Multiparametric Mixed-integer Quadratic Programming Research Articles

A hierarchical multi-parametric programming approach for dynamic risk-based model predictive quality control

In this work, we present a hierarchical batch quality control strategy with real-time process safety management. It features a multi-time-scale decision-making framework augmenting: (i) Risk-aware model predictive controller for short-term set point tracking and dynamic risk control under disturbances; (ii) Control-aware optimizer for long-term quality and safety optimization over the entire batch operation; (iii) Intermediate surrogate model to bridge the timescale gap by readjusting the optimizer operating decisions for the controller. All of the above problems are solved via multi-parametric mixed-integer quadratic programming with a key advantage to generate offline explicit control/optimization laws as affine functions of process and risk variables. This allows for the design of a fit-for-purpose risk management plan prior to real-time implementation, while reducing the need for repetitive online dynamic optimization. A unified process model is used to underpin the consistency of hierarchical operational optimization. The proposed approach offers a flexible strategy to integrate distinct decision-making time scales which can be selected separately tailored to the process-specific need of control, fault prognosis, and end-batch quality control. A T2 batch reactor case study is presented to showcase this approach to systematically address the interactions and trade-offs of multiple decision layers toward improving process efficiency and safety.

Lifted Evaluation of mp-MIQP Solutions

This note presents an efficient approach for the evaluation of multi-parametric mixed integer quadratic programming (mp-MIQP) solutions, occurring for instance in control problems involving discrete time hybrid systems with quadratic cost. Traditionally, the online evaluation requires a sequential comparison of piecewise quadratic value functions. We introduce a lifted parameter space in which the piecewise quadratic value functions become piecewise affine and can be merged to a single value function defined over a single polyhedral partition without any overlaps. This enables efficient point location approaches using a single binary search tree. Numerical experiments with a power electronics application demonstrate an online speedup up to an order of magnitude. We also show how the achievable online evaluation time can be traded off against the offline computational time.

Explicit hybrid model-predictive control: The exact solution

This article presents an algorithm for the exact solution of explicit hybrid model-predictive control problems of time-invariant, discrete-time mixed logical dynamical systems. Using multiparametric programming, this control problem is formulated as a multiparametric mixed-integer quadratic programming problem where the initial states of the system are treated as parameters. In conjunction with decomposition type or branch-and-bound type approaches, the proposed solution strategy first creates affine relaxations of nonconvex critical regions that result from the comparison of two quadratic objective functions. These relaxations are then used to generate an affine outer approximation for the critical regions. Upon termination, the bounded space of the initial states is partitioned into possibly nonconvex critical regions and corresponding optimal control laws.

An explicit Hybrid Model Predictive Control Strategy for Intravenous Anaesthesia

The paper describes a strategy for the control of intravenous depth of anaesthesia for the induction and maintenance phase. Based on the anaesthesia model, described by a piece-wise affine system, a hybrid model predictive control problem results, which is solved explicitly via the solution of a novel multi-parametric mixed integer quadratic programming algorithm. The control strategy is successfully tested on a set of 7 patients.

A multiparametric programming approach for mixed-integer quadratic engineering problems

In this work we propose algorithms for the solution of multiparametric quadratic programming (mp-QP) problems and multiparametric mixed-integer quadratic programming (mp-MIQP) problems with a convex and quadratic objective function and linear constraints. For mp-QP problems it is shown that the optimal solution, i.e. the vector of continuous variables and Lagrange multipliers, is an affine function of parameters. The basic idea of the algorithm is to use this affine expression for the optimal solution to systematically characterize the space of parameters by a set of regions of optimality. The solution of the mp-MIQP problems is approached by decomposing it into two subproblems, which converge based upon an iterative methodology. The first subproblem, which is an mp-QP, is obtained by fixing the integer variables and its solution represents a parametric upper bound. The second subproblem is formulated as a mixed-integer non-linear programming (MINLP) problem and its solution provides a new integer vector, which can be fixed to obtain a parametric solution, which is better than the current upper bound. The algorithm terminates with an envelope of parametric profiles corresponding to different optimal integer solutions. Examples are presented to illustrate the basic ideas of the algorithms and their application in model predictive and hybrid control problems.