ProxNLP a primal-dual augmented Lagrangian solver for nonlinear programming in Robotics and beyond

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ProxNLP: a primal-dual augmented Lagrangian solver

for nonlinear programming in Robotics and beyond

Wilson Jalleta,b,*, Antoine Bambadeb,c, Nicolas Mansardaand Justin Carpentierb

Abstract— Mathematical optimization is the workhorse be-

hind several aspects of modern robotics and control. In these

applications, the focus is on constrained optimization, and the

ability to work on manifolds (such as the classical matrix

Lie groups), along with a speciﬁc requirement for robustness

and speed. In recent years, augmented Lagrangian methods

have seen a resurgence due to their robustness and ﬂexibility,

their connections to (inexact) proximal-point methods, and

their interoperability with Newton or semismooth Newton

methods. In the sequel, we present primal-dual augmented

Lagrangian method for inequality-constrained problems on

manifolds, which we introduced in our recent work, as well

as an efﬁcient C++ implementation suitable for use in robotics

applications and beyond.

Paper Type – Recent Work [1] under review. (Extended with

open-sourced implementation)

I. INTRODUCTION

The setting of optimization on manifolds is of great inter-

est in the ﬁeld of robotics, where generalized coordinates are

naturally represented using Lie groups [2]. Further, solvers

for robotics need to account for physical constraints such as

joint angle and torque limits as well as friction cones, but

also for task-based constraints which could replace penalties

or costs. Problems such as trajectory optimization or inverse

dynamics with various task and physical constraints are

naturally expressed as nonlinear programs (NLP).

A generic nonlinear program on a manifold Mreads as

follows: min

x∈M f(x)

s.t. c(x)∈ C (1)

where c:M → Rmis a (potentially nonlinear) mapping and

C ⊂ Rmis the constraint set.

Equality and inequality-constrained case: We consider

the following generic problem, which captures most prob-

lems in nonlinear optimization, including in robotics:

min

x∈M f(x) s.t. g(x)=0, h(x)60.(2)

Most problems of interest in robotics can be expressed this

way: dynamics as equality constraints, target reaching, ob-

stacle avoidance and friction cones as inequality constraints.

Our proposed approach is based on the augmented La-

grangian method of multipliers [3], [4], [5], and its primal-

dual variant [6]. It was ﬁrst introduced in1[1] where we

aLAAS-CNRS, 7 Avenue du Colonel Roche, F-31400 Toulouse, France

bInria, D´

epartement d’informatique de l’ENS, ´

Ecole normale sup´

erieure,

CNRS, PSL Research University, Paris, France

cENPC, France, *corresponding author: wjallet@laas.fr

1Paper under review.

provide an application to constrained numerical optimal

control with a novel variant of the differential dynamic pro-

gramming (DDP) algorithm. The applicability of augmented

Lagrangians to equality-constrained DDP was recently inves-

tigated in the robotics literature [7], [8], with an extension

to multiple-shooting implicit dynamics in [9].

Overall, our key contribution is an open-source C++

solver for constrained optimization on manifolds for robotics,

named proxnlp2, which relies on a novel variant of the

augmented Lagrangian method.

II. METHODOLOGY

A. Generalized primal-dual augmented Lagrangians

This approach was ﬁrst introduced for equality-constrained

problems in [6]. We recently provided an extension to

inequality-constrained problems in [1] with an application to

constrained DDP. This method was further applied to convex

QPs in Bambade et al. [10].

The classical (Hestenes-Powell-Rockafellar) augmented

Lagrangian function for the problem (2) reads:

Lµ(x;ye, ze) = f(x)+ 1

2µkg(x)+µyek2

2+1

2µk[h(x)+µze]+k.

(3)

Augmented Lagrangians are known to be exact penalty

functions for constrained optimization, as in their exists an

estimate (¯y, ¯z)and penalty parameter ¯µ > 0such that a

minimizer x∗of L¯µ(·; ¯y, ¯z)is a solution of (2).

Method of multipliers: The method of multipliers al-

gorithm consists in iteratively minimizing the augmented

Lagrangian and taking a (projected) dual ascent step in the

multipliers:

xl+1 = argmin

Lµ(x;yl, zl),

yl+1 =ye+1

µg(xl+1)

zl+1 = [ze+1

µh(xl+1)]+

(4)

This process can also be seen as a proximal-point algorithm

for the dual problem to the initial NLP [11].

Primal-dual function: The primal-dual augmented La-

grangian (pdAL) adds a penalty term for dual variables:

Mµ(x, y, z;ye, ze)def

=Lµ(x;ye, ze)

2µkg(x) + µ(ye−y)k2

2+1

2µk[h(x) + µze]+−µzk2

(5)

Any stationary point (x∗, y∗, z∗)of Mµ(·;ye, ze)will satisfy

the KKT conditions of the iteration (4) where yl+1 =y∗.

2https://github.com/Simple-Robotics/proxnlp.

arXiv:2210.02109v1 [cs.RO] 5 Oct 2022

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摘要：

ProxNLP:aprimal-dualaugmentedLagrangiansolverfornonlinearprogramminginRoboticsandbeyondWilsonJalleta,b,*,AntoineBambadeb,c,NicolasMansardaandJustinCarpentierbAbstractMathematicaloptimizationistheworkhorsebe-hindseveralaspectsofmodernroboticsandcontrol.Intheseapplications,thefocusisonconstrainedopti...

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ProxNLP a primal-dual augmented Lagrangian solver for nonlinear programming in Robotics and beyond

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