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Summary: We develop an approximate multiparametric convex programming approach with its application to control constrained linear parameter‐varying systems. Recently, the application of the real‐time model predictive control (MPC) for various engineering systems has been significantly increased by using the multiparametric convex programming tool, known as explicit MPC approach. The main idea of explicit MPC is to move the major parts of the computations to offline phase and to provide an explicit piecewise affine solution of the constrained MPC problem, which is defined over a set of convex polyhedral partitions. In the proposed method, the idea of convex programming and partitioning is applied for linear parameter‐varying control systems. The feasible space of the time‐varying parameters is divided into simplices in which approximate solutions are calculated such that the approximation error is kept limited by solving sequences of linear programs. The approximate optimal solution within each simplex is obtained by linear interpolation of the optimal solutions in the simplex vertices, and then multiparametric programming tool is utilized to compute an explicit state feedback solution of linear quadratic optimal control problem for simplex vertices subject to state and input constraints. The proposed method is illustrated by a numerical example and the simulation results show the advantages of this approach. [ABSTRACT FROM AUTHOR]

Abstract: This work introduces two algorithms for the solution of pure integer and mixed-integer bilevel programming problems by multiparametric programming techniques. The first algorithm addresses the integer case of the bilevel programming problem where integer variables of the outer optimization problem appear in linear or polynomial form in the inner problem. The algorithm employs global optimization techniques to convexify nonlinear terms generated by a reformulation linearization technique (RLT). A continuous multiparametric programming algorithm is then used to solve the reformulated convex inner problem. The second algorithm addresses the mixed-integer case of the bilevel programming problem where integer and continuous variables of the outer problem appear in linear or polynomial forms in the inner problem. The algorithm relies on the use of global multiparametric mixed-integer programming techniques at the inner optimization level. In both algorithms, the multiparametric solutions obtained are embedded in the outer problem to form a set of single-level (M)(I)(N)LP problems – which are then solved to global optimality using standard fixed-point (global) optimization methods. Numerical examples drawn from the open literature are presented to illustrate the proposed algorithms. [ABSTRACT FROM AUTHOR]

The multiparametric 0-1-Integer Programming (0-1-IP) problem relative to the objective function is a family of 0-1-IP problems which are related by having identical constraint matrix and righthand-side vector. In this paper we present an algorithm to perform a complete multiparametric analysis relative to a generalized min max objective function such that the min sum and min max are particular cases. [ABSTRACT FROM AUTHOR]

The article focuses on a study which proposed the use of a multiparametric programming method in solving the process scheduling problems during uncertainty. Uncertainties, such as demands, prices and processing duration affect process scheduling. The study investigated the different approaches to addressing such process scheduling problems. It proposed a novel solution model for process scheduling problems. The idea behind the proposed solution is to divide the problem into subproblems and address each using a multiparametric linear programming and mixed integer linear/nonlinear programming methods.

For multiparametric convex nonlinear programming problems we propose a recursive algorithm for approximating, within a given suboptimality tolerance, the value function and an optimizer as functions of the parameters. The approximate solution is expressed as a piecewise affine function over a simplicial partition of a subset of the feasible parameters, and it is organized over a tree structure for efficiency of evaluation. Adaptations of the algorithm to deal with multiparametric semidefinite programming and multiparametric geometric programming are provided and exemplified. The approach is relevant for real-time implementation of several optimization-based feedback control strategies. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the solution to optimal control problems for constrained discrete-time linear hybrid systems based on quadratic or linear performance criteria. The aim of the paper is twofold. First, we give basic theoretical results on the structure of the optimal state-feedback solution and of the value function. Second, we describe how the state-feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming. [Copyright &y& Elsevier]

Multiparametric programming considers optimization problems where the data are functions of a parameter vector and describes the optimal value and an optimizer as explicit functions of the parameters. In this paper, we consider a linear program where the right-hand side is an affine function of a parameter vector; we propose an algorithm for approximating its solution. Given a full-dimensional simplex in the parameter space and an optimizer for each simplex vertex, the algorithm formulates the linear interpolation of the given solutions as an explicit function of the parameters, giving a primal feasible approximation of an optimizer inside the simplex. If the resulting absolute error in the objective exceeds a prescribed tolerance, then the algorithm subdivides the simplex into smaller simplices where it applies recursively. We propose both a basic version and a refined version of the algorithm. The basic version is polynomial in the output size, provided a polynomial LP solver is used; the refined version may give a smaller output. A global error bound for the optimizer is derived and some computational tests are discussed. [ABSTRACT FROM AUTHOR]

Dynamic flux balance models (DFBM) are used in this study to infer metabolite concentrations that are difficult to measure online. The concentrations are estimated based on few available measurements. To account for uncertainty in initial conditions the DFBM is converted into a variable structure system based on a multiparametric linear programming (mpLP) where different regions of the state space are described by correspondingly different state space models. Using this variable structure system, a special set membership-based estimation approach is proposed to estimate unmeasured concentrations from few available measurements. For unobservable concentrations, upper and lower bounds are estimated. The proposed set membership estimation was applied to batch fermentation of E. coli based on DFBM. [ABSTRACT FROM AUTHOR]

Deep learning models are a class of approximate models that are proven to have strong predictive capabilities for representing complex phenomena. The introduction of deep learning models into an optimization formulation provides a means to reduce the problem complexity and maintain model accuracy. Recently it has been shown that deep learning models in the form of neural networks with rectified linear units can be exactly recast as a mixed-integer linear programming formulation. However, developing the optimal solution of problems involving mixed-integer decisions in online applications remains challenging. Multiparametric programming alleviates the online computational burden of solving an optimization problem involving bounded uncertain parameters. In this work, a strategy is presented to integrate deep learning and multiparametric programming. This integration yields a unified methodology for developing accurate surrogate models based on deep learning and their offline, explicit optimal solution. The proposed strategy is demonstrated on the optimal operation of a chemostat. [ABSTRACT FROM AUTHOR]