On Polyhedral Projection and Parametric Programming

Jones, Colin N.; Kerrigan, Eric C.; Maciejowski, Jan M.

doi:10.1007/s10957-008-9384-4

Cited by 57 publications

(50 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Unfortunately, their implementation is not available. We took a step further and developed a generic PLP-solver exploiting insights by [18,17]. Our solver, implemented in OCAML, works over rationals and generates COQ-certificates of correctness of its computations, similar to those in VPL [8,9,10].…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

2017

View full text Add to dashboard Cite

Convex polyhedra capture linear relations between variables. They are used in static analysis and optimizing compilation. Their high expressiveness is however barely used in verification because of their cost, often prohibitive as the number of variables involved increases. Our goal in this article is to lower this cost. Whatever the chosen representation of polyhedra -as constraints, as generators or as bothexpensive operations are unavoidable. That cost is mostly due to four operations: conversion between representations, based on Chernikova's algorithm, for libraries in double description; convex hull, projection and minimization, in the constraintsonly representation of polyhedra. Libraries operating over generators incur exponential costs on cases common in program analysis. In the Verimag Polyhedra Library this cost was avoided by a constraints-only representation and reducing all operations to variable projection, classically done by Fourier-Motzkin elimination. Since Fourier-Motzkin generates many redundant constraints, minimization was however very expensive. In this article, we avoid this pitfall by expressing projection as a parametric linear programming problem. This dramatically improves efficiency, mainly because it avoids the post-processing minimization. We show how our new approach can be up to orders of magnitude faster than the previous approach implemented in the Verimag Polyhedra Library that uses only constraints and Fourier-Motzkin elimination, and on par with the conventional double description approach, as implemented in well-known libraries.

show abstract

Section: Related Workmentioning

confidence: 99%

“…Jones et al [17] then Howe et al [15] noticed that the projection of a polyhedron can be expressed as a Parametric Linear Programming problem. In fact, PLP naturally arises when trying to generalize FourierMotzkin method to eliminate several variables simultaneously.…”

Section: Projection Via Parametric Linear Programmingmentioning

confidence: 99%

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

2017

View full text Add to dashboard Cite

show abstract

“…The projection of a polyhedron can be numerically constructed in several ways [13] such as Fourier elimination, [21] (http://people.ee.ethz.ch/ mpt/3/), which contains some computational geometry routines including the projection via several algorithms. Alternatively, one can use the projection tool from QSKELETON (https://github.com/sbastrakov/qskeleton), which is based on an original modification of the FourierMotzkin elimination with Chernikov rules (the algorithm is described in [19]).…”

Section: Explicit Representation Of Optimal Cost Function Value mentioning

confidence: 99%

“…The resulting optimization problem is solved using a linear programming (LP) algorithm. This setup allows us to represent the optimal value of the cost function as an explicit function of the current system state, and this function appears to be polyhedral as well (more precisely, it is the maximum among finite number of piecewise-affine functions [13]). At each particular time instance the cost function is affine (with probability 1) in the absence of sliding modes [20], therefore the safeguarding controller implementation (with each control input constrained to an interval) leads to the bangbang control.…”

Section: Introductionmentioning

confidence: 99%

Relay-type safeguarding regulator for MPC with unknown computational time

Demenkov¹

2015

2015 International Conference on Event-Based Control, Communication, and Signal Processing (EBCCSP)

View full text Add to dashboard Cite

A hot topic of research in the model predictive control community is the real-time implementation of MPC controllers for "fast" systems (e.g. in automotive and aerospace applications). The controller needs to optimize a control sequence on-line while maintaining strict constraints on the time of computations. In this work-in-progress paper we discuss the possibility of asynchronous MPC computations while maintaining the stability of the closed-loop system. In case the computations cannot be performed within the available time slot and this can affect the stability of the closed-loop system, a second controller is activated, which is based on the explicitly constructed MPC cost function. In the case of continuous linear plant dynamics and polyhedral cost, this approach leads to the easily implementable relay-type control.

show abstract

“…In this section the goal is to compute an approximate solutionũ(x) that maps from the measured state to a feasible solution of pLP (8). We do this by approximating the epigraph of the optimal cost function J (x), which is a polyhedron defined implicitly through a projection operation [19] …”

Section: Application To Model Predictive Controlmentioning

confidence: 99%

The double description method for the approximation of explicit MPC control laws

Jones

Morari

2008

2008 47th IEEE Conference on Decision and Control

View full text Add to dashboard Cite

Abstract-A standard model predictive controller (MPC) can be written as a parametric optimization problem whose solution is a piecewise affine (PWA) map from the measured state to the optimal control input. The primary limitation of this optimal 'explicit solution' is that the complexity can grow quickly with problem size, and so in this paper we seek to compute approximate explicit control laws that can tradeoff complexity for approximation error. This computation is accomplished in a two-phase process: First, inner and outer polyhedral approximations of the the convex cost function of the parametric problem are computed with an algorithm based on an extension to the classic double-description method; a convex hull approach. The proposed method has two main advantages from a control point of view: it is an incremental approach, meaning that an approximation of any specified complexity can be produced and it operates on implicitly-defined convex sets, meaning that the optimal solution of the parametric problem is not required. In the second phase of the algorithm, a feasible approximate control law is computed that has the cost function derived in the first phase. For this purpose, a new interpolation method is introduced based on recent work on barycentric interpolation. The resulting control law is continuous, although non-linear and defined over a non-simplical polytopic partition of the state space. The non-simplical nature of the partition generates significantly simpler approximate control laws than current competing methods, as demonstrated on computational examples.

show abstract

On Polyhedral Projection and Parametric Programming

Cited by 57 publications

References 29 publications

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

Scalable Minimizing-Operators on Polyhedra via Parametric Linear Programming

Relay-type safeguarding regulator for MPC with unknown computational time

The double description method for the approximation of explicit MPC control laws

Contact Info

Product

Resources

About