Sebastian Sager

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.

Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function.

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bocks direct multiple shooting method. In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.

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 for an efficient implementation, including tailored numerical linear algebra, step size computation, parallelization, and infeasibility handling. We prove convergence of the algorithm for the considered problem class. A numerical study based on the open-source implementation qpDUNES shows that the algorithm outperforms both well-established general purpose QP solvers as well as state-of-the-art tailored control QP solvers significantly on the considered benchmark problems.

Efficient Numerics for Nonlinear Model Predictive Control

We review a closely connected family of algorithmic approaches for fast and real–time capable nonlinear model predictive control (NMPC) of dynamic processes described by ordinary differential equations or index-1 differential-algebraic equations. Focusing on active–set based algorithms, we present emerging ideas on adaptive updates of the local quadratic subproblems (QPs) in a multi–level scheme. Structure exploiting approaches for the solution of these QP subproblems are the workhorses of any fast active–set NMPC method. We present linear algebra tailored to the QP block structures that act both as a preprocessing and as block structured factorization methods.

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 a very particular structure for the analysis of optimal sampling decisions. We introduce the information gain function, motivated by an analysis of necessary conditions of optimality. We highlight differences between problem formulations and propose to use a linear penalization of sampling decisions to overcome the intrinsic ill-conditioning of OED. The results of this paper are independent from the actual numerical method to compute the solution to the OED problem and of the question of local and global optima.

Relaxation methods for mixed-integer optimal control of partial differential equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.

Block-structured quadratic programming for the direct multiple shooting method for optimal control

In this contribution, we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, for example, from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. We review the classical condensing algorithm that preprocesses the large but structured quadratic programs (QPs) to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present a new complementary condensing algorithm for QPs with many controls. This algorithm is based on a hybrid null-space range-space approach to exploit the block structure of the QPs that is due to direct multiple shooting. An assessment of the theoretical run-time complexity reveals significant advantages of the proposed algorithm. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally, we demonstrate the merit of the new complementary condensing approach by comparing the behaviour of both methods for a vehicle control problem in which the integer gear decision is convexified.

Discovering functional gene expression patterns in the metabolic network of Escherichia coli with wavelets transforms

BackgroundMicroarray technology produces gene expression data on a genomic scale for an endless variety of organisms and conditions. However, this vast amount of information needs to be extracted in a reasonable way and funneled into manageable and functionally meaningful patterns. Genes may be reasonably combined using knowledge about their interaction behaviour. On a proteomic level, biochemical research has elucidated an increasingly complete image of the metabolic architecture, especially for less complex organisms like the well studied bacterium Escherichia coli.ResultsWe sought to discover central components of the metabolic network, regulated by the expression of associated genes under changing conditions. We mapped gene expression data from E. coli under aerobic and anaerobic conditions onto the enzymatic reaction nodes of its metabolic network. An adjacency matrix of the metabolites was created from this graph. A consecutive ones clustering method was used to obtain network clusters in the matrix. The wavelet method was applied on the adjacency matrices of these clusters to collect features for the classifier. With a feature extraction method the most discriminating features were selected. We yielded network sub-graphs from these top ranking features representing formate fermentation, in good agreement with the anaerobic response of hetero-fermentative bacteria. Furthermore, we found a switch in the starting point for NAD biosynthesis, and an adaptation of the l-aspartate metabolism, in accordance with its higher abundance under anaerobic conditions.ConclusionWe developed and tested a novel method, based on a combination of rationally chosen machine learning methods, to analyse gene expression data on the basis of interaction data, using a metabolic network of enzymes. As a case study, we applied our method to E. coli under oxygen deprived conditions and extracted physiologically relevant patterns that represent an adaptation of the cells to changing environmental conditions. In general, our concept may be transferred to network analyses on biological interaction data, when data for two comparable states of the associated nodes are made available.

Minimization of sewage network overflow

We are interested in the optimal control of sewage networks. It is of high public interest to minimize the overflow of sewage onto the streets and to the natural environment that may occur during periods of heavy rain. The assumption of linear flow in a discrete time setting has proven to be adequate for the practical control of larger systems. However, the possibility of overflow introduces a nonlinear and nondifferentiable element to the formulation, by means of a maximum of linear terms. This particular challenge can be addressed by smoothing methods that result in a nonlinear program (NLP) or by logical constraints that result in a mixed integer linear program (MILP). We discuss both approaches and present a novel tailored branch-and-bound algorithm that outperforms competing methods from the literature for a set of realistic rain scenarios.

Numerical Methods for Optimal Control with Binary Control Functions Applied to a Lotka-Volterra Type Fishing Problem

We investigate possibilities to deal with optimal control problems that have special integer restrictions on the time dependent control functions, namely to take only the values of 0 or 1 on given time intervals. A heuristic penalty term homotopy and a Branch and Bound approach are presented, both in the context of the direct multiple shooting method for optimal control. A tutorial example from population dynamics is introduced as a benchmark problem for optimal control with 0 –1 controls and used to compare the numerical results of the different approaches.