# Multi-objective Variational Curves

Authors  C. Yalçın Kaya111Mathematics, UniSA STEM, University of South
Australia, Mawson Lakes, S.A. 5095, Australia. E-mail:
<EMAIL_ADDRESS>. J. Lyle Noakes222Department of Mathematics and
Statistics, University of Western Australia, Crawley, W.A. 6009, Australia.
E-mail<EMAIL_ADDRESS>. Erchuan Zhang333Department of Mathematics and
Statistics, University of Western Australia, Crawley, W.A. 6009, Australia.
E-mail<EMAIL_ADDRESS>.

###### Abstract

Riemannian cubics in tension are critical points of the linear combination of
two objective functionals, namely the squared norms of the velocity and
acceleration of a curve on a Riemannian manifold. We view this variational
problem of finding a curve as a multi-objective optimization problem and
construct the Pareto fronts for some given instances where the manifold is a
sphere and where the manifold is a torus. The Pareto front for the curves on
the torus turns out to be particularly interesting: the front is disconnected
and it reveals two distinct Riemannian cubics with the same boundary data,
which is the first known nontrivial instance of this kind. We also discuss
some convexity conditions involving the Pareto fronts for curves on general
Riemannian manifolds.

> Key words: Riemannian cubics in tension, Multi-objective optimal control,
> Pareto front, Numerical methods.

> Mathematics Subject Classification: 58E17; 49M25; 53Z99

##  1 Introduction

Variational curves are curves which minimize an objective functional such as
the length or some other quantity obtained from its velocity or acceleration,
and so on. Very often more than just one of these (competing) functionals are
to be minimized simultaneously, giving rise to a multi-objective optimization
problem. A typical example is cubic curves in tension, or simply cubics in
tension, where a critical point of a linear combination of the squared
$L^{2}$-norms of the velocity and the acceleration is found [18, 19, 21].
Although, clearly, two objective functionals are involved here, this problem
is not treated as a multi-objective optimization problem in the literature in
that it is only posed as a (single-objective) weighted-sum scalarization of
the two objective functionals.

The aim of a multi-objective optimization problem is to find the set of all
trade-off solutions. A trade-off solution, or a Pareto solution, is a solution
which cannot be improved any further to make an objective functional value
better (smaller), without making some of the other objective functional values
worse (larger). Such a solution set is referred to as the Pareto set, or the
efficient set, and in the objective value space, the Pareto front, or the
efficient front, respectively; see for example [9, 16].

It is of great interest to construct the Pareto front for inspection, so that
a desirable solution can be selected by the practitioner. The most popular
approach in constructing a Pareto front is first to scalarize the problem via
parameters, or weights, so as to obtain a continuum of single-objective
optimization problems. These parametric problems are then solved to get a
solution in the Pareto front for each parameter value [8, 9, 16].

To be able to construct the whole Pareto front the parametric scalarization is
required to be a surjection from the space of weights to the set of Pareto
solutions. If the problem is convex, i.e., the objective functionals and the
constraint set are convex, then the weighted-sum scalarization suffices to
furnish a surjection [3]. If, on the other hand, the problem is nonconvex, the
weighted-sum scalarization is not guaranteed to be surjective anymore.

It is well-known that variational problems can often be cast as optimal
control problems; see for example [14]. In [12, Theorem 1], it is stated that
the so-called Chebyshev scalarization (the weighted max-norm of the objective
functionals) is surjective, guaranteeing construction of the whole Pareto
front.

In this paper, we consider variational curves with two objective functionals
to minimize, motivated by cubics in tension. We state a similar result to that
in [12, Theorem 1]. We construct the Pareto fronts of certain instances of two
Riemannian manifolds, namely a sphere and a torus. To the best knowledge of
the authors, this is the first time multi-objective variational curves on
Riemannian manifolds are studied.

We observe that, on the torus, the Pareto front generated by using the
Chebychev scalarization is disconnected and Riemannian cubics with the same
boundary conditions are not unique. Non-uniqueness of Riemannian cubics under
given nontrivial boundary conditions as presented in this paper has not been
encountered in the literature before.

The paper is organized as follows. In Section 2, we introduce the multi-
objective optimization problem motivated by cubics in tension. Then we sketch
the scalarization techniques, namely the Chebychev and weighted-sum
scalarizations. In Section 3, we present some necessary and sufficient
conditions for the epigraph of Pareto fronts to be convex on general
Riemannian manifolds. Numerical experiments in Section 4 illustrate the Pareto
fronts and Pareto curves on a sphere and torus endowed with the standard
Euclidean metric. Finally, we conclude the paper in Section 5 and present some
future directions.

##  2 Problem Statement and Preliminaries

We are motivated by the problem of finding Riemannian cubics in tension. Let
$M$ be a path-connected Riemannian manifold with a Riemannian metric
$\langle\cdot,\cdot\rangle$ and a Levi–Civita connection $\nabla$, $TM$ the
tangent bundle of $M$. Given two points $x_{0},x_{1}$ and their associated
velocities $v_{0}\in T_{x_{0}}M$, $v_{1}\in T_{x_{1}}M$, we denote by
$\mathcal{X}$ the space of all smooth curves $x:[t_{0},t_{f}]\rightarrow M$
satisfying $x(t_{0})=x_{0}$, $\dot{x}(t_{0})=v_{0}$, $x(t_{f})=x_{1}$,
$\dot{x}(t_{f})=v_{1}$, where $t_{0}$ and $t_{f}$ are fixed. Riemannian cubics
in tension are defined to be critical points of the functional

$\int_{t_{0}}^{t_{f}}\bigg{(}\tau\,\langle\dot{x}(t),\dot{x}(t)\rangle+\langle\nabla_{t}\dot{x}(t),\nabla_{t}\dot{x}(t)\rangle\bigg{)}\,dt\,,$
(1)

where $x\in\mathcal{X}$, $\tau>0$ is a constant. Equivalently, Riemannian
cubics in tension are solutions of the following Euler–Lagrange equations [18,
19, 21]

$\displaystyle\nabla_{t}^{3}\dot{x}(t)+R(\nabla_{t}\dot{x}(t),\dot{x}(t))\dot{x}(t)-\tau\nabla_{t}\dot{x}(t)={\bf
0}$ (2)

over $\mathcal{X}$, where $R$ is the Riemannian curvature tensor associated
with the Levi-Civita connection $\nabla$, i.e.,
$R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$ for
vector fields $X,Y,Z$ on $M$.

We pose the problem of finding a critical point of the above functional in a
more general form as a multi-objective optimization problem:

$\mbox{(MOP)}\qquad\min_{x\in\mathcal{X}}\
\left[\int_{t_{0}}^{t_{f}}\langle\dot{x}(t),\dot{x}(t)\rangle\,dt\,,\ \
\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x}(t),\nabla_{t}\dot{x}(t)\rangle\,dt\right]\,.$

Let

$F_{1}(x):=\int_{t_{0}}^{t_{f}}\langle\dot{x}(t),\dot{x}(t)\rangle\,dt\qquad\mbox{and}\qquad
F_{2}(x):=\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x}(t),\nabla_{t}\dot{x}(t)\rangle\,dt\,.$

Define the vector of objective functionals, $F(x):=(F_{1}(x),F_{2}(x))$. The
point $x^{*}\in\cal{X}$ is said to be a Pareto minimum if there exists no
$x\in\cal{X}$ such that $F(x)\not=F(x^{*})$ and

$F_{i}(x)\leq F_{i}(x^{*})\quad\mbox{for }i=1,2\,.$

On the other hand, $x^{*}\in\cal{X}$ is said to be a weak Pareto minimum if
there exists no $x\in\cal{X}$ such that

$F_{i}(x)<F_{i}(x^{*})\quad\mbox{for }i=1,2\,.$

The set of all vectors of objective functional values at the weak Pareto
minima is said to be the Pareto front (or efficient front) of Problem (MOP) in
the 2-dimensional objective value space. Note that the coordinates of a point
in the Pareto front are simply $F_{i}(x^{*})$, $i=1,2$. The Pareto front is
usually a curve (connected or disconnected).

###  2.1 Scalarization

For computing a solution of the nonconvex multi-objective problem (MOP), we
will consider the following single-objective problem, referred to as
scalarization. While this scalarization is often considered for nonconvex
multi-objective finite-dimensional optimization problems [16], it has also
been considered for infinite-dimensional (optimal control) problems in [12,
13].

$\mbox{(MOP${}_{w}$)}\qquad\min_{x\in\cal{X}}\
\max\\{w_{1}\,F_{1}(x),\,w_{2}\,F_{2}(x)\\}\,,$

where $w_{1}$ and $w_{2}$ are referred to as weights. Problem (MOPw) is
referred to as the weighted Chebychev problem (or Chebychev
scalarization)444The more commonly used weighted-sum scalarization will be
considered below.. It is required that $w_{1},w_{2}\geq 0$. For brevity in
notation, we set $w_{1}=w$ and $w_{2}=1-w$ with $w\in[0,1]$.

The following theorem is a direct consequence of [11, Corollary 5.35]. It can
also be concluded by generalizing the proofs given in a finite-dimensional
setting in [16, Theorems 3.4.2 and 3.4.5] to the infinite-dimensional setting
in a straightforward manner.

###### Theorem 1

The point $x^{*}$ is a weak Pareto minimum of (MOP) if, and only if, $x^{*}$
is a solution of (MOPw) for some $w_{1},w_{2}>0$, $w_{1}+w_{2}=1$.

An ideal cost $F^{*}_{i}$, $i=1,2$, associated with Problem (MOPw) is the
optimal value of the optimal control problem,

$\mbox{(P${}_{i}$)}\qquad\inf_{x\in\cal{X}}\ F_{i}(x)\,.$

###### Remark 1

Since $F_{i}(x)$ for $x\in\mathcal{X}$ is bounded below, the problem $(P_{i})$
is well-defined. Note that for the optimization problem
$\min_{x\in\mathcal{X}}F_{1}(x)$ it is customary to specify two endpoints, but
problems ($P_{1}$) and ($P_{2}$) share the same set of inputs. For the
optimization problem $\min_{x\in\mathcal{X}}F_{2}(x)$, we usually specify two
endpoints and their associated end-velocities which usually determines a
Riemannian cubic curve. The difficulty is that $F_{1}$ usually does not have a
minimizer when the curves are restricted in this way, which is why we ask for
an infimum instead of a minimizer. $\Box$

Let $\overline{x}$ be a minimizer of the single-objective problem (Pi) above.
Then we also define $\overline{F}_{j}:=F_{j}(\overline{x})$, for $j\neq i$ and
$j=1,2$.

In general, it is useful to add a positive number to each objective in order
to obtain an even spread of the Pareto points approximating the Pareto
front—see, for example, [8], for further discussion and geometric
illustration. For this purpose, we define next the so-called utopian objective
values.

A utopian objective vector $\beta^{*}$ associated with Problem (MOPw) consists
of the components $\beta^{*}_{i}$ given as
$\beta^{*}_{i}=F_{i}^{*}-\varepsilon_{i}$ where $\varepsilon_{i}>0$ for
$i=1,2$. Problem (MOPw) can be equivalently written as

$\min_{x\in\cal{X}}\
\max\\{w\,(F_{1}(x)-\beta_{1}^{*}),\,(1-w)\,(F_{2}(x)-\beta_{2}^{*})\\}\,.$

We note that although the objective functionals in Problem (MOP) are
differentiable in their arguments, Problem (MOPw), or its equivalent above, is
still a non-smooth optimization problem, because of the $\max$ operator in the
objective. However, it is well-known that Problem (MOPw) can be re-formulated
in the following (smooth) form (cf. the formulation in [14]) by introducing a
new variable $\alpha\geq 0$.

$\mbox{(SP${}_{w}$) }\left\\{\begin{array}[]{rl}\displaystyle\min_{{\alpha\geq
0}\atop{x\in\cal{X}}}&\ \ \alpha\\\\[11.38109pt] \mbox{subject to}&\ \
w\,(F_{1}(x)-\beta_{1}^{*})\leq\alpha\,,\\\\[5.69054pt] &\ \
(1-w)\,(F_{2}(x)-\beta_{2}^{*})\leq\alpha\,.\end{array}\right.$

Problem (SPw) is referred to as goal attainment method [16], as well as
Pascoletti–Serafini scalarization [9]. We will solve Problem (SPw) for the two
examples we want to study.

Note that the popular weighted-sum scalarization, given below, fails to
generate the nonconvex parts of a Pareto front.

$\mbox{(MOPws)}\qquad\min_{x\in\cal{X}}\ (w\,F_{1}(x)+(1-w)\,F_{2}(x))\,.$

We will illustrate this with the torus problem in Section 4.2. We note that
the above weighted-sum scalarization is equivalent to the minimization of (1)
(or the Riemannian cubics in tension problem) with $\tau=w/(1-w)$,
$w\in[0,1)$.

###  2.2 Essential interval of weights

With the Chebyshev scalarization, it would usually be enough for the weight
$w$ to take values over a (smaller) subinterval $[w_{0},w_{f}]\subset[0,1]$,
with $w_{0}>0$ and $w_{f}<1$, for the generation of the whole front. Figure 1
illustrates the geometry to compute the subinterval end-points, $w_{0}$ and
$w_{f}$. In the illustration, the points $(F_{1}^{*},\overline{F}_{2})$ and
$(\overline{F}_{1},F_{2}^{*})$ represent the boundary of the Pareto front. The
equations of the rays which emanate from the utopia point
$(\beta_{1}^{*},\beta_{2}^{*})$ and pass through the boundary points are also
shown. By substituting the boundary values of the Pareto curve into the
respective equations, and solving each equation for $w_{0}$ and $w_{f}$ one
simply gets

$w_{0}=\frac{(F_{2}^{*}-\beta_{2}^{*})}{(\overline{F}_{1}-\beta_{1}^{*})+(F_{2}^{*}-\beta_{2}^{*})}\qquad\mbox{and}\qquad
w_{f}=\frac{(\overline{F}_{2}-\beta_{2}^{*})}{(F_{1}^{*}-\beta_{1}^{*})+(\overline{F}_{2}-\beta_{2}^{*})}\,.$
(3)

Figure 1: Determination of the essential subinterval of weights
$[w_{0},w_{f}]$ [12].

From the geometry depicted in Figure 1, as also discussed in [12], one can
deduce that with every $w\in[0,w_{0}]$ the solution of (MOPw) will yield the
same boundary point $(\overline{F}_{1},F_{2}^{*})$ on the Pareto front.
Likewise with every $w\in[w_{f},1]$ the same boundary point
$(F_{1}^{*},\overline{F}_{2})$ is generated. This observation justifies the
avoidance of the weights $w\in[0,w_{0})\cup(w_{f},1]$ in order not to keep
getting the boundary points of the Pareto front, as otherwise one would end up
wasting valuable computational effort and time.

##  3 Convexity Conditions of Pareto Fronts

The main purpose of this section is to present some necessary and sufficient
conditions for Pareto Fronts resulting from the problem (MOP) to be convex.
Throughout this paper, a Pareto Front is called convex when its epigraph is
convex in the standard sense. In detail, if the Pareto Front $PF$ can be
written as $F_{2}=PF(F_{1})$, then $PF$ is convex if and only if
$PF(sF_{1}+(1-s)\tilde{F}_{1})\leq sPF(F_{1})+(1-s)PF(\tilde{F}_{1})$ for any
$F_{1},\tilde{F}_{1}$ on the front and $s\in[0,1]$ such that
$sF_{1}+(1-s)\tilde{F}_{1}$ is also on the front.

Suppose the optimal curve of the problem (MOP) with respect to some
scalarization (such as weighted-sum, Chebychev) is $x=x(t,w)$, where $0<w<1$
is the weight parameter. Denote
$F_{1}(w):=\int_{t_{0}}^{t_{f}}\langle\dot{x}(t,w),\dot{x}(t,w)\rangle dt$,
$F_{2}(w):=\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x}(t,w),\nabla_{t}\dot{x}(t,w)\rangle
dt$ and their derivatives with respect to $w$ by $F_{1}^{\prime}$ and
$F_{2}^{\prime}$.

###### Theorem 2

Suppose the Pareto front $PF$ of the problem (MOP) with respect to some weight
$w$ can be denoted by $F_{2}(w)=PF(F_{1}(w))$ and $F_{1},F_{2},PF$ are all
twice-differentiable. Then, $PF$ is convex if and only if

$\displaystyle
F_{1}^{\prime}(F_{1}^{\prime}F_{2}^{\prime\prime}-F_{1}^{\prime\prime}F_{2}^{\prime})\geq
0.$ (4)

Proof. This proof can be seen from the fact that $PF$ is convex if and only if
$PF^{\prime\prime}\geq 0$. By straightforward calculations, we have

$\displaystyle F_{2}^{\prime}$ $\displaystyle=PF^{\prime}F_{1}^{\prime},$
$\displaystyle F_{2}^{\prime\prime}$
$\displaystyle=PF^{\prime\prime}(F_{1}^{\prime})^{2}+PF^{\prime}F_{1}^{\prime\prime}.$

Then substituting the first identity into the second one, we get the relation
(4). $\Box$

If $F_{1}^{\prime}\neq 0$, then we find

$\displaystyle
PF^{\prime\prime}=\frac{1}{(F_{1}^{\prime})^{2}}\left(F_{2}^{\prime\prime}-\frac{F_{2}^{\prime}}{F_{1}^{\prime}}F_{1}^{\prime\prime}\right).$

Now we consider the convexity conditions of Pareto Fronts generated by the
problem (MOP) with respect to the weighted-sum scalarization.

###### Theorem 3

The Pareto front PF of the problem (MOP) generated with respect to the
weighted-sum scalarization ($MOP_{ws}$) is convex if and only if

$\displaystyle\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x}(t,w),x^{\prime}(t,w)\rangle
dt\geq 0,$ (5)

where $x$ satisfies the Euler–Lagrange equation (2) with $\tau=\frac{w}{1-w}$
$(w\neq 1)$ and $x^{\prime}$ is the derivative with respect to the weight $w$.

Proof. By integration by parts, we have

$\displaystyle F_{1}^{\prime}$
$\displaystyle=2\int_{t_{0}}^{t_{f}}\langle\nabla_{x^{\prime}}\dot{x},\dot{x}\rangle
dt=-2\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x},x^{\prime}\rangle dt,$
$\displaystyle F_{2}^{\prime}$
$\displaystyle=2\int_{t_{0}}^{t_{f}}\langle\nabla_{x^{\prime}}\nabla_{t}\dot{x},\nabla_{t}\dot{x}\rangle
dt=2\int_{t_{0}}^{t_{f}}\langle\nabla_{t}^{2}x^{\prime}+R(x^{\prime},\dot{x})\dot{x},\nabla_{t}\dot{x}\rangle
dt$
$\displaystyle=2\int_{t_{0}}^{t_{f}}\langle\nabla_{t}^{3}\dot{x}+R(\nabla_{t}\dot{x},\dot{x})\dot{x},x^{\prime}\rangle
dt=2\tau\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x},x^{\prime}\rangle
dt=-\tau F_{1}^{\prime},$

where we have used the Euler–Lagrange equation (2) in the second last identity
above. Then, condition (4) implies

$\displaystyle
F_{1}^{\prime}(F_{1}^{\prime}F_{2}^{\prime\prime}-F_{1}^{\prime\prime}F_{2}^{\prime})=-\frac{(F_{1}^{\prime})^{3}}{(1-w)^{2}}\geq
0\implies F_{1}^{\prime}\leq
0\implies\int_{t_{0}}^{t_{f}}\langle\nabla_{t}\dot{x}(t,w),x^{\prime}(t,w)\rangle
dt\geq 0,$

which completes this proof. $\Box$

Frankly speaking, the condition (5) is not easy to verify in practice. Partly
because it is difficult to solve the the Euler–Lagrange equation (2) to get
$x(t,w)$ and partly because evaluating the derivative of $x(t,w)$ with respect
to $w$ and its associated integration brings a lot of difficulties. Now we
apply Theorem 3 to the case where the Riemannian manifold $M$ is the standard
Euclidean space, i.e., the Euler–Lagrange equation (2) can be solved
explicitly.

###### Remark 2

When $M$ is specified as the Euclidean space $\mathbb{E}^{n}$, the Pareto
front PF of the problem (MOP) generated with respect to the weighted-sum
scalarization ($MOP_{ws}$) is convex for the weight $w$ on some interval
$(w_{0},1)$, where $0<w_{0}<1$. Refer to Appendix A for more detailed
discussions.

The following example gives a visualisation of the Pareto front of the multi-
objective optimisation problem in the Euclidean space and verifies Theorem 3.

###### Example 1

Choosing $T=1$, $x_{0}=(0,0,0)^{T}$, $v_{0}=(1,0,0)^{T}$, $x_{T}=(2,2,1)^{T}$,
$v_{T}=(0,1,-1)^{T}$, we can get the analytical form of cubics in tension in
$\mathbb{E}^{3}$ (see expressions in Appendix A). Figure 2 shows the Pareto
front of the problem (MOP) with respect to the scalarization (MOPws) and plots
the convexity condition (5).

Figure 2: (Left) Pareto front (Right) Convexity condition

##  4 Numerical Experiments

For numerical experiments, we consider variational curves on two two-
dimensional manifolds, namely on sphere and torus. To construct an
approximation of the Pareto front of a given problem we implement Algorithm 1
in [12].

We employ the scalarize–discretize–then–optimize approach that was previously
used in [12]. Under this approach, one first scalarizes the multi-objective
problem in the infinite-dimensional space, and then discretizes the scalarized
problem directly (in time space) and applies a usually large-scale finite-
dimensional optimization method to find a discrete approximate solution of the
scalarized problem. By the existing theory of discretization (see for example
[1, 2, 5, 6, 7, 20]), under some general assumptions, the discrete approximate
solution converges to a solution of the continuous-time scalarization (SPw) of
the original problem, yielding a Pareto minimum of the original problem.

In the subproblems of the algorithm in [12], a direct discretization of
Problem (SPw) employing the trapezoidal rule is solved by using Knitro,
version 13.0.1. Knitro is a popular optimization software; see [4]. We use
AMPL [10] as an optimization modelling language, which employs Knitro as a
solver. We set the Knitro parameters feastol=1e-8 and opttol=1e-8. We also set
the number of discretization points to be 2,000 for the sphere and 5,000 for
the torus.

###  4.1 Sphere

Let
$\mathbb{S}^{2}:=\\{x=(x_{1},x_{2},x_{3})\in\mathbb{E}^{3}~{}|~{}\|x\|=1\\}$
be the unit-sphere endowed with the standard Euclidean metric/norm. With
respect to the induced metric, the covariant derivative $\nabla_{t}\dot{x}(t)$
on $\mathbb{S}^{2}$ can be written as

$\displaystyle\nabla_{t}\dot{x}(t)=\ddot{x}(t)-\langle\ddot{x}(t),x(t)\rangle
x(t)=\ddot{x}(t)+\langle\dot{x}(t),\dot{x}(t)\rangle x(t),$ (6)

where we have used the relation of the twice differentiation of the constraint
$\langle x(t),x(t)\rangle=1$. Then the squared norm of the covariant
acceleration on $\mathbb{S}^{2}$ is given by

$\displaystyle\|\nabla_{t}\dot{x}(t)\|^{2}=\|\ddot{x}(t)\|^{2}-\|\dot{x}(t)\|^{4}.$
(7)

Now the general multi-objective optimization problem (MOP) on $\mathbb{S}^{2}$
can be specified as follows,

$\mbox{(S) }\left\\{\begin{array}[]{rl}\displaystyle\min_{x(\cdot)}&\ \
\displaystyle\left[\int_{t_{0}}^{t_{f}}\|\dot{x}(t)\|^{2}\,dt\,,\ \
\int_{t_{0}}^{t_{f}}\left(\|\ddot{x}(t)\|^{2}-\|\dot{x}(t)\|^{4}\right)\,dt\right]\\\\[11.38109pt]
\mbox{subject to}&\ \ x_{1}(t)^{2}+x_{2}(t)^{2}+x_{3}(t)^{2}=1\,,\mbox{\ \ for
all }t\in[0,t_{f}]\,,\\\\[5.69054pt] &\ \ x(t_{0})=x_{0}\,\ \
\dot{x}(t_{0})=v_{0}\,,\ \ x(t_{f})=x_{f}\,\ \
\dot{x}(t_{f})=v_{f}\,.\end{array}\right.$

We have

$F_{1}(x):=\int_{t_{0}}^{t_{f}}\|\dot{x}(t)\|^{2}\,dt\qquad\mbox{and}\qquad
F_{2}(x):=\int_{t_{0}}^{t_{f}}\left(\|\ddot{x}(t)\|^{2}-\|\dot{x}(t)\|^{4}\right)\,dt\,.$

Then the smooth version of the Chebyshev scalarization of the sphere problem
(S) can simply be stated as

$\mbox{(S${}_{w}$)
}\left\\{\begin{array}[]{rl}\displaystyle\min_{x(\cdot),\,\alpha}&\ \
\alpha\\\\[11.38109pt] \mbox{subject to}&\ \
w\,F_{1}(x)\leq\alpha\,,\\\\[5.69054pt] &\ \
(1-w)\,F_{2}(x)\leq\alpha\,,\\\\[5.69054pt] &\ \
x_{1}(t)^{2}+x_{2}(t)^{2}+x_{3}(t)^{2}=1\,,\mbox{\ \ for all
}t\in[0,t_{f}]\,,\\\\[5.69054pt] &\ \ x(t_{0})=x_{0}\,\ \
\dot{x}(t_{0})=v_{0}\,,\ \ x(t_{f})=x_{f}\,\ \
\dot{x}(t_{f})=v_{f}\,,\end{array}\right.$

where $w\in[0,1)$ is fixed and we have taken $\beta_{1}^{*}=\beta_{2}^{*}=0$.

Figure 3: The Pareto front, the speed and acceleration on the sphere for a
Riemannian cubic in tension, $x:[0,1]\to\mathbb{S}^{2}$, from
$(x_{0},v_{0})=((1,0,0),(0,0.1,0))$ to $(x_{f},v_{f})=((-1,0,0),(0,0,-0.1))$.

In Figure 3, “speed on sphere” is simply $\|\dot{x}(t)\|$ and the
“acceleration on sphere” is $\sqrt{\|\ddot{x}(t)\|^{2}-\|\dot{x}(t)\|^{4}}$ .

In generating the Pareto solutions and an approximation of the Pareto front,
we have solved (Sw) in the subproblems of Algorithm 1 of [12], with $0\leq
w<1$.

The colour-coded Pareto solutions in Figure 3 are given as blue ($0\leq w\leq
0.9059$), green ($w=0.96$), magenta ($w=0.985$), cyan ($w=0.9940$), red ($w\to
1^{-}$). The meaningful interval of the $w$ values is $[0.9059,1)$. The
solution denoted by blue is nothing but a (single) Riemannian cubic with
$(F_{1},F_{2})=(11.76,113.2)$.

Numerical observations suggest that the solution represented by red is
asymptotically given by $(F_{1},F_{2})=(9.88,\infty)$, with $F_{1}$ correct to
three decimal places, as $w\to 1^{-}$. For practical (illustration) purposes
we depict $F_{2}<\infty$ in Figure 3; otherwise the Pareto curve extends to
infinity to the left.

The curves on the sphere appear as if they have different end-velocities,
which is not the case except for the red curve. The end-accelerations of the
solution coded in red seem to be impulsive, resulting in a jump in the end-
speeds. This can be observed as instantaneous rotation of the end-velocities
of the curves on the sphere.

The next example illustrates that not only the Pareto front can be
disconnected but also can contain two distinct Riemannian cubics.

###  4.2 Torus

Torus $\mathbb{T}^{2}$ is a geometric object described by

$\displaystyle\big{\\{}x=(x_{1},x_{2},x_{3})\in{\rm{I\ \kern-5.39993ptR}}^{3}\
$ $\displaystyle|\ $ $\displaystyle x_{1}=(c+a\cos v)\cos u,\ x_{2}=(c+a\cos
v)\sin u,\ x_{3}=a\sin v,$ $\displaystyle u,v\in[0,2\pi)\big{\\}}\,$

with the induced standard Euclidean metric, where the constants $a$ and $c$
are respectively the smaller and greater radii of the torus.

By straightforward calculations, we get the derivatives

$\displaystyle x_{u}=$ $\displaystyle(-(c+a\cos v)\sin u,(c+a\cos v)\cos
u,0),$ $\displaystyle x_{v}=$ $\displaystyle(-a\sin v\cos u,-a\sin v\sin
u,a\cos v).$

and the first fundamental form

$\displaystyle ds^{2}$ $\displaystyle=\langle x_{u},x_{u}\rangle
du^{2}+2\langle x_{u},x_{v}\rangle dudv+\langle x_{v},x_{v}\rangle dv^{2}$
$\displaystyle=(c+a\cos v)^{2}du^{2}+a^{2}dv^{2},$

which implies the Riemannian metric $g$ and its inverse $g^{-1}$ as follows,

$\displaystyle g=\begin{bmatrix}(c+a\cos v)^{2}&0\\\
0&a^{2}\end{bmatrix},~{}~{}g^{-1}=\begin{bmatrix}(c+a\cos v)^{-2}&0\\\
0&a^{-2}\end{bmatrix}.$

Further, the non-zero Christoffel symbols are given by

$\displaystyle\Gamma_{uu}^{v}$
$\displaystyle=-\frac{1}{2}g^{vv}\frac{g_{uu}}{\partial
v}=\left(\frac{c}{a}+\cos v\right)\sin v,$ $\displaystyle\Gamma_{uv}^{u}$
$\displaystyle=\Gamma_{vu}^{u}=\frac{1}{2}g^{uu}\frac{\partial
g_{uu}}{\partial v}=-\frac{a\sin v}{c+a\cos v}.$

Therefore, the covariant derivative
$\nabla_{t}\dot{x}(t)=\ddot{x}_{i}(t)+\Gamma_{jk}^{i}\dot{x}_{j}(t)\dot{x}_{k}(t)$
on the torus $\mathbb{T}^{2}$ can be denoted by

$\nabla_{t}\dot{x}(t)=(a_{1}(t),a_{2}(t))\,,$

where

$\displaystyle a_{1}$
$\displaystyle=\ddot{u}+\Gamma_{uv}^{u}\dot{u}\dot{v}+\Gamma_{vu}\dot{v}\dot{u}=\ddot{u}-\frac{2a\sin
v}{c+a\cos v}\,\dot{u}\dot{v},$ $\displaystyle a_{2}$
$\displaystyle=\ddot{v}+\Gamma_{uu}^{v}\dot{u}\dot{u}=\ddot{v}+\frac{1}{a}\sin
v(c+a\cos v)\,\dot{u}^{2}.$

The problem of interest is the simultaneous minimization of the objective
functionals

$F_{1}:=\int_{0}^{t_{f}}\left(\dot{x}_{1}^{2}(t)+\dot{x}_{3}^{2}(t)+\dot{x}_{3}^{2}(t)\right)dt\quad\mbox{and}\quad
F_{2}:=\int_{0}^{t_{f}}\left(a_{1}^{2}(t)+a_{2}^{2}(t)\right)dt\,.$

In other words, we are interested in solving the problem

$\mbox{(T) }\left\\{\begin{array}[]{rl}\displaystyle\min_{u(\cdot),v(\cdot)}&\
\ \displaystyle\left[F_{1}\,,\ F_{2}\right]\\\\[11.38109pt] \mbox{subject
to}&\ \ (u(0),v(0))=(u_{0},v_{0})\,,\ \
(u(t_{f}),v(t_{f}))=(u_{f},v_{f})\,,\\\\[2.84526pt] &\ \
(\dot{u}(0),\dot{v}(0))=(\dot{u}_{0},\dot{v}_{0})\,,\ \
(\dot{u}(t_{f}),\dot{v}(t_{f}))=(\dot{u}_{f},\dot{v}_{f})\,.\end{array}\right.$

Recall that a certain trade-off solution set of Problem (T) is referred to as
the Pareto set.

The Chebyshev scalarization of Problem (T) is given by

$\mbox{(T${}_{w}$)
}\left\\{\begin{array}[]{rl}\displaystyle\min_{u(\cdot),v(\cdot),\alpha}&\ \
\alpha\\\\[11.38109pt] \mbox{subject to}&\ \
w\,F_{1}(x)\leq\alpha\,,\\\\[5.69054pt] &\ \
(1-w)\,F_{2}(x)\leq\alpha\,,\\\\[5.69054pt] &\ \ (u(0),v(0))=(u_{0},v_{0})\,,\
\ (u(t_{f}),v(t_{f}))=(u_{f},v_{f})\,,\\\\[2.84526pt] &\ \
(\dot{u}(0),\dot{v}(0))=(\dot{u}_{0},\dot{v}_{0})\,,\ \
(\dot{u}(t_{f}),\dot{v}(t_{f}))=(\dot{u}_{f},\dot{v}_{f})\,.\end{array}\right.$

Take an example instance: Let $u_{0}=v_{0}=0$, $u_{f}=\pi$, $v_{f}=\pi/2$. Let
$\dot{u}_{0}=1$, $\dot{v}_{0}=0$, $\dot{u}_{f}=0$, $\dot{v}_{f}=-1$, with
$t_{f}=1$. See Figure 4 for the Pareto front constructed by using the
scalarization (Tw), after weeding out the dominated points found as local
minima of (Tw). The essential interval of the weights is found to be
$[0.593,1)$. The right-most (boundary) point in the front represents the
Riemannian cubic found with $w=w_{0}=0.5931$ as
$(F_{1},F_{2})=(170.143,248.049)$, indicated with the colour green. As can be
seen the Pareto front is disconnected with a large gap. Moreover, the right-
hand segment’s epigraph is nonconvex, prohibiting the use of the classical
weighted sum as scalarization.

Numerical observations suggest that the left-hand part of the Pareto front is
asymptotic to $(F_{1},F_{2})=(80.2,\infty)$, correct to three significant
figures, as $w\to 1^{-}$. The related curves are shown in the colour red.
Moreover, the Pareto solution $(F_{1},F_{2})=(95.15,457.2)$, found with
$w=0.8259$ and indicated with the colour blue, represents another Riemannian
cubic. To the best knowledge of the authors this is the first nontrivial
example of multiple Riemannian cubics with the same boundary data.

The remaining boundary point is indicated with the colour light blue, computed
as $(F_{1},F_{2})=(138.8,456.3)$ with $w=0.7667$.

Figure 4: Torus—Pareto front constructed via (Tw). Curves are obtained between
the oriented points $(u_{0},v_{0},\dot{u}_{0},\dot{v}_{0})=(0,0,1,0)$ and
$(u_{f},v_{f},\dot{u}_{f},\dot{v}_{f})=(\pi,\pi/2,0,-1)$.

The weighted-sum scalarization of Problem (T) (or the classical expression for
the Riemannian cubics in tension) is given by

$\mbox{(Tws)
}\left\\{\begin{array}[]{rl}\displaystyle\min_{u(\cdot),v(\cdot)}&\ \
w\,F_{1}+(1-w)F_{2}\\\\[11.38109pt] \mbox{subject to}&\ \
(u(0),v(0))=(u_{0},v_{0})\,,\ \
(u(t_{f}),v(t_{f}))=(u_{f},v_{f})\,,\\\\[2.84526pt] &\ \
(\dot{u}(0),\dot{v}(0))=(\dot{u}_{0},\dot{v}_{0})\,,\ \
(\dot{u}(t_{f}),\dot{v}(t_{f}))=(\dot{u}_{f},\dot{v}_{f})\,.\end{array}\right.$

See Figure 5 for the Pareto front constructed by using the scalarization
(Tws). When compared with the Pareto front in Figure 4 the classical
expression for the Riemannian cubics in tension clearly misses an important
segment of compromise/Pareto solutions.

We note that the Euler–Lagrange equations as optimality conditions of Problem
(Tws) is provided in Appendix B.

Figure 5: Torus—Pareto front constructed via (Tws).

##  5 Conclusion

Measurements from variational curves, for example, kinetic energy and squared
norm of higher-order derivatives, are seldom minimized at the same time in the
literature. Motivated by the problem of finding Riemannian cubics in tension,
we have formulated a multi-objective optimisation problem, that is, we have
posed the minimization of the vector of total kinetic energy and squared norm
of acceleration of a curve on a Riemannian manifold. To obtain a sequence of
single-objective optimization problems, we adopted the Chebychev and weighted-
sum scalarizations in this paper. Some numerical experiments presented on a
sphere and a torus illustrated disconnected Pareto fronts and non-uniqueness
of Riemannian cubics. To the authors’ best knowledge, these are non-trivial
observations that may bring the optimization and differential geometry
communities’ attention. Furthermore, we derived some convexity conditions for
the Pareto fronts on general Riemannian manifolds.

Future research may include more theoretical studies on the phenomenon of non-
unique Riemannian cubics on a torus and the disconnected Pareto fronts that
they belong to. We have investigated the local optimality of the two
Riemannian cubics by utilizing the test looking for bounded solutions of
certain Riccati equations presented in [15]. However, it was not possible for
us to obtain bounded Riccati solutions deeming the test inconclusive. Further
investigation in search for bounded Riccati solutions might also be a
direction interesting to pursue.

Data Availability
The full resolution Matlab graph/plot files that support the findings of this
study are available from the corresponding author upon request.

Conflict of Interest
The authors have no competing, or conflict of, interests to declare that are
relevant to the content of this article.

##  Appendix A A Discussion on Remark 2

Cubics in tension are solutions of the following Euler–Lagrange equation

$\displaystyle\begin{cases}x^{(4)}(t)-\tau\ddot{x}(t)=0,\\\
x(0)=x_{0},\dot{x}(0)=v_{0},x(T)=x_{T},\dot{x}(T)=v_{T},\end{cases}$

which can be analytically written as

$\displaystyle
x=x_{0}+v_{0}t+(\sinh(\sqrt{\tau}t)-\sqrt{\tau}t)c_{1}+(\cosh(\sqrt{\tau}t)-1)c_{2},$

where $c_{1}$ and $c_{2}$ satisfy

$\displaystyle\begin{pmatrix}c_{1}\\\
c_{2}\end{pmatrix}=\frac{1}{\Delta}\begin{pmatrix}\sqrt{\tau}T\sinh(\sqrt{\tau}T)&1-\cosh(\sqrt{\tau}T)\\\
\sqrt{\tau}T(1-\cosh(\sqrt{\tau}T))&\sinh(\sqrt{\tau}T)-\sqrt{\tau}T\end{pmatrix}\begin{pmatrix}x_{T}-x_{0}-v_{0}T\\\
(v_{T}-v_{0})T\end{pmatrix}$

with
$\Delta=\sqrt{\tau}T(2\cosh(\sqrt{\tau}T)-2-\sqrt{\tau}T\sinh(\sqrt{\tau}T))$.

To simplify notations, we denote by $c=(c_{1},c_{2})^{T}$,
$A=\begin{pmatrix}\sinh(\tau_{T})-\tau_{T}&\cosh(\tau_{T})-1\\\
\tau_{T}(\cosh(\tau_{T})-1)&\tau_{T}\sinh(\tau_{T})\end{pmatrix}$,
$b=(x_{T}-x_{0}-v_{0}T,(v_{T}-v_{0})T)^{T}$, $\tau_{T}=\sqrt{\tau}T$,
$\tau_{t}=\sqrt{\tau}t$. Then, $Ac=b$, which implies
$c^{\prime}=-A^{-1}A^{\prime}c$.

Straightforward calculations yield

$\displaystyle\ddot{x}$
$\displaystyle=\tau(\sinh(\tau_{t}),\cosh(\tau_{t}))c,$ $\displaystyle
x^{\prime}$
$\displaystyle=\frac{\tau^{\prime}t}{2\sqrt{\tau}}(\cosh(\tau_{t})-1,\sinh(\tau_{t}))c-(\sinh(\tau_{t})-\tau_{t},\cosh(\tau_{t})-1)A^{-1}A^{\prime}c.$

Note that

$\displaystyle\int_{0}^{T}\tau_{t}\begin{pmatrix}\sinh(\tau_{t})\\\
\cosh(\tau_{t})\end{pmatrix}(\cosh(\tau_{t})-1,\sinh(\tau_{t}))dt$
$\displaystyle=\frac{1}{\sqrt{\tau}}\begin{pmatrix}\frac{\tau_{T}}{4}\cosh(2\tau_{T})-\frac{\sinh(2\tau_{T})}{8}-\tau_{T}\cosh(\tau_{T})+\sinh(\tau_{T})&\frac{\tau_{T}}{4}\sinh(2\tau_{T})-\frac{\sinh^{2}(\tau_{T})}{4}-\frac{\tau_{T}^{2}}{4}\\\
-\frac{7}{8}+\frac{\tau_{T}^{2}}{4}+\frac{\tau_{T}}{4}\sinh(2\tau_{T})-\tau_{T}\sinh(\tau_{T})-\frac{\cosh(2\tau_{T})}{8}+\cosh(\tau_{T})&\frac{\tau_{T}}{4}\cosh(2\tau_{T})-\frac{\sinh(2\tau_{T})}{8}\end{pmatrix}$
$\displaystyle=:\frac{B_{1}}{\sqrt{\tau}},$
$\displaystyle\int_{0}^{T}\begin{pmatrix}\sinh(\tau_{t})\\\
\cosh(\tau_{t})\end{pmatrix}(\sinh(\tau_{t})-\tau_{t},\cosh(\tau_{t})-1)dt$
$\displaystyle=\frac{1}{\sqrt{\tau}}\begin{pmatrix}\frac{\sinh(2\tau_{T})}{4}-\frac{\tau_{T}}{2}-\tau_{T}\cosh(\tau_{T})+\sinh(\tau_{T})&\frac{(\cosh(\tau_{T})-1)^{2}}{2}\\\
\frac{\sinh^{2}(\tau_{T})}{2}-\tau_{T}\sinh(\tau_{T})+\cosh(\tau_{T})-1&\frac{\tau_{T}}{2}+\frac{\sinh(2\tau_{T})}{4}-\sinh(\tau_{T})\end{pmatrix}$
$\displaystyle=:\frac{B_{2}}{\sqrt{\tau}}.$

Then, the integration $\int_{0}^{T}\langle\ddot{x},x^{\prime}\rangle dt$
becomes

$\displaystyle\frac{\tau^{\prime}}{2\sqrt{\tau}}c^{T}\left(B_{1}-\frac{2\tau}{\tau^{\prime}}B_{2}A^{-1}A^{\prime}\right)c,$

where $\tau^{\prime}=\frac{1}{(1-w)^{2}}>0$ for $w\neq 1$ and

$\displaystyle
B_{1}-\frac{2\tau}{\tau^{\prime}}B_{2}A^{-1}A^{\prime}=:B_{3}=:\begin{pmatrix}b_{11}&b_{12}\\\
b_{21}&b_{22}\end{pmatrix},$ $\displaystyle b_{11}=$
$\displaystyle-\tau_{T}\cosh(\tau_{T})+\frac{\tau_{T}}{4}\cosh(2\tau_{T})+\sinh(\tau_{T})-\frac{1}{8}\sinh(2\tau_{T})+$
$\displaystyle\frac{\tau_{T}}{2\tilde{\Delta}}(\cosh(\tau_{T})-1)\sinh(\tau_{T})(\tau_{T}-2\sinh(\tau_{T})-\cosh(\tau_{T})(\sinh(\tau_{T})-2\tau_{T})),$
$\displaystyle b_{12}=$
$\displaystyle-\frac{\tau_{T}^{2}}{4}-\frac{\sinh^{2}(\tau_{T})}{4}+\frac{\tau_{T}\sinh(2\tau_{T})}{4}+\frac{1}{2\tilde{\Delta}}(\cosh(\tau_{T})-1)^{3}\sinh(\tau_{T}),$
$\displaystyle b_{21}=$
$\displaystyle-\frac{7}{8}+\frac{\tau_{T}^{2}}{4}+\cosh(\tau_{T})-\frac{\cosh(2\tau_{T})}{8}-\tau_{T}\sinh(\tau_{T})+\frac{\tau_{T}}{4}\sinh(2\tau_{T})-$
$\displaystyle\frac{\tau_{T}}{2\tilde{\Delta}}(\cosh(\tau_{T})-1+\tau_{T}\sinh(\tau_{T}))(2-2\cosh(\tau_{T})+2\tau_{T}\sinh(\tau_{T})-\sinh^{2}(\tau_{T})),$
$\displaystyle b_{22}=$
$\displaystyle\frac{\tau_{T}}{4}\cosh(2\tau_{T})-\frac{\sinh(2\tau_{T})}{8}-$
$\displaystyle\frac{1}{4\tilde{\Delta}}(\sinh(\tau_{T})-\tau_{T})(\sinh(\tau_{T})+\tau_{T}\cosh(\tau_{T}))(2\tau_{T}-4\sinh(\tau_{T})+\sinh(2\tau_{T})),$
$\displaystyle\tilde{\Delta}=$ $\displaystyle\Delta/\tau_{T}.$

It is a bit messy to write down the closed forms of the eigenvalues of
$B_{3}$, instead, we plot the eigenvalues for straightforward visualization.
In Figure 6, we choose $T=1$, from which we can see that eigenvalues of
$B_{3}$ are positive at least on $(0.2,1)$. Therefore, the convexity condition
5 is verified at least on $(0.2,1)$.

Figure 6: Eigenvalues of $B_{3}$ (Left: smaller, Right: larger)

##  Appendix B B Riemannian cubics in tension on torus

In this section, we consider Riemannian cubics in tension on torus using
variational methods. To short notations, we denote by $p:=\frac{2a\sin
v}{c+a\cos v}$, $q:=\sin v\left(\frac{c}{a}+\cos v\right)$, $r:=c+a\cos v$ and
their derivatives with respect to $v$ by upper notation prime. Then, we have

$\displaystyle F=$
$\displaystyle\int_{0}^{T}\|\nabla_{t}\dot{x}\|^{2}+\lambda\|\dot{x}\|^{2}dt$
$\displaystyle=$
$\displaystyle\int_{0}^{T}(\ddot{u}-p\dot{u}\dot{v})^{2}+(\ddot{v}+q\dot{u}^{2})^{2}+\lambda(r^{2}\dot{u}^{2}+a^{2}\dot{v}^{2})dt.$

Considering the variation of $F$, we find

$\displaystyle\delta F=$ $\displaystyle
2\int_{0}^{T}(\ddot{u}-p\dot{u}\dot{v})(\delta\ddot{u}-p^{\prime}\delta
v\dot{u}\dot{v}-p\delta\dot{u}\dot{v}-p\dot{u}\delta\dot{v})+(\ddot{v}+q\dot{u}^{2})(\delta\ddot{v}+q^{\prime}\delta
v\dot{u}^{2}+2q\dot{u}\delta\dot{u})+$ $\displaystyle\lambda(rr^{\prime}\delta
v\dot{u}^{2}+r^{2}\dot{u}\delta\dot{u}+a^{2}\dot{v}\delta\dot{v})dt$
$\displaystyle=$ $\displaystyle
2\int_{0}^{T}\left(\frac{d^{2}}{dt^{2}}\left(\ddot{u}-p\dot{u}\dot{v}\right)+\frac{d}{dt}\left(\left(\ddot{u}-p\dot{u}\dot{v}\right)p\dot{v}\right)-\frac{d}{dt}\left(\left(\ddot{v}+q\dot{u}^{2}\right)2q\dot{u}\right)-\frac{d}{dt}\left(\lambda
r^{2}\dot{u}\right)\right)\delta u+$
$\displaystyle\left(\frac{d^{2}}{dt^{2}}\left(\ddot{v}+q\dot{u}^{2}\right)+p\frac{d}{dt}\left((\ddot{u}-p\dot{u}\dot{v})\dot{u}\right)+(\ddot{v}+q\dot{u}^{2})q^{\prime}\dot{u}^{2}+\lambda
rr^{\prime}\dot{u}^{2}-\frac{d}{dt}\left(\lambda
a^{2}\dot{v}\right)\right)\delta vdt,$

which implies the following Euler–Lagrange equations,

$\displaystyle u^{(4)}$
$\displaystyle-(p^{2}+p^{\prime})\ddot{u}\dot{v}^{2}-(p+2q)\ddot{u}\ddot{v}-(p^{\prime\prime}+2pp^{\prime})\dot{u}\dot{v}^{3}-(2p^{2}+3p^{\prime}+2q^{\prime})\dot{u}\dot{v}\ddot{v}$
$\displaystyle-(p+2q)\dot{u}\dddot{v}-4qq^{\prime}\dot{u}^{3}\dot{v}-6q^{2}\dot{u}^{2}\ddot{u}-2\lambda
rr^{\prime}\dot{u}\dot{v}-\lambda r^{2}\ddot{u}=0,$ $\displaystyle v^{(4)}$
$\displaystyle+(q^{\prime\prime}-pp^{\prime})\dot{u}^{2}\dot{v}^{2}+(2q^{\prime}-p^{2})(2\dot{u}\ddot{u}\dot{v}-\dot{u}^{2}\ddot{v})+(p+2q)(\ddot{u}^{2}+\dot{u}\dddot{u})$
$\displaystyle+qq^{\prime}\dot{u}^{4}+\lambda rr^{\prime}\dot{u}^{2}-\lambda
a^{2}\ddot{v}=0.$

Introducing the variables $u_{1}:=u$, $u_{2}:=\dot{u}$, $u_{3}:=\ddot{u}$,
$u_{4}:=\dddot{u}$, $v_{1}:=v$, $v_{2}:=\dot{v}$, $v_{3}:=\ddot{v}$,
$v_{4}:=\dddot{v}$, we have the following first-order ODE,

$\displaystyle\begin{cases}\dot{u}_{1}=&u_{2},\\\ \dot{u}_{2}=&u_{3},\\\
\dot{u}_{3}=&u_{4},\\\
\dot{u}_{4}=&(p^{2}+p^{\prime})u_{3}v_{2}^{2}+(p+2q)u_{3}v_{3}+(p^{\prime\prime}+2pp^{\prime})u_{2}v_{2}^{3}+(2p^{2}+3p^{\prime}+2q^{\prime})u_{2}v_{2}v_{3}\\\
&+(p+2q)u_{2}v_{4}+4qq^{\prime}u_{2}^{3}v_{2}+6q^{2}u_{2}^{2}u_{3}-2\lambda
a^{2}qu_{2}v_{2}+\lambda r^{2}u_{3},\\\ \dot{v}_{1}=&v_{2},\\\
\dot{v}_{2}=&v_{3},\\\ \dot{v}_{3}=&v_{4},\\\
\dot{v}_{4}=&(pp^{\prime}-q^{\prime\prime})u_{2}^{2}v_{2}^{2}+(p^{2}-2q^{\prime})(2u_{2}u_{3}v_{2}-u_{2}^{2}v_{3})-(p+2q)(u_{3}^{2}+u_{2}u_{4})\\\
&-qq^{\prime}u_{2}^{4}-\lambda rr^{\prime}u_{2}^{2}+\lambda
a^{2}v_{3}.\end{cases}$ (8)

Therefore, in order to solve the problem (Tws), we only have to solve the
above ODE system.

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# Radflow: Decomposable Time Series Modeling with Network Information

Alasdair Tran1,2, Alexander Mathews1, Cheng Soon Ong1,2, Lexing Xie1 1
Australian National University 2 Data61, CSIRO
<EMAIL_ADDRESS>
(2021)

# Radflow: A Recurrent, Aggregated, and Decomposable Model for Networks of
Time Series

Alasdair Tran1,2, Alexander Mathews1, Cheng Soon Ong1,2, Lexing Xie1 1
Australian National University 2 Data61, CSIRO
<EMAIL_ADDRESS>
(2021)

###### Abstract.

ACM Reference Format:

Alasdair Tran, Alexander Mathews, Cheng Soon Ong, Lexing Xie. 2021. Radflow: A
Recurrent, Aggregated, and Decomposable Model for Networks of Time Series. In
Proceedings of the Web Conference 2021 (WWW ’21), April 19–23, 2021,
Ljubljana, Slovenia. ACM, New York, NY, USA, 12 pages.
https://doi.org/10.1145/3442381.3449945

time series, networks, graphs, sequence models, wikipedia

††copyright: acmcopyright††journalyear: 2021††copyright: iw3c2w3††conference:
Proceedings of the Web Conference 2021; April 19–23, 2021; Ljubljana,
Slovenia††booktitle: Proceedings of the Web Conference 2021 (WWW ’21), April
19–23, 2021, Ljubljana, Slovenia††doi: 10.1145/3442381.3449945††isbn:
978-1-4503-8312-7/21/04