Multi-objective Variational Curves

Abstract

Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared $L^2$ 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.

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 [19, 20, 23]. 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 [10, 18].

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 [9, 10, 18].

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 [15]. In [13, 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 [13, 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 Chebyshev 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 Sect. 2, we introduce the multi-objective optimization problem motivated by cubics in tension. Then we sketch the scalarization techniques, namely the Chebyshev and weighted-sum scalarizations. In Sect. 3, we present some necessary and sufficient conditions for the epigraph of Pareto fronts to be convex on general Riemannian manifolds. Numerical experiments in Sect. 4 illustrate the Pareto fronts and Pareto curves on a sphere and a torus, endowed with the restriction of the standard Euclidean metric from ${\mathbb {R}}^3$. Finally, we conclude the paper in Sect. 5 and present some future directions.

2 Problem Statement and Preliminaries

In this section, we start with some basic definitions and notations in differential geometry. The readers who are familiar with differential geometry, should feel free to skip the first subsection below. Then, we formulate the problem of finding Riemannian cubics in tension as a multi-objective optimization problem. Following this, scalarization methods for a multi-objective optimization problem, namely the weighted-sum and Chebyshev scalarization techniques, will be introduced.

2.1 Basics in differential geometry

Differential geometry investigates the geometric properties of smooth curved spaces, known as Riemannian manifolds, which plays a crucial role in understanding the intrinsic geometry of curves, surfaces and higher-dimensional spaces. Classical textbooks on differential geometry may include [5, 16, 21].

A manifold is a topological space that locally looks like the Euclidean space. This space is also equipped with a smooth structure, which means it has a collection of coordinate charts that smoothly transition from one to another. A Riemannian manifold is a manifold equipped with a Riemannian metric, which is smoothly varying inner product defined on the tangent space at each point of this manifold. This metric allows the measurement of distances, angles, and other geometric properties.

Let M be a Riemannian manifold, $T_xM$ the tangent space of the manifold M at point $x\in M$, and TM the tangent bundle defined as $TM:=\{(x,v)\vert x\in M,v\in T_xM\}$. A Riemannian metric on M is a bilinear and positive-definite inner product $\langle \cdot ,\cdot \rangle _x:T_xM\times T_xM\rightarrow {\mathbb {R}}$. A connection on a manifold allows one to differentiate vector fields along smooth curves on the manifold. The Levi–Civita connection is a specific type of connection that is compatible with a given Riemannian metric. With respect to the Riemannian metric $\langle \cdot ,\cdot \rangle$, the uniquely determined Levi–Civita connection is given by the Koszul formula

$$\begin{aligned} 2\langle \nabla _XY,Z\rangle&= X\langle Y,Z\rangle +Y\langle Z,X\rangle -Z\langle X,Y\rangle +\langle [X,Y],Z\rangle \\&\quad -\langle [Y,Z],X\rangle -\langle [X,Z],Y\rangle , \end{aligned}$$

where [X, Y] is the Lie bracket of X and Y, X, Y and Z are vector fields on the manifold M.

Choose local coordinates $\{x^i\}_{i=1}^n$ with coordinate basis vector fields $\partial _i:=\frac{\partial }{\partial x^i}$, the Christoffel symbols $\Gamma _{ij}^k$ of $\nabla$ with respect to these coordinates are defined as $\nabla _{\partial _i}\partial _j=\Gamma _{ij}^k\partial _k$. The Koszul expression of the Levi–Civita connection above is equivalent to a definition of the Christoffel symbols in terms of the metric $g:=\langle \cdot ,\cdot \rangle$ as

$$\begin{aligned} \Gamma _{ij}^k=\frac{1}{2}g^{kl}(\partial _j g_{li}+\partial _i g_{lj}-\partial _l g_{ij}), \end{aligned}$$

where g with upper indexes are the coefficients of the dual metric tensor, i.e., the entries of the inverse of the matrix g, g with lower indexes are the entries of the matrix g, and we have used the Einstein summation convention.

The Riemannian curvature tensor provides a way to quantify the curvature of a space at each point, which is crucial in understanding the geometry of curved spaces. For vector fields X, Y, Z on M, the Riemannian curvature tensor associated with the Levi–Civita connection $\nabla$ is defined as $R(X,Y)Z=\nabla _X\nabla _YZ-\nabla _Y\nabla _XZ-\nabla _{[X,Y]}Z$.

2.2 Problem statement

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

$$\begin{aligned} \int _{t_0}^{t_f} \bigg (\tau \,\langle {\dot{x}}(t),{\dot{x}}(t)\rangle + \langle \nabla _t{\dot{x}}(t),\nabla _t{\dot{x}}(t)\rangle \bigg )\,dt, \end{aligned}$$

(1)

where $x\in \mathcal {X}$, $\tau >0$ is a constant. (When $\tau = 0$ the critical point is called a Riemannian cubic.) Equivalently, Riemannian cubics in tension are solutions of the following Euler–Lagrange equation [19, 20, 23]

$$\begin{aligned} \nabla _t^3{\dot{x}}(t)+R(\nabla _t{\dot{x}}(t),{\dot{x}}(t)){\dot{x}}(t)-\tau \nabla _t{\dot{x}}(t)=\textbf{0} \end{aligned}$$

(2)

over $\mathcal {X}$, where R is the Riemannian curvature tensor associated with the Levi–Civita connection $\nabla$.

We now consider the following related multi-objective optimization problem:

$$\begin{aligned} \text{(MOP) }\qquad \min _{x\in \mathcal {X}}\ \left[ \int _{t_0}^{t_f} \langle {\dot{x}}(t),{\dot{x}}(t)\rangle \,dt,\ \ \int _{t_0}^{t_f} \langle \nabla _t{\dot{x}}(t),\nabla _t{\dot{x}}(t)\rangle \,dt\right] . \end{aligned}$$

Let

$$\begin{aligned} F_1(x):= \int _{t_0}^{t_f} \langle {\dot{x}}(t),{\dot{x}}(t)\rangle \,dt \qquad \text{ and }\qquad F_2(x):= \int _{t_0}^{t_f} \langle \nabla _t{\dot{x}}(t),\nabla _t{\dot{x}}(t)\rangle \,dt. \end{aligned}$$

Define the vector of objective functionals, $F(x):= (F_1(x),F_2(x))$. The point $x^*\in {\mathcal {X}}$ is said to be a Pareto minimum if there exists no $x\in {\mathcal {X}}$ such that $F(x) \not = F(x^*)$ and

$$\begin{aligned} F_i(x) \le F_i(x^*)\quad \text{ for } i=1,2. \end{aligned}$$

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

$$\begin{aligned} F_i(x) < F_i(x^*)\quad \text{ for } i=1,2. \end{aligned}$$

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 of the form $(F_1(x^*),F_2(x^*))$. The Pareto front is usually a (connected or disconnected) curve for the bi-objective case.

2.3 Scalarization

For computing a solution of the nonconvex multi-objective problem (MOP), it is common practice to solve a single-objective optimization problem, in particular the popular weighted-sum scalarization, posed as

$$\begin{aligned} \text{(MOPws) }\qquad \min _{x\in {\mathcal {X}}}\ (w\,F_1(x) + (1-w)\,F_2(x)), \end{aligned}$$

where w is some weight chosen in the interval (0, 1). We note that the above weighted-sum scalarization can be interpreted as the minimization of (1) (or the Riemannian cubics in tension problem) with $\tau = w/(1-w)$, $w\in (0,1)$, although in (1) the combination of the two objective functionals is not a convex combination.

One common approach in multi-objective optimization is that (MOPws) is solved with the range of values of w coming from a partition of (0, 1) in order to obtain an approximation of the Pareto front. When the epigraph of the Pareto front is convex (i.e. the “Pareto curve” is convex) the weighted-sum scalarization can be used to construct an approximation of the Pareto front. However it is also well known that this approach fails to generate the nonconvex parts of a Pareto front [18]. We will illustrate this with the torus problem in Sect. 4.2.

To generate an approximation of the Pareto front we will instead use what is referred to as the weighted Chebyshev problem, or Chebyshev scalarization:

$$\begin{aligned} \text{(MOP }_w\text{) }\qquad \min _{x\in {\mathcal {X}}}\ \max \{ w\,F_1(x),\, (1-w)\,F_2(x)\}, \end{aligned}$$

with the weight $w\in (0,1)$. Solutions in the cases $w=0$ and $w=1$ are Riemannian cubics and geodesics, respectively. While this scalarization is often considered for nonconvex multi-objective finite-dimensional optimization problems [18], it has also been considered for infinite-dimensional (optimal control) problems in [13, 14].

The following theorem is a direct consequence of [12, Corollary 5.35]. It can also be concluded by generalizing the proofs given in a finite-dimensional setting in [18, 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 (MOP$_w$) for some $w\in (0,1)$.

An ideal cost $F^*_i$, $i=1,2$, associated with Problem (MOP$_w$) is the optimal value of the optimal control problem,

$$\begin{aligned} \text{(P }_i\text{) }\qquad \inf _{x\in {\mathcal {X}}}\ F_i(x). \end{aligned}$$

Remark 1

Since $F_i(x)$ for $x\in \mathcal {X}$ is bounded below, 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. $\square$

Let ${\overline{x}}$ be a minimizer of the single-objective problem (P$_i$) above. Then we also define ${\overline{F}}_j:= F_j({\overline{x}})$, for $j\ne i$ 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, [9], 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 (MOP$_w$) consists of the components $\beta ^*_i$ given as $\beta ^*_i = F_i^* - \varepsilon _i$ where $\varepsilon _i > 0$ for $i = 1, 2$. Problem (MOP$_w$) can be equivalently written as

$$\begin{aligned} \min _{x\in {\mathcal {X}}}\ \max \{w\,(F_1(x) - \beta _1^*),\, (1-w)\,(F_2(x) - \beta _2^*)\}. \end{aligned}$$

We note that although the objective functionals in Problem (MOP) are differentiable in their arguments, Problem (MOP$_w$), 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 (MOP$_w$) can be re-formulated in the following (smooth) form (cf. the formulation in [15]) by introducing a new variable $\alpha \ge 0$.

$$\begin{aligned} \text{(SP }_w\text{) } \left\{ \begin{array}{rl} \displaystyle \min _{{\begin{array}{c} {\alpha \ge 0} \\ {x\in {\mathcal {X}}} \end{array}}} &{} \ \ \alpha \\ \text{ subject } \text{ to } &{} \ \ w\,(F_1(x) - \beta _1^*)\le \alpha , \\ {} &{} \ \ (1-w)\,(F_2(x) - \beta _2^*) \le \alpha . \end{array} \right. \end{aligned}$$

Problem (SP$_w$) is referred to as goal attainment method [18], as well as Pascoletti–Serafini scalarization [10]. We will solve Problem (SP$_w$) for the two examples we want to study.

2.4 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

$$\begin{aligned} w_0 = \frac{(F_2^* - \beta _2^*)}{({\overline{F}}_1 - \beta _1^*) + (F_2^* - \beta _2^*)} \qquad \text{ and }\qquad w_f = \frac{({\overline{F}}_2 - \beta _2^*)}{(F_1^* - \beta _1^*) + ({\overline{F}}_2 - \beta _2^*)}. \end{aligned}$$

Fig. 1

Determination of the essential subinterval of weights $[w_0,w_f]$ [13]

From the geometry depicted in Fig. 1, as also discussed in [13], one can deduce that with every $w\in [0,w_0]$ the solution of (MOP$_w$) 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 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 $F_2=\Phi (F_1)$, then PF is convex if and only if $\Phi (sF_1+(1-s){\tilde{F}}_1)\le s\Phi (F_1)+(1-s)\Phi ({\tilde{F}}_1)$ 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.

Theorem 2

Suppose the Pareto front PF of Problem (MOP) with respect to some weight w can be denoted by $F_2(w)=\Phi (F_1(w))$ and $F_1,F_2,\Phi$ are all twice-differentiable. Then, PF is convex if and only if

$$\begin{aligned} F_1^\prime (F_1^\prime F_2^{\prime \prime }-F_1^{\prime \prime }F_2^{\prime })\ge 0\,. \end{aligned}$$

(3)

Proof

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

$$\begin{aligned} F_2^\prime&=\Phi ^\prime F_1^\prime ,\\ F_2^{\prime \prime }&=\Phi ^{\prime \prime } (F_1^\prime )^2+\Phi ^\prime F_1^{\prime \prime }\\ \implies&F_2^{\prime \prime }(F_1^\prime )^2 ={ \Phi }^{\prime \prime } (F_1^\prime )^4+{ \Phi }^\prime (F_1^\prime )^2F_1^ {\prime \prime } = \Phi ^{\prime \prime } (F_1^\prime )^4+F_2^\prime F_1^\prime F_1^ {\prime \prime } \\ \implies&\Phi ^{\prime \prime } (F_1^\prime )^4=F_1^\prime (F_1^\prime F_2^{\prime \prime }-F_1^{\prime \prime }F_2^{\prime })\ge 0. \end{aligned}$$

The proof is completed. $\square$

If $F_1^\prime \ne 0$, then we find

$$\begin{aligned} \Phi ^{\prime \prime }= \frac{1}{(F_1^{\prime })^2}\left( F_2^{\prime \prime }-\frac{F_2^\prime }{F_1^\prime }F_1^{\prime \prime }\right) . \end{aligned}$$

Suppose the optimal curve of Problem (MOP) with respect to some scalarization (such as weighted-sum, Chebyshev) 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 $F_2^\prime$.

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

Theorem 3

The Pareto front PF of Problem (MOP) generated with respect to the weighted-sum scalarization (MOPws) is convex if and only if

$$\begin{aligned} \int _{t_0}^{t_f}\langle \nabla _t{\dot{x}}(t,w),x^\prime (t,w)\rangle dt\ge 0, \end{aligned}$$

(4)

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

Proof

By integration by parts, we have

$$\begin{aligned} F_1^\prime&= 2\int _{t_0}^{t_f}\langle \nabla _{x^\prime }{\dot{x}},{\dot{x}}\rangle dt=-2\int _{t_0}^{t_f}\langle \nabla _{t}{\dot{x}},x^\prime \rangle dt,\\ F_2^\prime&= 2\int _{t_0}^{t_f}\langle \nabla _{x^\prime }\nabla _t {\dot{x}},\nabla _t{\dot{x}}\rangle dt=2\int _{t_0}^{t_f}\langle \nabla _t^2 x^\prime +R(x^\prime ,{\dot{x}}){\dot{x}},\nabla _t{\dot{x}}\rangle dt\\&= 2\int _{t_0}^{t_f}\langle \nabla _t^3{\dot{x}}+R(\nabla _t{\dot{x}},{\dot{x}}){\dot{x}},x^\prime \rangle dt = 2\tau \int _{t_0}^{t_f}\langle \nabla _t{\dot{x}},x^\prime \rangle dt=-\tau F_1^\prime , \end{aligned}$$

where $\tau =w/(1-w)$, we have used the Euler–Lagrange equation (2) in the second last identity above. Then, condition (3) implies

$$\begin{aligned} F_1^\prime (F_1^\prime F_2^{\prime \prime }-F_1^{\prime \prime } F_2^\prime )&=-\frac{(F_1^\prime )^3}{(1-w)^2}\ge 0\implies F_1^\prime \le 0\\&\implies \int _{t_0}^{t_f}\langle \nabla _t{\dot{x}}(t,w),x^\prime (t,w)\rangle dt\ge 0, \end{aligned}$$

which completes this proof. $\square$

Frankly speaking, the condition (4) is not easy to verify in practice. Partly because it is difficult to solve 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 (MOPws) 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. $\square$

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

Example 1

Choosing $t_0=0$, $t_f=1$, $x_0=(0,0,0)^T$, $v_0=(1,0,0)^T$, $x_f=(2,2,1)^T$, $v_f=(0,1,-1)^T$, 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 (4).

Fig. 2

(Left) Pareto front (Right) Convexity condition

4 Numerical Experiments

For numerical experiments, we consider variational curves on two two-dimensional manifolds, namely on a sphere and a torus. To construct an approximation of the Pareto front of a given problem we implement Algorithm 1 in [13]. In the numerical examples we worked on, it was good enough to take the utopian objective vector $\beta ^* = \textbf{0}$ as the spread of the approximating points in the Pareto front was uniform enough.

We employ the scalarize–discretize–then–optimize approach that was previously used in [13]. 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, 6,7,8, 22]), under some general assumptions, the discrete approximate solution converges to a solution of the continuous-time scalarization (SP$_w$) of the original problem, yielding a Pareto minimum of the original problem.

In the subproblems of the algorithm in [13], a direct discretization of Problem (SP$_w$) employing the trapezoidal rule is solved by using Knitro, version 13.0.1. Knitro is a popular optimization software; see [4]. We use AMPL [11] 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 ${\mathbb {S}}^2:=\{x=(x_1,x_2,x_3)\in {\mathbb {R}}^3~\vert ~ \Vert x\Vert =1\}$ be the unit-sphere endowed with the restriction of 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

$$\begin{aligned} \nabla _t{\dot{x}}(t)=\ddot{x}(t)-\langle \ddot{x}(t),x(t)\rangle x(t)=\ddot{x}(t)+\langle {\dot{x}}(t),{\dot{x}}(t)\rangle x(t), \end{aligned}$$

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

$$\begin{aligned} \Vert \nabla _t{\dot{x}}(t)\Vert ^2=\Vert \ddot{x}(t)\Vert ^2-\Vert {\dot{x}}(t)\Vert ^4. \end{aligned}$$

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

$$\begin{aligned} \text{(S) } \left\{ \begin{array}{rl} \displaystyle \min _{x(\cdot )} &{} \ \ \displaystyle \left[ \int _{t_0}^{t_f} \Vert {\dot{x}}(t)\Vert ^2\,dt,\ \ \int _{t_0}^{t_f} \left( \Vert \ddot{x}(t)\Vert ^2 - \Vert {\dot{x}}(t)\Vert ^4\right) \,dt\right] \\ \text{ subject } \text{ to } &{} \ \ x_1(t)^2 + x_2(t)^2 + x_3(t)^2 = 1, \text{ for } \text{ all } t\in [0,t_f], \\ &{} \ \ x(t_0) = x_0\,\ \ {\dot{x}}(t_0) = v_0,\ \ x(t_f) = x_f\,\ \ {\dot{x}}(t_f) = v_f. \end{array} \right. \end{aligned}$$

We have

$$\begin{aligned} F_1(x):= \int _{t_0}^{t_f} \Vert {\dot{x}}(t)\Vert ^2\,dt \qquad \text{ and }\qquad F_2(x):= \int _{t_0}^{t_f} \left( \Vert \ddot{x}(t)\Vert ^2 - \Vert {\dot{x}}(t)\Vert ^4\right) \,dt. \end{aligned}$$

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

$$\begin{aligned} \text{(S }_w\text{) } \left\{ \begin{array}{rl} \displaystyle \min _{x(\cdot ),\,\alpha } &{} \ \ \alpha \\ \text{ subject } \text{ to } &{} \ \ w\,F_1(x) \le \alpha , \\ &{} \ \ (1-w)\,F_2(x) \le \alpha , \\ &{}\ \ x_1(t)^2 + x_2(t)^2 + x_3(t)^2 = 1, \text{ for } \text{ all } t\in [0,t_f], \\ &{} \ \ x(t_0) = x_0\,\ \ {\dot{x}}(t_0) = v_0,\ \ x(t_f) = x_f\,\ \ {\dot{x}}(t_f) = v_f, \end{array} \right. \end{aligned}$$

where $w\in [0,1)$ is fixed and we have taken $\beta _1^* = \beta _2^* = 0$.

Fig. 3

The Pareto front, the speed and acceleration on the sphere for a Riemannian cubic in tension, $x:[0,1]\rightarrow {\mathbb {S}}^2$, 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 Fig. 3, “speed on sphere” is simply $\Vert {\dot{x}}(t)\Vert$ and the “acceleration on sphere” is $\sqrt{\Vert \ddot{x}(t)\Vert ^2 - \Vert {\dot{x}}(t)\Vert ^4}$ .

In generating the Pareto solutions and an approximation of the Pareto front, we have solved (S$_w$) in the subproblems of Algorithm 1 of [13], with $0\le w < 1$.

The colour-coded Pareto solutions in Fig. 3 are given as blue ($0\le w\le 0.9059$), green ($w = 0.96$), magenta ($w = 0.985$), cyan ($w = 0.9940$), red ($w \rightarrow 1^-$). 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 $w \rightarrow 1^-$. For practical (illustration) purposes we depict $F_2 < \infty$ in Fig. 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.

Remark 3

The underlying problem is not convex because of the non-affine constraints. However, the epigraph of the Pareto front shown in Fig. 3 appears/happens to be convex. So, it should in principle be possible to use the weighted-sum scalarization (MOPws) to construct an approximation of the front. On the other hand, there would be two main issues with weighted sum compared to Chebyshev: (i) the essential interval of weights would be the whole interval [0, 1) as opposed to the subinterval [0.9059, 1) and (ii) the discrete approximations to the front would be not as uniformly distributed as otherwise (for non-uniformity of approximation see examples in [9]). Non-uniformity with weighted sum is also evident when one looks at the “left-hand side” of the Pareto front approximation shown in Fig. 5 and compares that with the left-hand side of the Pareto front approximation in Fig. 4. $\square$

Remark 4

The convexity condition (3) in Theorem 2 for the Pareto front can be numerically verified by computing $F_i'(w)$ and $F_i''(w)$, $i = 1,2$, via finite differences as in [14, Equation (8)]. Note that a solution of Problem (MOP$_w$) for w, and $w+\delta$ or $w-\delta$, for some small $\delta > 0$ can be computed to yield $F_i(w)$, and $F_i(w+\delta )$ or $F_i(w-\delta )$. Then approximations of $F_i'(w)$, $i = 1,2$, can be obtained by the finite difference formula (as a result of certain solution differentiability arguments used for optimal control problems in [14])

$$\begin{aligned} F_i'(w) \approx \left\{ \begin{array}{rl} \dfrac{F(w+\delta ) - F(w)}{\delta }, &{}\ \text{ if } w\in [w_0,w_f-\delta ), \\ \dfrac{F(w) - F(w-\delta )}{\delta }, &{}\ \text{ if } w\in [w_f-\delta ,w_f], \end{array} \right. \end{aligned}$$

over the essential interval $[w_0,w_f]$. Finite difference approximations of $F_i''(w)$, $i = 1,2$, can be obtained using a similar procedure (this time evaluating approximations of $F_i'(w)$, and $F_i'(w+\delta )$ or $F_i'(w-\delta )$). Carrying out these computations is quite demanding and is beyond the scope of the current paper. $\square$

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

$$\begin{aligned}{} & {} \big \{x=(x_1,x_2,x_3)\in {\mathbb {R}}^3\ |\ x_1 = (c+a\cos v)\cos u,\ x_2 \\{} & {} \quad = (c+a\cos v)\sin u,\ x_3 = a\sin v, u,v\in [0,2\pi ) \big \}, \end{aligned}$$

with the Riemannian metric induced by the 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

$$\begin{aligned} x_u&= (-(c+a\cos v)\sin u,(c+a \cos v)\cos u,0),\\ x_v&= (-a\sin v\cos u,-a\sin v\sin u, a\cos v). \end{aligned}$$

and the first fundamental form

$$\begin{aligned} ds^2&=\langle x_u,x_u\rangle du^2+2\langle x_u,x_v\rangle dudv+\langle x_v,x_v\rangle dv^2\\&= (c+a\cos v)^2du^2+a^2dv^2, \end{aligned}$$

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

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

Further, the non-zero Christoffel symbols are given by

$$\begin{aligned} \Gamma _{uu}^v&=-\frac{1}{2}g^{vv}\frac{g_{uu}}{\partial v}=\left( \frac{c}{a}+\cos v\right) \sin v,\\ \Gamma _{uv}^u&=\Gamma _{vu}^u=\frac{1}{2}g^{uu}\frac{\partial g_{uu}}{\partial v}=-\frac{a\sin v}{c+a\cos v}. \end{aligned}$$

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

$$\begin{aligned} \nabla _t{\dot{x}}(t) = (a_1(t),a_2(t)), \end{aligned}$$

where

$$\begin{aligned} a_1&= \ddot{u}+\Gamma _{uv}^u{\dot{u}}{\dot{v}}+\Gamma _{vu}{\dot{v}}{\dot{u}}=\ddot{u} - \frac{2a\sin v}{c + a\cos v}\,{\dot{u}}{\dot{v}},\\ a_2&= \ddot{v}+\Gamma _{uu}^v{\dot{u}}{\dot{u}}=\ddot{v} + \frac{1}{a} \sin v (c + a\cos v)\,{\dot{u}}^2. \end{aligned}$$

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

$$\begin{aligned} F_1:= \int _0^{t_f}\left( {\dot{x}}_1^2(t) + {\dot{x}}_3^2(t) + {\dot{x}}_3^2(t)\right) dt \quad \text{ and }\quad F_2:= \int _0^{t_f}\left( a_1^2(t) + a_2^2(t)\right) dt. \end{aligned}$$

In other words, we are interested in solving the problem

$$\begin{aligned} \text{(T) } \left\{ \begin{array}{rl} \displaystyle \min _{u(\cdot ),v(\cdot )} &{} \ \ \displaystyle \left[ F_1,\ F_2\right] \\ \text{ subject } \text{ to } &{} \ \ (u(0), v(0)) = (u_0,v_0),\ \ (u(t_f), v(t_f)) = (u_f,v_f), \\ &{} \ \ ({\dot{u}}(0), {\dot{v}}(0)) = ({\dot{u}}_0,{\dot{v}}_0),\ \ ({\dot{u}}(t_f), {\dot{v}}(t_f)) = ({\dot{u}}_f,{\dot{v}}_f). \end{array} \right. \end{aligned}$$

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

$$\begin{aligned} \text{(T }_w\text{) } \left\{ \begin{array}{rl} \displaystyle \min _{u(\cdot ),v(\cdot ),\alpha } &{} \ \ \alpha \\ \text{ subject } \text{ to } &{} \ \ w\,F_1(x) \le \alpha , \\ &{} \ \ (1-w)\,F_2(x) \le \alpha , \\ &{} \ \ (u(0), v(0)) = (u_0,v_0),\ \ (u(t_f), v(t_f)) = (u_f,v_f), \\ &{} \ \ ({\dot{u}}(0), {\dot{v}}(0)) = ({\dot{u}}_0,{\dot{v}}_0),\ \ ({\dot{u}}(t_f), {\dot{v}}(t_f)) = ({\dot{u}}_f,{\dot{v}}_f). \end{array} \right. \end{aligned}$$

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 $t_0=0$ and $t_f = 1$. See Fig. 4 for the Pareto front constructed by using the scalarization (T$_w$), after weeding out the dominated points found as local minima of (T$_w$). 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, therefore cannot be recovered by the classical weighted-sum scalarization.

Fig. 4

Torus—Pareto front constructed via (T$_w$). Curves are obtained between the oriented points $(u_0,v_0,{\dot{u}}_0,{\dot{v}}_0) = (0,0,1,0)$ and $(u_f,v_f,{\dot{u}}_f,{\dot{v}}_f) = (\pi ,\pi /2,0,-1)$

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 $w\rightarrow 1^-$. 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 $w = 0.7667$.

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

$$\begin{aligned} \text{(Tws) } \left\{ \begin{array}{rl} \displaystyle \min _{u(\cdot ),v(\cdot )} &{} \ \ w\,F_1 + (1-w) F_2 \\ \text{ subject } \text{ to } &{} \ \ (u(0), v(0)) = (u_0,v_0),\ \ (u(t_f), v(t_f)) = (u_f,v_f), \\ &{} \ \ ({\dot{u}}(0), {\dot{v}}(0)) = ({\dot{u}}_0,{\dot{v}}_0),\ \ ({\dot{u}}(t_f), {\dot{v}}(t_f)) = ({\dot{u}}_f,{\dot{v}}_f). \end{array} \right. \end{aligned}$$

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

By comparing Figs. 4 and 5, we see that the blue and green points both correspond to the weighted-sum solutions (i.e., the solutions to (Tws)) with $w=0$. So they correspond to Riemannian cubics.

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

Fig. 5

Torus—Pareto front constructed via (Tws)

5 Conclusion

In the literature, measurements from variational curves, for example, kinetic energy and squared norm of higher-order derivatives, are seldom minimized at the same time. Motivated by the problem of finding Riemannian cubics in tension, we have formulated a multi-objective optimization 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. In this paper, we adopted the Chebyshev and weighted-sum scalarizations in order to obtain a sequence of single-objective optimization problems. 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 [17]. 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.

Appendices

Appendix A: Discussion on Remark 2

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

$$\begin{aligned}&x^{(4)}(t)-\tau \ddot{x}(t)=0,\\&x(0)=x_0,{\dot{x}}(0)=v_0,x(t_f)=x_f,{\dot{x}}(t_f)=v_f, \end{aligned}$$

which can be analytically written as

$$\begin{aligned} x=x_0+v_0 t+(\sinh (\sqrt{\tau }t)-\sqrt{\tau }t)c_1+(\cosh (\sqrt{\tau }t)-1)c_2, \end{aligned}$$

where $c_1$ and $c_2$ satisfy

$$\begin{aligned} \begin{pmatrix} c_1\\ c_2 \end{pmatrix}=\frac{1}{\Delta }\begin{pmatrix} \sqrt{\tau }t_f\sinh (\sqrt{\tau }t_f) &{} 1-\cosh (\sqrt{\tau }t_f)\\ \sqrt{\tau }t_f(1-\cosh (\sqrt{\tau }t_f)) &{} \sinh (\sqrt{\tau }t_f)-\sqrt{\tau }t_f \end{pmatrix}\begin{pmatrix} x_f-x_0-v_0t_f\\ (v_f-v_0)t_f \end{pmatrix} \end{aligned}$$

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

To simplify notations, we denote by $c=(c_1,c_2)^T$, $A=\begin{pmatrix}\sinh (\tau _f)-\tau _f &{} \cosh (\tau _f)-1\\ \tau _f(\cosh (\tau _f)-1) &{} \tau _f\sinh (\tau _f)\end{pmatrix}$, $b=(x_f-x_0-v_0t_f,(v_f-v_0)t_f)^T$, $\tau _f=\sqrt{\tau }t_f$, $\tau _t=\sqrt{\tau }t$. Then, $Ac=b$, which implies $c^\prime =-A^{-1}A^\prime c$.

Straightforward calculations yield

$$\begin{aligned} \ddot{x}&=\tau (\sinh (\tau _t),\cosh (\tau _t))c,\\ x^\prime&=\frac{\tau ^\prime t}{2\sqrt{\tau }}(\cosh (\tau _t)-1,\sinh (\tau _t))c-(\sinh (\tau _t)-\tau _t,\cosh (\tau _t)-1)A^{-1}A^\prime c. \end{aligned}$$

Note that

$$\begin{aligned}&\int _0^{t_f} \tau _t\begin{pmatrix} \sinh (\tau _t)\\ \cosh (\tau _t) \end{pmatrix}(\cosh (\tau _t)-1,\sinh (\tau _t)) dt\\&=\frac{1}{\sqrt{\tau }}\begin{pmatrix} \frac{\tau _f}{4}\cosh (2\tau _f)-\frac{\sinh (2\tau _f)}{8}-\tau _f\cosh (\tau _f)+\sinh (\tau _f) &{} \frac{\tau _f}{4}\sinh (2\tau _f)-\frac{\sinh ^2(\tau _f)}{4}-\frac{\tau _f^2}{4}\\ -\frac{7}{8}+\frac{\tau _f^2}{4}+\frac{\tau _f}{4}\sinh (2\tau _f)-\tau _f\sinh (\tau _f)-\frac{\cosh (2\tau _f)}{8}+\cosh (\tau _f) &{} \frac{\tau _f}{4}\cosh (2\tau _f)-\frac{\sinh (2\tau _f)}{8} \end{pmatrix}\\&=:\frac{B_1}{\sqrt{\tau }},\\&\int _0^{t_f} \begin{pmatrix} \sinh (\tau _t)\\ \cosh (\tau _t) \end{pmatrix}(\sinh (\tau _t)-\tau _t,\cosh (\tau _t)-1) dt\\&=\frac{1}{\sqrt{\tau }}\begin{pmatrix} \frac{\sinh (2\tau _f)}{4}-\frac{\tau _f}{2}-\tau _f\cosh (\tau _f)+\sinh (\tau _f) &{} \frac{(\cosh (\tau _f)-1)^2}{2}\\ \frac{\sinh ^2(\tau _f)}{2}-\tau _f\sinh (\tau _f)+\cosh (\tau _f)-1 &{} \frac{\tau _f}{2}+\frac{\sinh (2\tau _f)}{4}-\sinh (\tau _f) \end{pmatrix}\\&=:\frac{B_2}{\sqrt{\tau }}. \end{aligned}$$

Then, the integration $\int _0^{t_f}\langle \ddot{x},x^\prime \rangle dt$ becomes

$$\begin{aligned} \frac{\tau ^\prime }{2\sqrt{\tau }}c^T\left( B_1-\frac{2\tau }{\tau ^\prime }B_2A^{-1}A^\prime \right) c, \end{aligned}$$

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

$$\begin{aligned} B_1-\frac{2\tau }{\tau ^\prime }B_2A^{-1}A^\prime =:B_3=:\begin{pmatrix} b_{11} &{} b_{12}\\ b_{21} &{} b_{22} \end{pmatrix}, \end{aligned}$$

$$\begin{aligned} b_{11}&=-\tau _f \cosh (\tau _f)+\frac{\tau _f}{4}\cosh (2\tau _f)+\sinh (\tau _f)-\frac{1}{8}\sinh (2\tau _f)+\\&\frac{\tau _f}{2\tilde{\Delta }}(\cosh (\tau _f)-1)\sinh (\tau _f)(\tau _f-2\sinh (\tau _f)-\cosh (\tau _f)(\sinh (\tau _f)-2\tau _f)),\\ b_{12}&=-\frac{\tau _f^2}{4}-\frac{\sinh ^2(\tau _f)}{4}+\frac{\tau _f\sinh (2\tau _f)}{4}+\frac{1}{2\tilde{\Delta }}(\cosh (\tau _f)-1)^3\sinh (\tau _f),\\ b_{21}&=-\frac{7}{8}+\frac{\tau _f^2}{4}+\cosh (\tau _f)-\frac{\cosh (2\tau _f)}{8}-\tau _f\sinh (\tau _f)+\frac{\tau _f}{4}\sinh (2\tau _f)-\\&\frac{\tau _f}{2\tilde{\Delta }}(\cosh (\tau _f)-1+\tau _f\sinh (\tau _f))(2-2\cosh (\tau _f)+2\tau _f\sinh (\tau _f)-\sinh ^2(\tau _f)),\\ b_{22}&=\frac{\tau _f}{4}\cosh (2\tau _f)-\frac{\sinh (2\tau _f)}{8}-\\&\frac{1}{4\tilde{\Delta }}(\sinh (\tau _f)-\tau _f)(\sinh (\tau _f)+\tau _f\cosh (\tau _f))(2\tau _f-4\sinh (\tau _f)+\sinh (2\tau _f)),\\ \tilde{\Delta }&=\Delta /\tau _f. \end{aligned}$$

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 Fig. 6, we choose $t_f=1$, from which we can see that eigenvalues of $B_3$ are positive at least on (0.2, 1). Therefore, the convexity condition (4) is verified at least on (0.2, 1).

Fig. 6

Eigenvalues of $B_3$ (Left: smaller, Right: larger)

Appendix B: Riemannian cubics in tension on torus

In this section, we consider Riemannian cubics in tension on torus using variational methods. To shorten 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

$$\begin{aligned} F&=\int _0^T\Vert \nabla _t{\dot{x}}\Vert ^2+\lambda \Vert {\dot{x}}\Vert ^2 dt\\&=\int _0^T(\ddot{u}-p{\dot{u}}{\dot{v}})^2+(\ddot{v}+q{\dot{u}}^2)^2+\lambda (r^2{\dot{u}}^2+a^2{\dot{v}}^2) dt. \end{aligned}$$

Considering the variation of F, we find

$$\begin{aligned} \delta F&=2\int _0^T(\ddot{u}-p{\dot{u}}{\dot{v}})(\delta \ddot{u}-p^\prime \delta v{\dot{u}}{\dot{v}}-p\delta {\dot{u}} {\dot{v}}-p{\dot{u}}\delta {\dot{v}})\\&\quad +(\ddot{v}+q{\dot{u}}^2)(\delta \ddot{v}+q^\prime \delta v{\dot{u}}^2+2q{\dot{u}}\delta {\dot{u}})+\\&\lambda (rr^\prime \delta v{\dot{u}}^2+r^2{\dot{u}}\delta {\dot{u}}+a^2{\dot{v}}\delta {\dot{v}})dt\\&=2\int _0^T\left( \frac{d^2}{dt^2}\left( \ddot{u}-p{\dot{u}}{\dot{v}}\right) +\frac{d}{dt}\left( \left( \ddot{u}-p{\dot{u}}{\dot{v}}\right) p{\dot{v}}\right) \right. \\&\quad \left. -\frac{d}{dt}\left( \left( \ddot{v}+q{\dot{u}}^2\right) 2q{\dot{u}}\right) -\frac{d}{dt}\left( \lambda r^2{\dot{u}}\right) \right) \delta u+\\&\left( \frac{d^2}{dt^2}\left( \ddot{v}+q{\dot{u}}^2\right) +p\frac{d}{dt}\left( (\ddot{u}-p{\dot{u}}{\dot{v}}){\dot{u}}\right) +(\ddot{v}+q{\dot{u}}^2)q^\prime {\dot{u}}^2\right. \\&\quad \left. +\lambda rr^\prime {\dot{u}}^2-\frac{d}{dt}\left( \lambda a^2{\dot{v}}\right) \right) \delta v dt, \end{aligned}$$

which implies the following Euler–Lagrange equations,

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

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,

$$\begin{aligned} {\dot{u}}_2&= u_3, \nonumber \\ {\dot{u}}_3&= u_4, \nonumber \\ {\dot{u}}_4&= (p^2+p^\prime )u_3v_2^2+(p+2q)u_3v_3+(p^{\prime \prime }+2pp^\prime )u_2v_2^3+(2p^2+3p^\prime +2q^\prime ) \nonumber \\&\,\, u_2v_2v_3+(p+2q)u_2v_4+4qq^\prime u_2^3v_2+6q^2u_2^2u_3-2\lambda a^2qu_2v_2+\lambda r^2u_3, \nonumber \\ {\dot{v}}_1&= v_2, \nonumber \\ {\dot{v}}_2&= v_3, \nonumber \\ {\dot{v}}_3&= v_4, \nonumber \\ {\dot{v}}_4&= (pp^\prime -q^{\prime \prime })u_2^2v_2^2+(p^2-2q^\prime )(2u_2u_3v_2-u_2^2v_3)-(p+2q)(u_3^2+u_2u_4) \nonumber \\&-qq^\prime u_2^4-\lambda rr^\prime u_2^2+\lambda a^2v_3. \end{aligned}$$

(5)

Therefore, in order to solve the problem (T$_{ws}$), we only have to solve the above ODE system.