Global optimal control with the direct multiple shooting method

We propose to solve global optimal control problems with a new algorithm that is based on Bock’s direct multiple shooting method. We provide conditions and numerical evidence for a significant overall runtime reduction compared to the standard single shooting approach. Citation Optimal Control Applications and Methods, Vol. 39 (2), pp. 449–470, 2017 DOI 10.1002/oca.2324 Article … Read more

A Parallel Quadratic Programming Method for Dynamic Optimization Problems

Quadratic programming problems (QPs) that arise from dynamic optimization problems typically exhibit a very particular structure. We address the ubiquitous case where these QPs are strictly convex and propose a dual Newton strategy that exploits the block-bandedness similarly to an interior-point method. Still, the proposed method features warmstarting capabilities of active-set methods. We give details … Read more

Efficient upper and lower bounds for global mixed-integer optimal control

We present a control problem for an electrical vehicle. Its motor can be operated in two discrete modes, leading either to acceleration and energy consumption, or to a recharging of the battery. Mathematically, this leads to a mixed-integer optimal control problem (MIOCP) with a discrete feasible set for the controls taking into account the electrical … Read more

On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

Logical implications appear in a number of important mixed-integer nonlinear optimal control problems (MIOCPs). Mathematical optimization offers a variety of different formulations that are equivalent for boolean variables, but result in different relaxations. In this article we give an overview over a variety of different modeling approaches, including outer versus inner convexification, generalized disjunctive programming, … Read more

Pricing Conspicuous Consumption Products in Recession Periods with Uncertain Strength

We compare different approaches of optimization under uncertainty in the context of pricing strategies for conspicuous consumption products in recession periods of uncertain duration and strength. We consider robust worst-case ideas and how the concepts of Value at Risk (VaR) and Conditional Value at Risk (CVaR) can be incorporated efficiently. The approaches are generic in … Read more

The Lagrangian Relaxation for the Combinatorial Integral Approximation Problem

We are interested in methods to solve mixed-integer nonlinear optimal control problems (MIOCPs) constrained by ordinary di erential equations and combinatorial constraints on some of the control functions. To solve these problems we use a rst discretize, then opti- mize approach to get a specially structured mixed-integer nonlinear program (MINLP). We decompose this MINLP into an … Read more

Sampling Decisions in Optimum Experimental Design in the Light of Pontryagin’s Maximum Principle

Optimum Experimental Design (OED) problems are optimization problems in which an experimental setting and decisions on when to measure – the so-called sampling design – are to be determined such that a follow-up parameter estimation yields accurate results for model parameters. In this paper we use the interpretation of OED as optimal control problems with … Read more

A parametric active set method for quadratic programs with vanishing constraints

Combinatorial and logic constraints arising in a number of challenging optimization applications can be formulated as vanishing constraints. Quadratic programs with vanishing constraints (QPVCs) then arise as subproblems during the numerical solution of such problems using algorithms of the Sequential Quadratic Programming type. QPVCs are nonconvex problems violating standard constraint qualifications. In this paper, we … Read more

Combinatorial Integral Approximation

We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel … Read more

A Factorization with Update Procedures for a KKT Matrix Arising in Direct Optimal Control

Quadratic programs obtained for optimal control problems of dynamic or discrete–time processes usually involve highly block structured Hessian and constraints matrices. Efficient numerical methods for the solution of such QPs have to respect and exploit this block structure. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite … Read more