Jacobian of the mixed constraint functional, with respectto the control variable, have full rank. We show that it canbe replaced by a weaker ‘interiority’ hypothesis.This refinement broadens the scope of the optimality conditions,to cover some optimal control problems involving differentialalgebraic constraints, with index greater than unity.     

Sufficient conditions for optimal control problems with time delay

Various first-order and second-order sufficient conditions of optimality for nonlinear optimal control problems with delayed argument are formulated. The functions involved are not required to be convex. Second-order sufficient conditions are shown to be related to the existence of solutions of a Riccati-type matrix differential inequality. Their relation with the second variation is discussed.The authors are indebted to an anonymous referee for valuable suggestions that lead to various improvements in the paper.     

Scheduling staff using mixed integer programming

This paper describes the solution of a problem of scheduling a workforce so as to meet demand which varies markedly with the time of day and moderately with the day of week. The main objectives are determining how many staff to employ and the times at which shifts should start. The problem is expressed as a large MIP problem initially presenting computational difficulties. The difficulties vanish when the formulation is modified and a package allowing the use of reduce and (especially) special ordered sets becomes available. The client has commissioned the study primarily to benchmark its existing schedule by comparing it with a theoretical optimum. The optimal schedule and comparison are very sensitive to technical and cost coefficients which are not precisely known.     

Enhanced mixed integer programming techniques and routing problems

This is a summary of the author’s PhD thesis supervised by Andrea Lodi and Paolo Toth and defended on 16 April 2009 at the Università di Bologna. The thesis is written in English and is available from the author upon request. This work is focused on Mixed Integer Programming (MIP). In particular, the first part of the thesis deals with general purpose cutting planes, which are probably the key ingredient behind the success of the current generation of MIP solvers. The second part is instead focused on the heuristic and exact exploitation of integer programming techniques for hard combinatorial optimization problems in the context of routing applications.     

The reformulation of two mixed integer programming problems

Two practical problems are described, each of which can be formulated in more than one way as a mixed integer programming problem. The computational experience with two formulations of each problem is given. It is pointed out how in each case a reformulation results in the associated linear programming problem being more constrained. As a result the reformulated mixed integer problem is easier to solve. The problems are a multi-period blending problem and a mining investment problem.     

Error bounds for mixed integer nonlinear optimization problems

We introduce a-posteriori and a-priori error bounds for optimality and feasibility of a point generated as the rounding of an optimal point of the NLP relaxation of a mixed-integer nonlinear optimization problem. Our analysis mainly bases on the construction of a tractable approximation of the so-called grid relaxation retract. Under appropriate Lipschitz assumptions on the defining functions, we thereby generalize and slightly improve results for the mixed-integer linear case from Stein (Mathematical Programming, 2015, doi: 10.1007/s10107-015-0872-7). In particular, we identify cases in which the optimality and feasibility errors tend to zero at an at least linear rate for increasingly refined meshes.     

Reflected BSDEs with random default time and related mixed optimal stopping-control problems

In this paper we study the one-dimensional reflected backward stochastic differential equations which are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. Using a penalization method, we prove the existence and uniqueness of the solutions to these equations. As an application, we show that under proper assumptions the solution of the reflected equation is the value of the related mixed optimal stopping-control problem.     

A note on a pair of nonlinear mixed integer programming problems

In a recent paper some duality results were proved for a pair of nonsymmetric and nonlinear mixed integer programming problems under pseudo-convexity/pseudo-concavity, separability and an additional feasibility assumption. In this note the same results have been obtained under strong pseudoconvexity/strong pseudo-concavity and separability assumptions only.     

Lipschitz stability for elliptic optimal control problems with mixed control-state constraints

A family of linear-quadratic optimal control problems with pointwise mixed state-control constraints governed by linear elliptic partial differential equations is considered. All data depend on a vector parameter of perturbations. Lipschitz stability with respect to perturbations of the optimal control, the state and adjoint variables, and the Lagrange multipliers is established.