Theoretical and algorithmic advances in multi-parametric optimization and control

This thesis discusses recent advances in a variety of areas in multi-parametric programming and explicit model predictive control (MPC). First, novel theoretical and algorithmic results for multi-parametric quadratic and mixed-integer quadratic programming (mp-QP/mp- MIQP) problems extend the current state-of-the-art: for mp-QP problems, it is shown that its solution is given by a connected graph, based on which a novel solution procedure is developed. Furthermore, several computational studies investigate the performance of different mp-QP algorithms, and a new parallelization strategy is presented, together with an application of mp-QP algorithms to multi-objective optimization. For mp-MIQP problems, it is shown that it is possible to obtain the exact solution of a mp-MIQP problem without resorting to the use of envelopes of solutions, whose computational performance is compared in a computational study with different mp-MIQP algorithms. Then, the concept of robust counterparts in robust explicit MPC for discrete-time linear systems is revisited and an elegant reformulation enables the solution of closed-loop robust explicit MPC problems with a series of projection operations. This approach is extended to hybrid systems, where the same properties are proven to hold. Finally, a new approach towards unbounded and binary parameters in multi-parametric programming is introduced, and several examples highlight its potential.Open Acces

Experimental analysis of algebraic modelling languages for mathematical optimization

In this work, we perform an extensive theoretical and experimental analysis of the characteristics of five of the most prominent algebraic modelling languages (AMPL, AIMMS, GAMS, JuMP, and Pyomo) and modelling systems supporting them. In our theoretical comparison, we evaluate how the reviewed modern algebraic modelling languages match the current requirements. In the experimental analysis, we use a purpose-built test model library to perform extensive benchmarks. We provide insights on which algebraic modelling languages performed the best and the features that we deem essential in the current mathematical optimization landscape. Finally, we highlight possible future research directions for this work

POP – Parametric Optimization Toolbox

In this paper, we describe POP, a MATLAB toolbox for parametric optimization. It features (a) efficient implementations of multiparametric programming problem solvers for multiparametric linear and quadratic programming problems and their mixed-integer counter-parts, (b) a versatile problem generator capable of creating random multiparametric programming problems of arbitrary size, and (c) a comprehensive library of multiparametric programming test problems featuring benchmark test sets for multiparametric linear, quadratic, mixed-integer linear, and mixed-integer quadratic programming problems. In addition, POP is equipped with a graphical user interface which enables the user-friendly use of all functionalities of POP and a link to the solvers of the Multi-Parametric Toolbox (MPT), as well as the ability to design explicit MPC problems. These features are demonstrated in detailed computational studies providing insights into the versatility and applicability of POP. Additionally, the example of a periodic chromatographic system is used to show the scalability of multiparametric programming in general and POP, in particular