Mixed-integer programming in motion planning

Abstract

This paper presents a review of past and present results and approaches in the area of motion planning using MIP (Mixed-integer Programming). Although in the early 2000s MIP was still seen with reluctance as method for solving motion planning-related problems, nowadays, due to increases in computational power and theoretical advances, its extensive modeling capabilities and versatility are coming to the fore and enjoy increased application and appreciation. This class of control problems involves, essentially, either a selection from a limited number of alternatives or a constrained optimization problem over a non-convex domain. In both situations, MIP has proven to be an efficient modeling technique as it will be shown in the present review paper. Furthermore, an emphasis is laid on the existing alternatives for implementation and on various experimental validations documented in the literature.

Introduction

This section introduces some general real-world problems/situations actually reducing to optimization problems which contain both integer and real variables and can be assimilated to open-loop or closed-loop control problems. Also, for a more general perspective it presents the class of problems which are tackled through MIP formulations.

During more than 60 years of existence, the field of integer programming was extensively studied in the mathematics community due to its promising modeling capability and flexibility. In recent years (mainly the last two decades), mostly owing to the growing computational capabilities, the integer programming was brought to the attention of the control and robotic communities. There exists a broad variety of decision making problems that can be dealt through a MIP framework/approach.

MIP (Mixed-integer Programming) is a mathematical optimization problem in which some or all the variables are integers.

As its name indicates, MIP (Mixed-integer Programming) represents a mathematical optimization problem in which the objective is a linear, quadratic function or sometimes a more general criterion to be minimized or maximized, the constraints are linear (or non-linear) equalities (or inequalities) and there exist some (non-empty) subsets of integer and real variables playing the role of arguments (Jünger, Liebling, Naddef, Nemhauser, Pulleyblank, Reinelt, Rinaldi, Wolsey, 2009, Williams, 2013).

MIP is used to model several design problems and decision processes.

In a larger perspective, MIP is used to model several design problems and decision processes. Consider a typical logistics problem: an airport, which serves on average 50 flights per hour. The airport has only four runways. The task assignment problem that appears is to assign flights to runways, such that the runways are efficiently and uniformly used, while respecting some regulations (e.g., time separation between two consecutive landings/takeoffs or a minimum distance between two runways for simultaneous takeoffs). Another classical situation is described by the well-known traveling salesman problem and its variations, where the salesman wants to visit a number of customers in a minimal time or to cover a minimal distance. This has applications in several domains (e.g, overhauling gas turbine engines or X-ray crystallography Matai, Singh, & Mittal (2010)). The above problems can be solved either intuitively, based on experience or by a trial and error method, but for critical situations an accurate mathematical formulation is necessary in view of certification. There are of course many other use cases which may employ MIP. For example, the optimal power flow in the energy transmission networks Bahiense, Oliveira, Pereira, and Granville (2001) or the transportation problems in a cluttered environment. Consider a boat moving within a fjord region. In order to safely arrive to its destination, the boat should follow a given path and avoid collision with the fjords. Thus, the feasible region is non-convex and should be efficiently described.

In the following, a brief classification of the types of problems, which can be modeled through MIP, is provided. A first class of problems is designated by those that involve integer quantities (i.e. discrete/quantified inputs or outputs), e.g. the knapsack problem (Williams, 2013). For this type of problem, MIP does not seem the obvious, natural, first choice, but, usually, it represents a better solution than a classical approach (e.g., use of the classical linear programming and approximate the provided solution to the nearest integer value).

Another MIP-modelisable class of problems is the one involving logical conditions, extensively treated in: Bemporad and Morari (1999), Williams (2013), Smith and Taskin (2008). For example, in Bemporad and Morari (1999), using the notations from Williams (2013), boolean algebra tools are aggregated, which allows to transform logical conditions on continuous variables into mixed-integer inequalities (linear inequalities involving continuous and binary variables).

As well, MIP is a popular modeling tool for sequencing and/or allocation problems (also, named combinatorial problems) (Smith & Taskin, 2008), including here the typical task assignment problem and its variations (e.g travelling salesman problem Dantzig, Fulkerson, & Johnson (1954)). This class of problems can be easily extended to networks (and graph theory) problems: resource allocation on a PERT (Project Evaluation and Review Techniques) network (Williams, 2013).

Lastly, but most importantly for this paper’s purpose, MIP turned out to be a captivating method to model non-linearity (Bemporad, Morari, 1999, Vielma, 2015) and/or non-convexity (Prodan, Stoican, Olaru, Niculescu, 2015, Richards, How, 2005). A plethora of control engineering problem are naturally and intrinsically characterized by non-linearity and/or non-convexity. For this reason and due to the increasing interest in optimization-based control (Mayne, Rawlings, Rao, & Scokaert, 2000), MIP has became an essential technique, which allows to include logical decisions and non-convex constraints in the optimization problem. Therefore, MIP’s presence in control can be perceived in: piecewise-affine system identification (Bemporad, Roll, Ljung, 2001, Roll, Bemporad, Ljung, 2004), assignment problems (Alighanbari, Kuwata, & How, 2003), persisting exciting control (Marafioti, Stoican, Bitmead, & Hovd, 2012), control of hybrid systems (Bemporad & Morari, 1999), fault detection (Stoican, Olaru, Seron, & De Doná, 2012) or motion planning (Prodan, Stoican, Olaru, Niculescu, 2015, Richards, How, 2002). As the title suggests, in the present review, the main objective is to identify and summarize the state of the art of MIP-based motion planning. Hence, in what follows, we place less emphasis on the other control areas employing MIP, even if throughout the paper we occasionally refer the interested readers to the references covering the other MIP-based control topics.

This paper offers a detailed literature review of breakthrough research results and open issues in the field of multi-agent motion planning in a mixed-integer framework. This work can be employed to the benefit of both control and optimization research communities allowing to swiftly identify previous, timely and relevant research topics in the field and, at the same time, decreasing the time for literature review, although we acknowledge that the present review effort is not exhaustive but merely covers the experience of the authors in the last 10 years in these topics.

To the best of the authors’ knowledge, there are no other exhaustive surveys in this topic although valuable attempts with different objectives are to be found in the literature. For instance, the tutorial session Richards and How (2005) offers a brief overview on how MIP can be employed for (feedback) control. As well, the paper of Smith and Taskin (2008) represents a concise introduction to the MIP modeling, providing the basic concepts regarding MIP formulations (the principles and some handful recipes) and, at the same time, discussing the techniques widely-used in the resolution of MIP problems. Moreover, the work “50 Years of integer programming 1958–2008: From the early years to the state-of-the-art” (Jünger et al., 2009) presents a historical perspective of the field of integer programming and discusses the theoretical, algorithmic and computational aspects of MIP throughout more than five decades of existence. Additionally, but with significant influence, there exist surveys, as, e.g., Vielma (2015), reviewing the advanced MIP formulations techniques, aiming to provide the guidelines for obtaining stronger and/or smaller formulations for a certain class of decision making problems. These works either limit themselves looking to the programming side, either to decision making in general or to a narrow control design topic. We henceforth decide to focus our review on control problems based on MIP and in particular those emerging from the active research field of motion planning which has a plethora of applications in automotive, robotics or multi-agent systems to mention just a few.

The remainder of the paper is organized as follows. We begin in Section 2 with a brief delineation of the evolution of MIP mathematical descriptions, providing the necessary prerequisites used in these formulations. Section 3 presents and details the standard MIP-based problems in motion planning, introducing the generic control strategy employed in such problems. Next, Section 4 makes the transition to multi-agent systems and to the formation control problems involving MIP. Section 5 gives a concise overview of the control architectures exploited in MIP-based navigation problems. Further, we proceed in Section 6 with a brief presentation of both software and hardware implementations of MIP solutions in the literature. Section 7 succinctly presents the motion planning alternatives to MIP. We close our review with the conclusions and with some challenges in MIP-based motion planning and some suggestions for future research.

Throughout this paper we use the following standard notations. The logical operators: $\wedge$ (AND), $\vee$ (OR) and $\neg$ (negation). The Minkowski sum of two sets: $A\oplus B=\{x:x=a+b,a\in A,b\in B\}$. For $x\in {\mathbb{R}}^d$ we denote ${\parallel x\parallel }_Q^2=x^{\top }Qx$. Given a compact set $S\subset {\mathbb{R}}^n,$ ${\mathcal{C}}_X\left(S\right)$ denotes the complement of $S$ over $X,$ while $cl\left(S\right)$ is the closure of the subset $S$. For any polyhedron $P\subset {\mathbb{R}}^d,$ $\mathcal{V}\left(P\right)$ is the (finite) set of its vertices. Any polytope (i.e. a bounded polyhedron) has a dual representation in terms of intersection of half-spaces or convex hull of extreme points: $P=\{x:s_i^{\top }x\le r_i,\forall i\}=\{x:x=\sum {\alpha }_j{v}_j,\sum {\alpha }_j=1,{\alpha }_j\ge 0,\forall j\}$. For a discrete set $\mathcal{I},$ $\#\mathcal{I}$ represents its cardinality.