Ngoc Anh Nguyen

Inverse parametric convex programming problems via convex liftings

Abstract The present paper introduces a procedure to recover an inverse parametric linear or quadratic programming problem from a given polyhedral partition over which a continuous piecewise affine function is defined. The solution to the resulting parametric linear problem is exactly the initial piecewise affine function over the given original parameter space partition. We provide sufficient conditions for the existence of solutions for such inverse problems. Furthermore, the constructive procedure proposed here requires at most one supplementary variable in the vector of optimization arguments. The principle of this method builds upon an inverse map to the orthogonal projection, known as a convex lifting. Finally, we show that the theoretical results has a practical interest in Model Predictive Control (MPC) design. It is shown that any linear Model Predictive Controller can be obtained through a reformulated MPC problem with control horizon equal to two prediction steps.

On the lifting problems and their connections with piecewise affine control law design

Lifting as geometric operation can be defined as a pseudo-inverse of orthogonal projection. It has received attention in different fields and applications (mechanics, geometry, control, etc). Numerous studies have been dedicated to the existence conditions of a convex lifting in a higher dimension for a given cell complex. It is worth noting that this notion can be extended for a polyhedral partition for applications in control theory, and it is recently shown to be the key step in solving the inverse parametric linear/quadratic programming problems. The present paper presents in a succinct manner the main elements of this topological problem with specific attention to the case of polyhedral partitions and their liftings. Furthermore, we are interested from the practical point of view, in the use of these concepts in control system design. Practically, a construction for the Voronoi diagram class is presented. Secondly, a methodological result is presented which leads to the modification of partitions guaranteeing a theoretically liftable result. In addition, a generic constructive procedure for the partitions based on convexity, continuity and linear (or quadratic) programming is proposed. In order to bring the discussion closer to the control related formulations, the correspondence between convex liftings of a given partition in ℝd onto (d+1)-space and n-space with n > d+1 is provided. Finally an analysis with respect to predictive control related problems will conclude our contribution.

Implications of Inverse Parametric Optimization in Model Predictive Control

Recently, inverse parametric linear/quadratic programming problem was shown to be solvable via convex liftings approach [13]. This technique turns out to be relevant in explicit model predictive control (MPC) design in terms of reducing the prediction horizon to at most two steps. In view of practical applications, typically leading to problems that are not directly invertible, we show how to adapt the inverse optimality to specific, possibly convexly non-liftable partitions. Case study results moreover indicate that such an extension leads to controllers of lower complexity without loss of optimality. Numerical data are also presented for illustration.

Any discontinuous PWA function is optimal solution to a parametric linear programming problem

Recent studies have investigated the continuous functions in terms of inverse optimality. The continuity is a primordial structural property which is exploited in order to link a given piecewise affine (PWA) function to an optimization problem. The aim of this work is to deepen the study of the PWA functions in the inverse optimality context and specifically deal with the presence of discontinuities. First, it will be shown that a solution to the inverse optimality problem exists via a constructive argument. The loss of continuity will have an implication on the structure of the optimization problem which, albeit convex, turns to have a set-valued optimal solution. As a consequence, the original PWA function will represent an optimal solution but the uniqueness is lost. From the numerical point of view, we introduce an algorithm to construct an optimization problem that admits a given discontinuous PWA function as an optimal solution. This construction is shown to rely on convex liftings. A numerical example is considered to illustrate the proposal.

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.

Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra

Constructive solution to inverse parametric linear/quadratic programming problems has recently been investigated and shown to be solvable via convex liftings [15], [14]. These results were stated and solved starting from polytopic partitions of a polytope in the parameter space. Therefore, the case of polyhedral partitions of unbounded polyhedra, was not handled by this method and deserves a complete characterization to address the general inverse optimality problem. This paper has as main objective to overcome the unboundedness limitation of the given polyhedral partition and to extend the constructive solution put forward in [14] for this omitted case.

On the complexity of the convex liftings-based solution to inverse parametric convex programming problems

The link between linear model predictive control (MPC) and parametric linear/quadratic programming has reached maturity in terms of the characterization of the structural properties and the numerical methods available for the effective resolution. The computational complexity is one of the current bottlenecks for these control design methods and inverse optimality has been recently shown to provide a new perspective for this challenge. However, the question of the minimal complexity of inverse optimality formulation is still open and much under discussion. In this paper we revisit some recent results by pointing out unnecessary geometrical complications which can be avoided by the interpretation of the optimality conditions. Two algorithms for fine-tuning inverse optimality formulation will be proposed and the results will be interpreted via two illustrative examples in comparison with existing formulations.

Convex Lifting: Theory and Control Applications

This paper presents the concept of convex lifting, which will be proven to enable significant implementation benefits for the class of piecewise affine controllers. Accordingly, two different algorithms to construct a convex lifting for a given polyhedral/polytopic partition will be presented. These two algorithms rely on either the vertex or the halfspace representation of the related polyhedra. Also, we introduce an algorithm to refine a polyhedral partition, which does not admit a convex lifting, into a convexly liftable one. Furthermore, two different schemes will be put forward to considerably reduce both the memory footprint and the online evaluation effort, which play a key role in implementation of piecewise affine controllers. Finally, these results will be illustrated via numerical examples and a complexity analysis.

Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law

In this paper, we focus on the robustness and fragility problem for piecewise affine (PWA) control laws for discrete-time linear system dynamics in the presence of parametric uncertainty of the state space model. A generic geometrical approach will be used to obtain robustness/fragility margins with respect to the positive invariance properties. For PWA control laws defined over a bounded region in the state space, it is shown that these margins can be described in terms of polyhedral sets in parameter space. The methodology is further extended to the fragility problem with respect to the partition defining the controller. Finally, several computational aspects are presented regarding the transformation from the theoretical formulations to explicit representations (vertex/halfspace representation of polytopes) of these sets.

Recognition of additively weighted Voronoi diagrams and weighted Delaunay decompositions

Additively weighted Voronoi diagram and weighted Delaunay decomposition are a generalization of Voronoi diagram and Delaunay triangulation, respectively. They have received much attention due to their relevance in many domains see [1], [2], [3]. In particular, their interest in control theory has been recently shown in [4], [5] to solve inverse parametric linear/quadratic programming problem. This paper elaborates on the identification of suitable sites and weights of such particular partitions via two algorithms. It will be shown that these algorithms require solving a bi-linear programming problem.