# Bounds to the first eigenvalues of weighted p-Steklov and (p,q)-Laplacian
Steklov problems

Shahroud Azami  Department of Pure Mathematics, Faculty of Science, Imam
Khomeini International University, Qazvin, Iran.
Tel.: +98-28-33901321
Fax: +98-28-33780083
<EMAIL_ADDRESS>
###### Abstract.

We consider the Steklov problem associated with the weighted p-Laplace
operator and $(p,q)$-Laplacian on submanifolds with boundary of Euclidean
spaces and prove Reilly-type upper bounds for their first eigenvalues.

###### Key words and phrases:

Steklov problem, Eigenvalue, hypersurfaces

###### 2020 Mathematics Subject Classification:

53C20, 43C24, 53C42

##  1\. Introduction

Let $(M^{n},g)$ be a compact Riemannian manifold with a possibility non-empty
boundary $\partial M$. The triple $(M,g,d\mu_{g}=e^{-f}dv)$ is called a smooth
metric measure space, where $f:M\to\mathbb{R}$ is a smooth real-valued
function on $M$ and $dv$ is the Riemannian volume element related to $g$. We
also call $e^{-f}$ the density.
For $1<p<\infty$ and any $u\in W_{0}^{1,p}(M)$, the p-Laplacian $\Delta_{p}$
is defined by

$\Delta_{p}u={\rm div}(|\nabla u|^{p-2}\nabla u)=|\nabla u|^{p-2}\Delta
u+(p-2)|\nabla u|^{p-4}{\rm Hess}u(\nabla u,\nabla u),$

where ${\rm div}$ is the divergence operator, $\nabla$ is the gradient
operator, and ${\rm Hess}u$ is the hessian of $u$. For $p=2$, the p-Laplacian
is the Laplace-Beltrami operator of $(M^{n},g)$. Also, the weighted
p-Laplacian is defined by

$\Delta_{p,f}u=e^{f}{\rm div}(e^{-f}|\nabla u|^{p-2}\nabla u).$

The spectrum of the weighted p-Laplacian has been studied on smooth metric
measure spaces with Dirichlet or Neuman boundary conditions (see [10, 17, 18,
19]). In the present paper, we will consider the Steklov problem associated
with the weighted p-Laplace operator and $(p,q)$-Laplacian on submanifolds
with boundary of Euclidean spaces.

In following we will consider the weighted $p$-Steklov problem on submanifolds
with boundary of the Euclidean space

(1.1) $\begin{cases}\Delta_{p,f}u=0&\text{in}\,M\\\ |\nabla
u|^{p-2}\frac{\partial u}{\partial\nu}=\lambda|u|^{p-2}u&\text{on}\,\partial
M,\end{cases}$

where $\frac{\partial u}{\partial\nu}$ is the derivative of the function $u$
with respect to the outward unit normal $\nu$ to the boundary $\partial M$. If
$f$ be a constant function then the weighted p-Steklov problem (1.1) reduces
to the p-Steklov problem which it has been studied in [16]. This problem
arises from the following variational characterization of the first positive
eigenvalue given by

(1.2) $\lambda_{1}(M)=\inf\left\\{\frac{\int_{M}|\nabla
u|^{p}d\mu_{g}}{\int_{\partial M}|u|^{p}d\mu_{h}}\Big{|}u\in
W^{1,p}(M)\setminus\\{0\\},\,\int_{\partial M}|u|^{p-2}ud\mu_{h}=0\right\\}$

where $d\mu_{h}$ is the weighted measure on $\partial M$. Also, we consider a
Steklov problem associated with the $(p,q)$-Laplacian as follow

(1.3) $\begin{cases}\Delta_{p}u+\Delta_{q}u=0&\text{in}\,\,\,M,\\\
\left(|\nabla u|^{p-2}+|\nabla u|^{q-2}\right)\frac{\partial
u}{\partial\nu}=\sigma|u|^{r-2}u&\text{on}\,\,\,\partial M,\end{cases}$

where $1<p<q<r<\infty$, $r\in(1,\frac{q(N-1)}{N-q})$ if $p<N$ and
$r\in(1,\infty)$ if $p\geq N$. The first positive eigenvalue of the
$(p,q)$-Steklov problem (1.3) defined as

(1.4) $\sigma_{1}(M)=\inf\left\\{\frac{\int_{M}\left(|\nabla u|^{p}+|\nabla
u|^{q}\right)dv_{g}}{\int_{\partial M}|u|^{r}\,dv_{h}}\Big{|}u\in
W^{1,q}(M)\setminus\\{0\\},\,\,\int_{\partial M}|u|^{r-2}udv_{h}=0\right\\}.$

The $(p,q)$-Steklov problem has been studied in [4, 5, 6].

The aim of this paper is to obtain upper bounds for the first positive
eigenvalue of the problems (1.1) and (1.3), for submanifolds of Riemannian
manifolds, depending on the geometry of boundary in the spirit of the
classical Reilly upper bounds for the Laplacian on closed hypersurfaces.

For the first positive eigenvalue $\lambda_{1}$ of Laplacian, Reilly [12]
proved the following well-known upper bound

$\lambda_{1}\leq\frac{n}{Vol(M)}\int_{M}H^{2}dv_{g},$

where $H$ is the mean curvature of the immersion. Also, he [12] showed that
for $r\in\\{1,2,\cdots,n\\}$,

$\lambda_{1}\left(\int_{M}H_{r-1}dv_{g}\right)^{2}\leq
Vol(M)\int_{M}H_{r}^{2}dv_{g},$

where $H_{r}$ is the $r$-th mean curvature of the immersion and defined by the
$r$-th symmetric polynomial of the principal curvatures. Moreover, Reilly
studied the equality cases and proved that equality holds in one these
inequalities, if and only if $M$ is immersed in a geodesic sphere of radius
$\sqrt{\frac{n}{\lambda_{1}}}$. More generally, he show that if $(M^{n},g)$ is
isometrically immersed into $\mathbb{R}^{N}$, $N>n+1$, then

$\lambda_{1}\left(\int_{M}H_{r}dv_{g}\right)^{2}\leq
Vol(M)\int_{M}|{H_{r+1}}|^{2}dv_{g},$

for any even $r\in\\{0,1,\cdots,n\\}$ and equality holds if and only if $M$ is
minimally immersed in a geodesic sphere of $\mathbb{R}^{N}$. For codimension
greater than 1, $H_{r}$ is a function and ${H}_{r+1}$ is a normal vector
field. These inequalities have been generalized for other ambient spaces and
other operators (see [1, 2, 3, 7, 8, 11, 13, 14, 15, 16]).
Du and Mao [9] established the first positive eigenvalue of the $p$-Laplacian
on closed submanifold of $\mathbb{R}^{N}$ satisfies the follows inequalities.

$\lambda_{1}\leq\frac{n^{\frac{p}{2}}}{(Vol(M))^{p}}\left(\int_{M}|H|^{\frac{p}{p-1}}dv_{g}\right)^{p-1}\begin{cases}N^{\frac{2-p}{2}}&\text{if}\,\,\,1<p\leq
2,\\\ N^{\frac{p-2}{2}}&\text{if}\,\,\,p\geq 2.\end{cases}$

In addition, equality holds if and only if $p=2$ and $M$ is minimally immersed
into a geodesic hypersphere. On the other hand, Roth [16] proved Reilly-type
inequalities for the first eigenvalue of $p$-Steklov problem on submanifolds
of $\mathbb{R}^{N}$ and showed that

$\lambda_{1}\leq\frac{Vol(M)}{(Vol(\partial
M))^{p}}n^{\frac{p}{2}}\left(\int_{M}|H|^{\frac{p}{p-1}}\right)^{p-1}\begin{cases}N^{\frac{2-p}{2}}&\text{if}\,\,\,1<p\leq
2,\\\ N^{\frac{p-2}{2}}&\text{if}\,\,\,p\geq 2.\end{cases}$

Moreover, equality holds if and only if $p=2$ and $M$ is minimally immersed
into $B^{N}(\frac{1}{\lambda_{1}})$ such that $X(\partial M)\subset\partial
B^{N}(\frac{1}{\lambda_{1}})$, where $X$ is the isometric immersion.

##  2\. Main results

Motivated by above works, we prove that:

######  Theorem 2.1.

Let $(M^{n},g,d\mu=e^{-f}dv)$ be a compact connected and oriented Riemannian
manifold with nonempty boundary $\partial M$ and $p\in(1,+\infty)$. Assume
that $(M^{n},g,d\mu=e^{-f}dv)$ isometrically immersied into the Euclidean
space $\mathbb{R}^{N}$ by $X$. If $\lambda_{1}(M)$ is the eigenvalue of the
weighted $p$-Steklov problem (1.1) then for $1<p\leq 2$ we have

$\lambda_{1}(M)\leq
2^{\frac{1}{p-1}}n^{-\frac{p}{2}}N^{1-\frac{p}{2}}\left(\int_{\partial
M}\left(|nH|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}\frac{Vol_{\mu_{g}}(M)}{(Vol_{\mu_{h}}(\partial
M))^{p}}$

and for $p\geq 2$ we get

$\lambda_{1}(M)\leq
2^{\frac{1}{p-1}}n^{-\frac{p}{2}}N^{\frac{p}{2}-1}\left(\int_{\partial
M}\left(|nH|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}\frac{Vol_{\mu_{g}}(M)}{(Vol_{\mu_{h}}(\partial
M))^{p}}.$

where $Vol_{\mu_{g}}(\partial M)=\int_{M}d\mu_{g}$ and $Vol_{\mu_{h}}(\partial
M)=\int_{\partial M}d\mu_{h}$. Moreover,

  * (i)

If $f$ is constant, $H$ does not vanish identically then equality occurs in
both inequality if and only if $p=2$ and $M$ is minimally immersed into
$B^{N}(\frac{1}{\lambda_{1}(M)})$ so that $\partial M$ lies into geodesic
hypersphere $\partial B^{N}(\frac{1}{\lambda_{1}(M)})$.

  * (ii)

If $f$ is not constant and if equality occurs then $M$ is a self-shrinker for
the mean curvature flow and $f_{|_{M}}=a-\frac{b}{2}r_{p}^{2}$ for some
constants $a,b$, where $r_{p}$ is the Euclidean distance to the center of mass
$p$ of $M$. In particular, if $n=N-1$ and $H>0$ or $n=2$, $N=3$ and $M$ is
embedded and has genus $0$, then $M$ is a geodesic ball.

Let $T$ be a symmetric positive definite and divergence-free $(1,1)$-tensor on
$M$. We associated with $T$ the normal vecor field $H_{T}$ defined by

$H_{T}=\sum_{i,j=1}^{n}\langle Te_{i},e_{j}\rangle B(e_{i},e_{j}),$

where $\\{e_{1},\cdots,e_{n}\\}$ is a local orthonormal frame of $T\partial M$
and $B$ is the second fundamental form of the immersion of $M$ into
$\mathbb{R}^{N}$. We also, recall the generalized Hsiung-Minkowski formula
[11, 14, 15] as

(2.1) $\int_{\partial M}\left(\langle X,H_{T}-T(\nabla f)\rangle+{\rm
tr}(T)\right)d\mu_{h}=0.$

In following theorem we extended the theorem 2.1 to estimates with higher
order mean curvatures.

######  Theorem 2.2.

Let $(M^{n},g,d\mu=e^{-f}dv)$ be a compact connected and oriented Riemannian
manifold with nonempty boundary $\partial M$ and $p\in(1,+\infty)$. Assume
that $(M^{n},g,d\mu=e^{-f}dv)$ isometrically immersed into the Euclidean space
$\mathbb{R}^{N}$ by $X$ and let $T$ be a symmetric and divergence-free
$(0,2)$-tensor on $\partial M$. If $\lambda_{1}(M)$ is the eigenvalue of the
weighted $p$-Steklov problem (1.1) then for $1<p\leq 2$ we have

$\displaystyle\lambda_{1}(M)\left|\int_{\partial M}{\rm
tr}(T)d\mu_{h}\right|^{p}$ $\displaystyle\leq
2^{\frac{1}{p-1}}n^{\frac{p}{2}}N^{1-\frac{p}{2}}\left(\int_{\partial
M}\left(|H_{T}|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M)$

and for $p\geq 2$ we get

$\displaystyle\lambda_{1}(M)\left|\int_{\partial M}{\rm
tr}(T)d\mu_{h}\right|^{p}$ $\displaystyle\leq
2^{\frac{1}{p-1}}n^{\frac{p}{2}}N^{\frac{p}{2}-1}\left(\int_{\partial
M}\left(|H_{T}|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M).$

where $Vol_{\mu_{g}}(\partial M)=\int_{M}d\mu_{g}$ and $Vol_{\mu_{h}}(\partial
M)=\int_{\partial M}d\mu_{h}$. Moreover,

  * (i)

If $f$ is constant, $H_{T}$ does not vanish identically then equality occurs
in both inequality if and only if $p=2$ and $M$ is minimally immersed into
$B^{N}(\frac{1}{\lambda_{1}(M)})$ so that $\partial M$ lies into geodesic
hypersphere $\partial B^{N}(\frac{1}{\lambda_{1}(M)})$.

  * (ii)

If $f$ is not constant and if equality occurs then $M$ is a self-shrinker for
the mean curvature flow and $f_{|_{M}}=a-\frac{b}{2}r_{p}^{2}$ for some
constants $a,b$, where $r_{p}$ is the Euclidean distance to the center of mass
$p$ of $M$. In particular, if $n=N-1$ and $H>0$ or $n=2$, $N=3$ and $M$ is
embedded and has genus $0$, then $M$ is a geodesic ball.

For $r\in\\{1,\cdots,n\\}$, let

$T_{r}=\frac{1}{r!}\sum_{\begin{subarray}{l}i,{i_{1}},...,{i_{r}}\\\
j,{j_{1}},...,{j_{r}}\\\
\end{subarray}}{\varepsilon\left(\begin{gathered}i,{i_{1}},...,{i_{r}}\hfill\\\
j,{j_{1}},...,{j_{r}}\hfill\\\ \end{gathered}\right)}\langle
B_{i_{1}j_{1}},B_{i_{2}j_{2}}\rangle\cdots\langle
B_{i_{r-1}j_{r-1}},B_{i_{r}j_{r}}\rangle e_{i}^{*}\otimes e_{j}^{*}$

if $r$ is even and

$T_{r}=\frac{1}{r!}\sum_{\begin{subarray}{l}i,{i_{1}},...,{i_{r}}\\\
j,{j_{1}},...,{j_{r}}\\\
\end{subarray}}{\varepsilon\left(\begin{gathered}i,{i_{1}},...,{i_{r}}\hfill\\\
j,{j_{1}},...,{j_{r}}\hfill\\\ \end{gathered}\right)}\langle
B_{i_{1}j_{1}},B_{i_{2}j_{2}}\rangle\cdots\langle
B_{i_{r-2}j_{r-2}},B_{i_{r-1}j_{r-1}}\rangle B_{i_{r}j_{r}}\otimes
e_{i}^{*}\otimes e_{j}^{*}$

if $r$ is odd, where the $B_{ij}$’s are the coefficients of the second
fundamental form $B$ in a orthonormal frame $\\{e_{1},\cdots,e_{n}\\}$ with
the dual coframe $\\{e_{1}^{*},\cdots,e_{n}^{*}\\}$ and $\epsilon$ is the
standard signature for permutations. The $r$-th mean curvature is defined as
$H_{0}=0$ and $H_{r}=\frac{1}{(n-r){r\choose n}}{\rm tr}(T_{r})$. If $r$ is
even then $H_{r}$ is a real function and if $r$ is odd then $H_{r}$ is a
normal vector field, in this case, we will denote it ${\bf H}_{r}$. Also, the
Hsiung-Minkowski formula becomes

$\int_{\partial M}\left(\langle X,{\bf H}_{r+1}\rangle+H_{r}\right)d\mu_{h}=0$

for any even $r\in\\{0,1,\cdots,n\\}$ if $N>n+1$, and

$\int_{\partial M}\left(\langle X,\nu\rangle H_{r+1}+H_{r}\right)d\mu_{h}=0$

for any $r\in\\{0,1,\cdots,n\\}$ if $N=n+1$, where $\nu$ is the normal unit
vector field on $\partial M$ chosen to define the shape operator.
Now we obtain the following corollary from Theorem 2.2.

######  Corollary 2.3.

Let $(M^{n},g,d\mu=e^{-f}dv)$ be a compact connected and oriented Riemannian
manifold with nonempty boundary $\partial M$ and $p\in(1,+\infty)$. Assume
that $(M^{n},g,d\mu=e^{-f}dv)$ isometrically immersed into the Euclidean space
$\mathbb{R}^{N}$ by $X$. If $\lambda_{1}(M)$ is the eigenvalue of the weighted
$p$-Steklov problem (1.1)

  * (1)

If $N>n+1$, and $r\in\\{0,\cdots,n-1\\}$ is an even integer then we have

    * (a)

If $1<p\leq 2$ we have

$\displaystyle\lambda_{1}(M)\left|\int_{\partial M}H_{r}d\mu_{h}\right|^{p}$
$\displaystyle\leq
2^{\frac{1}{p-1}}n^{\frac{p}{2}}N^{1-\frac{p}{2}}\left(\int_{\partial
M}\left(|{\bf H}_{r+1}|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M)$

    * (b)

If $p\geq 2$ we have

$\displaystyle\lambda_{1}(M)\left|\int_{\partial M}H_{r}d\mu_{h}\right|^{p}$
$\displaystyle\leq
2^{\frac{1}{p-1}}n^{\frac{p}{2}}N^{\frac{p}{2}-1}\left(\int_{\partial
M}\left(|{\bf H}_{r+1}|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M)$

Moreover, if $f$ is constant, $H_{r}$ does not vanish identically then
equality occurs in both inequality if and only if $p=2$ and $M$ is minimally
immersed into $B^{N}(\frac{1}{\lambda_{1}(M)})$ so that $\partial M$ lies into
geodesic hypersphere $\partial B^{N}(\frac{1}{\lambda_{1}(M)})$.

  * (2)

If $N=n+1$, and $r\in\\{0,\cdots,n-1\\}$ is an even integer then we have

    * (a)

If $1<p\leq 2$ we have

$\displaystyle\lambda_{1}(M)\left|\int_{\partial M}H_{r}d\mu_{h}\right|^{p}$
$\displaystyle\leq
2^{\frac{1}{p-1}}n^{\frac{p}{2}}N^{1-\frac{p}{2}}\left(\int_{\partial
M}\left(|{H}_{r+1}|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M)$

    * (b)

If $p\geq 2$ we have

$\displaystyle\lambda_{1}(M)\left|\int_{\partial M}H_{r}d\mu_{h}\right|^{p}$
$\displaystyle\leq
2^{\frac{1}{p-1}}n^{\frac{p}{2}}N^{\frac{p}{2}-1}\left(\int_{\partial
M}\left(|{H}_{r+1}|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M)$

Moreover, if $f$ is constant, $H_{r+1}$ does not vanish identically then
equality occurs in both inequality if and only if $p=2$ and
$X(M)=B^{N}(\frac{1}{\lambda_{1}(M)})$.

In following we investigate the first eigenvalue of weighted $p$-Steklov
problem on Riemannian products $\mathbb{R}\times M$ where $M$ is a complete
Riemannian manifold.

######  Theorem 2.4.

Let $p\geq 2$ and $(M^{n},\bar{g})$ be a complete Riemannian manifold.
Consider $(\Sigma^{n},g)$ a closed oriented Riemannian manifold isometrically
immersed into the Riemannian product $(\mathbb{R}\times
M,\tilde{g}=dt^{2}\oplus\bar{g})$ with a density $e^{-f}$. Moreover, assume
that $\Sigma$ is mean-convex and bounds a domain $\Omega$ in $\mathbb{R}\times
M$. Let $\lambda_{1}(M)$ be the first eigenvalue of the weighted $p$-Steklov
problem on $\Omega$, then

$\lambda_{1}(\Omega)\leq\left(\frac{\kappa_{+}(\Sigma)|H|_{\infty}}{\mathop{\inf}\limits_{\Sigma}H}\right)^{\frac{p}{2}}\left(\frac{Vol_{\mu_{g}}(\Omega)}{Vol_{\mu_{h}}(\Sigma)}\right)^{1-\frac{p}{2}},$

where $\kappa_{+}(\Sigma)=\max\\{\kappa_{+}(x)|x\in M\\}$ with $\kappa_{+}$
the biggest principal curvature of $\Sigma$ at the point $x$.

Now, we obtain Reilly upper bounds for $(p,q)$-Steklov problem. Similar the
theorem 2.1 we have

######  Theorem 2.5.

Let $(M^{n},g,dv)$ be a compact connected and oriented Riemannian manifold
with nonempty boundary $\partial M$ and $1<p<q<r<\infty$. Assume that
$(M^{n},g,dv)$ isometrically immersed into the Euclidean space
$\mathbb{R}^{N}$ by $X$. If $\sigma_{1}(M)$ is the eigenvalue of the
$(p,q)$-Steklov problem (1.3) then

  * (1)

for $1<p<q<r\leq 2$ we have

$\sigma_{1}(M)\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}\frac{Vol(M)}{(Vol(\partial
M))^{r}}.$

  * (2)

For $1<p<q\leq 2$ and $r>2$ we get

$\sigma_{1}(M)\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}\frac{Vol(M)}{(Vol(\partial
M))^{r}}.$

  * (3)

For $1<p\leq 2$ and $2<q<r$ we get

$\sigma_{1}(M)\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}\frac{Vol(M)}{(Vol(\partial
M))^{r}}.$

  * (4)

For $2\leq p<q<r$ we get

$\sigma_{1}(M)\leq
N^{\frac{r}{2}-1}\left(n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}\frac{Vol(M)}{(Vol(\partial
M))^{r}}.$

In following theorem we extended the theorem 2.5 to estimates with higher
order mean curvatures.

######  Theorem 2.6.

Let $(M^{n},g,dv)$ be a compact connected and oriented Riemannian manifold
with nonempty boundary $\partial M$ and $1<p<q<r$. Assume that $(M^{n},g,dv)$
isometrically immersed into the Euclidean space $\mathbb{R}^{N}$ by $X$ and
let $T$ be a symmetric and divergence-free $(0,2)$-tensor on $\partial M$. If
$\sigma_{1}(M)$ is the eigenvalue of the $(p,q)$-Steklov problem (1.3) then

  * (1)

for $1<p<q<r\leq 2$ we have

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}{\rm
tr}(T)dv_{h}\right|^{r}$
$\displaystyle\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{T}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

  * (2)

For $1<p<q\leq 2$ and $r>2$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}{\rm
tr}(T)dv_{h}\right|^{r}$ $\displaystyle\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{T}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

  * (3)

For $1<p\leq 2$ and $2<q<r$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}{\rm
tr}(T)dv_{h}\right|^{r}$ $\displaystyle\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{T}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

  * (4)

For $2\leq p<q<r$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}{\rm
tr}(T)dv_{h}\right|^{r}$ $\displaystyle\leq
N^{\frac{r}{2}-1}\left(n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{T}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

Also, we have

######  Corollary 2.7.

Let $(M^{n},g,dv)$ be a compact connected and oriented Riemannian manifold
with nonempty boundary $\partial M$ and $1<p<q<r$. Assume that $(M^{n},g,dv)$
isometrically immersed into the Euclidean space $\mathbb{R}^{N}$ by $X$. If
$\sigma_{1}(M)$ is the eigenvalue of the $(p,q)$-Steklov problem (1.3)

  * (i)

If $N>n+1$, and $s\in\\{0,\cdots,n-1\\}$ is an even integer then we have

    * (1)

for $1<p<q<r\leq 2$ we have

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|{\bf H}_{s+1}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

    * (2)

For $1<p<q\leq 2$ and $r>2$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|{\bf H}_{s+1}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

    * (3)

For $1<p\leq 2$ and $2<q<r$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|{\bf H}_{s+1}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

    * (4)

For $2\leq p<q<r$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq
N^{\frac{r}{2}-1}\left(n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|{\bf H}_{s+1}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

  * (ii)

If $N=n+1$, and $s\in\\{0,\cdots,n-1\\}$ is an even integer then we have

    * (1)

for $1<p<q<r\leq 2$ we have

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{s}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

    * (2)

For $1<p<q\leq 2$ and $r>2$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{s}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

    * (3)

For $1<p\leq 2$ and $2<q<r$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{s}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

    * (4)

For $2\leq p<q<r$ we get

$\displaystyle\sigma_{1}(M)\left|\int_{\partial M}H_{s}dv_{h}\right|^{r}$
$\displaystyle\leq
N^{\frac{r}{2}-1}\left(n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H_{s}|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

In following we investigate the first eigenvalue of $(p,q)$-Steklov problem on
Riemannian products $\mathbb{R}\times M$ where $M$ is a complete Riemannian
manifold.

######  Theorem 2.8.

Let $2\leq p<q<r$ and $(M^{n},\bar{g})$ be a complete Riemannian manifold.
Consider $(\Sigma^{n},g)$ a closed oriented Riemannian manifold isometrically
immersed into the Riemannian product $(\mathbb{R}\times
M,\tilde{g}=dt^{2}\oplus\bar{g})$. Moreover, assume that $\Sigma$ is mean-
convex and bounds a domain $\Omega$ in $\mathbb{R}\times M$. Let
$\sigma_{1}(M)$ be the first eigenvalue of the weighted $p$-Steklov problem on
$\Omega$, then

$\sigma_{1}(\Omega)\leq
2\left(\frac{\kappa_{+}(\Sigma)|H|_{\infty}}{\mathop{\inf}\limits_{\Sigma}H}\right)^{\frac{r}{2}}\left(\frac{Vol(\Omega)}{Vol(\Sigma)}\right)^{1-\frac{r}{2}},$

where $\kappa_{+}(\Sigma)=\max\\{\kappa_{+}(x)|x\in M\\}$ with $\kappa_{+}$
the biggest principal curvature of $\Sigma$ at the point $x$.

##  3\.  Proof of main results

In this section we give the proof of our main results.

###### proof of theorem 2.1.

For coordinates functions $X^{k}$, by replacing if needed,
$|X^{i}|^{p-2}X^{i}$ by

$|X^{i}|^{p-2}X^{i}-\frac{\int_{\partial
M}|X^{i}|^{p-2}X^{i}d\mu_{h}}{Vol_{\mu_{h}}(\partial M)}$

we can assume without loss of generality,

$\int_{\partial M}|X^{i}|^{p-2}X^{i}d\mu_{h}=0$

for all $i\in\\{1,2,\cdots,N\\}$. Thus, we can use the coordinates functions
$X^{k}$ as test functions.

The case $1<p\leq 2$.
By the definition of $\lambda_{1}(M)$ we have

(3.1) $\lambda_{1}(M)\int_{\partial
M}\sum_{i=1}^{N}|X^{i}|^{p}d\mu_{h}\leq\int_{M}\sum_{i=1}^{N}|\nabla
X^{i}|^{p}d\mu_{g}.$

Since $p\leq 2$, we get
$\left(\sum_{i=1}^{N}|X^{i}|^{2}\right)^{\frac{1}{2}}\leq\left(\sum_{i=1}^{N}|X^{i}|^{p}\right)^{\frac{1}{p}}$,
then

(3.2)
$|X|^{p}=\left(\sum_{i=1}^{N}|X^{i}|^{2}\right)^{\frac{p}{2}}\leq\sum_{i=1}^{N}|X^{i}|^{p}.$

On the other hand, the concavity of $y\to y^{\frac{p}{2}}$ yields

(3.3) $\sum_{i=1}^{N}|\nabla X^{i}|^{p}=\sum_{i=1}^{N}\left(|\nabla
X^{i}|^{2}\right)^{\frac{p}{2}}\leq
N^{1-\frac{p}{2}}\left(\sum_{i=1}^{N}|\nabla
X^{i}|^{2}\right)^{\frac{p}{2}}=N^{1-\frac{p}{2}}n^{\frac{p}{2}},$

since we have $\sum_{i=1}^{N}|\nabla X^{i}|^{2}=n$ (see [14, Lemma 2.1]).
Hence, we obtain

(3.4) $\lambda_{1}(M)\int_{\partial M}|X|^{p}d\mu_{h}\leq
N^{1-\frac{p}{2}}n^{\frac{p}{2}}Vol_{\mu_{g}}(M).$

On the other hand, using Hölder inequality we have

$\displaystyle\int_{\partial M}\langle X,nH-\nabla f\rangle d\mu_{h}$
$\displaystyle\leq\left(\int_{\partial
M}|X|^{p}d\mu_{h}\right)^{\frac{1}{p}}\left(\int_{\partial M}|nH-\nabla
f|^{\frac{p}{p-1}}d\mu_{h}\right)^{\frac{p-1}{p}}$ $\displaystyle\leq$
$\displaystyle\left(\int_{\partial
M}|X|^{p}d\mu_{h}\right)^{\frac{1}{p}}\left(2^{\frac{1}{p-1}}\int_{\partial
M}(|nH|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}})d\mu_{h}\right)^{\frac{p-1}{p}}$

With multiply both sides of (3.4) by $\left(\int_{\partial
M}\left(|nH|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}$ and use the integral Hölder
inequality, we conclude that

(3.5) $\displaystyle 2^{-\frac{1}{p-1}}\lambda_{1}(M)\left|\int_{\partial
M}\langle X,nH-\nabla f\rangle d\mu_{h}\right|^{p}$ $\displaystyle\leq
N^{1-\frac{p}{2}}n^{\frac{p}{2}}\left(\int_{\partial
M}\left(|nH|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M).$

Now, using the Hsiung-Minkouski formula

(3.6) $\int_{\partial M}\left(\langle X,nH-\nabla f\rangle+n\right)d\mu_{h}=0$

we infer

(3.7) $\displaystyle
2^{-\frac{1}{p-1}}\lambda_{1}(M)\left(nVol_{\mu_{h}}(\partial M)\right)^{p}$
$\displaystyle\leq N^{1-\frac{p}{2}}n^{\frac{p}{2}}\left(\int_{\partial
M}\left(|nH|^{\frac{p}{p-1}}+|\nabla
f|^{\frac{p}{p-1}}\right)d\mu_{h}\right)^{p-1}Vol_{\mu_{g}}(M).$

The inequality is proven. First, assume that $f$ is constant, $H$ does not
vanish identically, and equality holds. Then the end of the proof is similar
to the proof of Roth [16] for the $p$-Steklov problem. Now, assume that $f$ is
not constant. If equality occurs, then the end of the proof is similar to the
proof of Roth [15].

The case $p\geq 2$.
It is straightforward that

(3.8) $\sum_{i=1}^{N}|\nabla X^{i}|^{p}=\sum_{i=1}^{N}\left(|\nabla
X^{i}|^{2}\right)^{\frac{p}{2}}\leq\left(\sum_{i=1}^{N}|\nabla
X^{i}|^{2}\right)^{\frac{p}{2}}=n^{\frac{p}{2}}.$

On the other hand, using the fact that $y\to y^{\frac{p}{2}}$ is convex, we
obtain

(3.9) $\sum_{i=1}^{N}|X^{i}|^{p}\geq
N^{1-\frac{p}{2}}\left(\sum_{i=1}^{N}|X^{i}|^{2}\right)^{\frac{p}{2}}=N^{1-\frac{p}{2}}|X|^{p}.$

Therefore, using the last two inequalities in the variational characterization
of $\lambda_{1}(M)$, we get

(3.10) $\lambda_{1}(M)\int_{\partial M}|X|^{p}d\mu_{h}\leq
N^{\frac{p}{2}-1}n^{\frac{p}{2}}Vol_{\mu_{g}}(M).$

The end of the proof is the same that in case $1<p\leq 2$. ∎

###### Proof of theorem 2.2.

Similar to the proof of Theorem 2.1, just enough to use the generalized
Hsiung-Minkowski formula (2.1) instead of the classical one. ∎

###### Proof of theorem 2.4.

Similar to [16], we assume that the function $t$ is a test function. Let
$v=\langle\partial_{t},\nu\rangle=\langle\tilde{\nabla}t,\nu\rangle$. Hence,
we have $\Delta t=-nHv$ and

(3.11) $\int_{\Sigma}|\nabla t|^{2}d\mu_{g}=\int_{\Sigma}nHvt\,d\mu_{g}.$

Also, since $\nabla v=-S\nabla t$ we have

(3.12) $\int_{\Sigma}\langle S\nabla t,\nabla t\rangle
d\mu_{g}=\int_{\Sigma}nHv^{2}\,d\mu_{g}.$

Then,

$\displaystyle n\mathop{\inf}\limits_{\Sigma}(H)\int_{\Sigma}v^{2}d\mu_{g}$
$\displaystyle\leq$
$\displaystyle\int_{\Sigma}nHv^{2}d\mu_{g}\leq\int_{\Sigma}\langle S\nabla
t,\nabla t\rangle d\mu_{g}\leq\kappa_{+}(\Sigma)\int_{\Sigma}|\nabla
t|^{2}d\mu_{g}$ $\displaystyle\leq$
$\displaystyle\kappa_{+}(\Sigma)\int_{\Sigma}nHvt\,d\mu_{g}\leq
n\kappa_{+}(\Sigma)|H|_{\infty}\int_{\Sigma}vt\,d\mu_{g}$ $\displaystyle\leq$
$\displaystyle
n\kappa_{+}(\Sigma)|H|_{\infty}\left(\int_{\Sigma}|t|^{p}\,d\mu_{g}\right)^{\frac{1}{p}}\left(\int_{\Sigma}|v|^{\frac{p}{p-1}}\,d\mu_{g}\right)^{\frac{p-1}{p}}.$

From the Hölder inequality we have

$\displaystyle\mathop{\inf}\limits_{\Sigma}(H)\left(\int_{\Sigma}|v|^{\frac{p}{p-1}}d\mu_{g}\right)^{\frac{2(p-1)}{p}}Vol_{\mu_{g}}(\Sigma)^{\frac{2-p}{p}}$
$\displaystyle\leq\mathop{\inf}\limits_{\Sigma}(H)\int_{\Sigma}v^{2}d\mu_{g}$
$\displaystyle\leq\kappa_{+}(\Sigma)|H|_{\infty}\left(\int_{\Sigma}|t|^{p}\,d\mu_{g}\right)^{\frac{1}{p}}\left(\int_{\Sigma}|v|^{\frac{p}{p-1}}\,d\mu_{g}\right)^{\frac{p-1}{p}},$

therefore,

(3.14)
$\frac{\left(\int_{\Sigma}|v|^{\frac{p}{p-1}}\,d\mu_{g}\right)^{\frac{p-1}{p}}}{\left(\int_{\Sigma}|t|^{p}\,d\mu_{g}\right)^{\frac{1}{p}}}\leq\frac{\kappa_{+}(\Sigma)|H|_{\infty}}{\mathop{\inf}\limits_{\Sigma}(H)}Vol_{\mu_{g}}(\Sigma)^{\frac{p-2}{p}}.$

On the other hand, from the variational characterization of $\lambda_{1}(M)$,
we have

(3.15)
$\lambda_{1}(\Omega)\int_{\Sigma}|t|^{p}d\mu_{g}\leq\int_{\Omega}|\tilde{\nabla}t|^{p}d\mu_{\tilde{g}}.$

Since $|\tilde{\nabla}t|=1$ and $\tilde{\Delta}t=0$ we have

(3.16)
$\int_{\Omega}|\tilde{\nabla}t|^{p}d\mu_{\tilde{g}}=Vol_{\mu_{\tilde{g}}}(\Omega)=\left(\int_{\Omega}|\tilde{\nabla}t|^{2}d\mu_{\tilde{g}}\right)^{\frac{p}{2}}Vol_{\mu_{\tilde{g}}}(\Omega)^{1-\frac{p}{2}}$

and

(3.17)
$\int_{\Omega}|\tilde{\nabla}t|^{2}d\mu_{\tilde{g}}=\int_{\Sigma}\langle
t\tilde{\nabla}t,\nu\rangle d\mu_{g}=\int_{\Sigma}tvd\mu_{g}.$

By Hölder inequality we get

(3.18)
$\int_{\Omega}|\tilde{\nabla}t|^{2}d\mu_{\tilde{g}}\leq\left(\int_{\Sigma}|t|^{p}\,d\mu_{g}\right)^{\frac{1}{p}}\left(\int_{\Sigma}|v|^{\frac{p}{p-1}}\,d\mu_{g}\right)^{\frac{p-1}{p}},$

thus, we obtain

(3.19)
$\lambda_{1}(\Omega)\leq\frac{\left(\int_{\Sigma}|v|^{\frac{p}{p-1}}\,d\mu_{g}\right)^{\frac{p-1}{2}}}{\left(\int_{\Sigma}|t|^{p}\,d\mu_{g}\right)^{\frac{1}{2}}}Vol_{\mu_{g}}(\Omega)^{1-\frac{p}{2}}.$

Therefore, substituting (3.14) in (3.19), we complete the proof of theorem. ∎

###### Proof of theorem 2.5.

For coordinates functions $X^{k}$, by replacing if needed,
$|X^{i}|^{r-2}X^{i}$ by

$|X^{i}|^{r-2}X^{i}-\frac{\int_{\partial
M}|X^{i}|^{r-2}X^{i}d\mu_{h}}{Vol_{\mu_{h}}(\partial M)}$

we can assume without loss of generality,

$\int_{\partial M}|X^{i}|^{r-2}X^{i}d\mu_{h}=0$

for all $i\in\\{1,2,\cdots,N\\}$. Thus, we can use the coordinates functions
$X^{k}$ as test functions.

  * (1)

The case $1<p<q<r\leq 2$.
By the definition of $\sigma_{1}(M)$ we have

(3.20) $\sigma_{1}(M)\int_{\partial
M}\sum_{i=1}^{N}|X^{i}|^{r}dv_{h}\leq\int_{M}\sum_{i=1}^{N}\left(|\nabla
X^{i}|^{p}+|\nabla X^{i}|^{q}\right)dv_{g}.$

Since $r\leq 2$, we get
$\left(\sum_{i=1}^{N}|X^{i}|^{2}\right)^{\frac{1}{2}}\leq\left(\sum_{i=1}^{N}|X^{i}|^{r}\right)^{\frac{1}{r}}$,
then

(3.21)
$|X|^{r}=\left(\sum_{i=1}^{N}|X^{i}|^{2}\right)^{\frac{r}{2}}\leq\sum_{i=1}^{N}|X^{i}|^{r}.$

On the other hand, the concavity of $y\to y^{\frac{p}{2}}$ yields

(3.22) $\sum_{i=1}^{N}|\nabla X^{i}|^{p}=\sum_{i=1}^{N}\left(|\nabla
X^{i}|^{2}\right)^{\frac{p}{2}}\leq
N^{1-\frac{p}{2}}\left(\sum_{i=1}^{N}|\nabla
X^{i}|^{2}\right)^{\frac{p}{2}}=N^{1-\frac{p}{2}}n^{\frac{p}{2}}.$

Similarly,

$\sum_{i=1}^{N}|\nabla X^{i}|^{q}\leq N^{1-\frac{q}{2}}n^{\frac{q}{2}},$

Hence, we obtain

(3.23) $\sigma_{1}(M)\int_{\partial
M}|X|^{r}dv_{h}\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)Vol(M),$

with multiply by $\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}$ and use the integral
Hölder inequality, we conclude that

(3.24) $\sigma_{1}(M)\left|\int_{\partial M}\langle X,H\rangle
dv_{h}\right|^{r}\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

Now, using the Hsiung-Minkouski formula

(3.25) $\int_{\partial M}\left(\langle X,H\rangle+1\right)dv_{h}=0$

we infer

(3.26) $\sigma_{1}(M)\left(Vol(\partial
M)\right)^{r}\leq\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)\left(\int_{\partial
M}\left(|H|^{\frac{r}{r-1}}\right)dv_{h}\right)^{r-1}Vol(M).$

The case $1<p<q\leq 2$ and $r>2$.
It is straightforward that

(3.27) $\sum_{i=1}^{N}\left(|\nabla X^{i}|^{p}+|\nabla X^{i}|^{q}\right)\leq
N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}.$

On the other hand, using the fact that $y\to y^{\frac{r}{2}}$ is convex, we
obtain

(3.28) $\sum_{i=1}^{N}|X^{i}|^{r}\geq
N^{1-\frac{r}{2}}\left(\sum_{i=1}^{N}|X^{i}|^{2}\right)^{\frac{r}{2}}=N^{1-\frac{r}{2}}|X|^{r}.$

Therefore, using the last two inequalities in the variational characterization
of $\sigma_{1}(M)$, we get

(3.29) $\sigma_{1}(M)\int_{\partial M}|X|^{r}dv_{h}\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+N^{1-\frac{q}{2}}n^{\frac{q}{2}}\right)Vol(M).$

The case $1<p\leq 2$ and $2<q<r$.
It is straightforward that

(3.30) $\sum_{i=1}^{N}|\nabla X^{i}|^{p}\leq
N^{1-\frac{p}{2}}n^{\frac{p}{2}},$

and

(3.31) $\sum_{i=1}^{N}|\nabla X^{i}|^{q}=\sum_{i=1}^{N}\left(|\nabla
X^{i}|^{2}\right)^{\frac{q}{2}}\leq\left(\sum_{i=1}^{N}|\nabla
X^{i}|^{2}\right)^{\frac{q}{2}}=n^{\frac{q}{2}}.$

On the other hand, using the fact that $y\to y^{\frac{r}{2}}$ is convex, we
obtain

(3.32) $\sum_{i=1}^{N}|X^{i}|^{r}\geq N^{1-\frac{r}{2}}|X|^{r}.$

Therefore, using the last two inequalities in the variational characterization
of $\sigma_{1}(M)$, we get

(3.33) $\sigma_{1}(M)\int_{\partial M}|X|^{r}dv_{h}\leq
N^{\frac{r}{2}-1}\left(N^{1-\frac{p}{2}}n^{\frac{p}{2}}+n^{\frac{q}{2}}\right)Vol(M).$

The case $2<p<q<r$.
It is straightforward that

(3.34) $\sum_{i=1}^{N}(\left(|\nabla X^{i}|^{p}+|\nabla X^{i}|^{q}\right)\leq
n^{\frac{p}{2}}+n^{\frac{q}{2}}.$

On the other hand, using the fact that $y\to y^{\frac{r}{2}}$ is convex, we
obtain

(3.35) $\sum_{i=1}^{N}|X^{i}|^{r}\geq N^{1-\frac{r}{2}}|X|^{r}.$

Therefore, using the last two inequalities in the variational characterization
of $\sigma_{1}(M)$, we get

(3.36) $\sigma_{1}(M)\int_{\partial M}|X|^{r}dv_{h}\leq
N^{\frac{r}{2}-1}(n^{\frac{p}{2}}+n^{\frac{q}{2}})Vol(M).$

∎

###### Proof of theorem 2.5.

Similar to the proof of Theorem 2.5, just enough to use the generalized
Hsiung-Minkowski formula (2.1) instead of the classical one. ∎

###### Proof of the Theorem 2.8.

Similar to [16], we assume that the function $t$ is a test function. Let
$v=\langle\partial_{t},\nu\rangle=\langle\tilde{\nabla}t,\nu\rangle$. Hence,
we have $\Delta t=-nHv$ and

(3.37) $\int_{\Sigma}|\nabla t|^{2}dv_{g}=\int_{\Sigma}nHvt\,dv_{g}.$

Also, since $\nabla v=-S\nabla t$ we have

(3.38) $\int_{\Sigma}\langle S\nabla t,\nabla t\rangle
dv_{g}=\int_{\Sigma}nHv^{2}\,dv_{g}.$

Then,

$\displaystyle n\mathop{\inf}\limits_{\Sigma}(H)\int_{\Sigma}v^{2}dv_{g}$
$\displaystyle\leq$
$\displaystyle\int_{\Sigma}nHv^{2}d\mu_{g}\leq\int_{\Sigma}\langle S\nabla
t,\nabla t\rangle dv_{g}\leq\kappa_{+}(\Sigma)\int_{\Sigma}|\nabla
t|^{2}dv_{g}$ $\displaystyle\leq$
$\displaystyle\kappa_{+}(\Sigma)\int_{\Sigma}nHvt\,d\mu_{g}\leq
n\kappa_{+}(\Sigma)|H|_{\infty}\int_{\Sigma}vt\,dv_{g}$ $\displaystyle\leq$
$\displaystyle
n\kappa_{+}(\Sigma)|H|_{\infty}\left(\int_{\Sigma}|t|^{r}\,dv_{g}\right)^{\frac{1}{r}}\left(\int_{\Sigma}|v|^{\frac{r}{r-1}}\,dv_{g}\right)^{\frac{r-1}{r}}.$

From the Hölder inequality we have

$\displaystyle\mathop{\inf}\limits_{\Sigma}(H)\left(\int_{\Sigma}|v|^{\frac{r}{r-1}}d\mu_{g}\right)^{\frac{2(r-1)}{r}}Vol_{v_{g}}(\Sigma)^{\frac{2-r}{r}}$
$\displaystyle\leq\mathop{\inf}\limits_{\Sigma}(H)\int_{\Sigma}v^{2}dv_{g}$
$\displaystyle\leq\kappa_{+}(\Sigma)|H|_{\infty}\left(\int_{\Sigma}|t|^{r}\,dv_{g}\right)^{\frac{1}{r}}\left(\int_{\Sigma}|v|^{\frac{r}{r-1}}\,dv_{g}\right)^{\frac{r-1}{r}},$

therefore,

(3.40)
$\frac{\left(\int_{\Sigma}|v|^{\frac{r}{r-1}}\,dv_{g}\right)^{\frac{r-1}{r}}}{\left(\int_{\Sigma}|t|^{r}\,dv_{g}\right)^{\frac{1}{r}}}\leq\frac{\kappa_{+}(\Sigma)|H|_{\infty}}{\mathop{\inf}\limits_{\Sigma}(H)}Vol(\Sigma)^{\frac{r-2}{r}}.$

On the other hand, from the variational characterization of $\sigma_{1}(M)$,
we have

(3.41)
$\sigma_{1}(\Omega)\int_{\Sigma}|t|^{r}dv_{g}\leq\int_{\Omega}\left(|\tilde{\nabla}t|^{p}+|\tilde{\nabla}t|^{q}\right)dv_{\tilde{g}}.$

Since $|\tilde{\nabla}t|=1$ and $\tilde{\Delta}t=0$ we have

(3.42)
$\int_{\Omega}|\tilde{\nabla}t|^{p}dv_{\tilde{g}}=Vol(\Omega)=\left(\int_{\Omega}|\tilde{\nabla}t|^{2}dv_{\tilde{g}}\right)^{\frac{r}{2}}Vol(\Omega)^{1-\frac{r}{2}}$

and

(3.43) $\int_{\Omega}|\tilde{\nabla}t|^{2}dv_{\tilde{g}}=\int_{\Sigma}\langle
t\tilde{\nabla}t,\nu\rangle dv_{g}=\int_{\Sigma}tvdv_{g}.$

By Hölder inequality we get

(3.44)
$\int_{\Omega}|\tilde{\nabla}t|^{2}dv_{\tilde{g}}\leq\left(\int_{\Sigma}|t|^{r}\,dv_{g}\right)^{\frac{1}{r}}\left(\int_{\Sigma}|v|^{\frac{r}{r-1}}\,dv_{g}\right)^{\frac{r-1}{r}},$

thus, we obtain

(3.45) $\sigma_{1}(\Omega)\leq
2\frac{\left(\int_{\Sigma}|v|^{\frac{r}{r-1}}\,dv_{g}\right)^{\frac{r-1}{2}}}{\left(\int_{\Sigma}|t|^{r}\,dv_{g}\right)^{\frac{1}{2}}}Vol(\Omega)^{1-\frac{r}{2}}.$

Therefore, substituting (3.40) in (3.45), we complete the proof of theorem. ∎

Acknowledgements
We thank the anonymous referees for their valuable comments, which improved
the paper.
Funding
Not available.
Availability of data and materials
Not applicable.
Competing interests
The authors have declared that no competing interests exist.
Authors’ information
Department of Pure Mathematics, Faculty of Science, Imam Khomeini
International University, Qazvin, Iran. E-mails<EMAIL_ADDRESS>
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.

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∎

11institutetext: Nathan Adelgren 22institutetext: Andlinger Center for Energy
and the Environment
Princeton University
Princeton, NJ, 08544
22email<EMAIL_ADDRESS>ORCHID: https://orcid.org/0000-0003-3836-9324

# A Unified Tool for Solving Uni-Parametric Linear Programs, Convex Quadratic
Programs, and Linear Complementarity Problems

Nathan Adelgren

(Received: date / Accepted: date)

###### Abstract

We introduce a new technique for solving uni-parametric versions of linear
programs, convex quadratic programs, and linear complementarity problems in
which a single parameter is permitted to be present in any of the input data.
We demonstrate the use of our method on a small, motivating example and
present the results of a small number of computational tests demonstrating its
utility for larger scale problems.

###### Keywords:

parametric optimization linear complementarity linear programming quadratic
programming

###### Acknowledgements.

The author would like to thank Jacob Adelgren for offering helpful advice and
feedback regarding the implementation.

##  1 Introduction

In this work we consider the uni-parametric form of the Linear Complementarity
Problem (LCP) in which all input data is permitted to be dependent on a single
parameter $\theta\in\Theta$, where

$\Theta:=\\{\theta\in\hbox{$\mathbb{R}$}:\alpha\leq\theta\leq\beta\text{ and
}\alpha,\beta\in\hbox{$\mathbb{R}$}\\}$ (1)

is a connected interval in $\mathbb{R}$ that represents the set of
“attainable” values for $\theta$. This problem is referred to as the uni-
parametric (or single-parametric) Linear Complementarity Problem (upLCP). Let
$\mathscr{A}=\\{\mu\theta+\sigma:\mu,\sigma\in\hbox{$\mathbb{R}$}\\}$, the set
of affine functions of $\theta$. Then upLCP is as follows:

> Given $M(\theta)\in\mathscr{A}^{h\times h}$ and
> $q(\theta)\in\mathscr{A}^{h}$, for each $\theta\in\Theta$ find vectors
> $w(\theta)$ and $z(\theta)$ that satisfy the system
>
> $\begin{array}[]{c}w-M(\theta)z=q(\theta)\\\\[2.84526pt]
> w^{\top}z=0\\\\[2.84526pt] w,z\geq 0\end{array}$ (2)
>
> or show that no such vectors exist.

upLCP is said to be _feasible at $\theta$_ if there exist $w(\theta)$ and
$z(\theta)$ that satisfy System (2) and _infeasible at $\theta$_ otherwise.
Similarly, upLCP is said to be _feasible_ if there exists a
$\hat{\theta}\in\Theta$ at which upLCP is feasible, and _infeasible_
otherwise.

Now, recognize that $\Theta$ must be an infinite set, otherwise upLCP reduces
to LCP. Hence, it is not possible to determine a solution to System (2) for
each $\theta\in\Theta$ individually. Instead, upLCP is solved by partitioning
the interval $\Theta$ into a set of _invariancy intervals_. As the name
“invariancy intervals” suggests, within each of these intervals the
representation of the solution vectors $w$ and $z$ as functions of $\theta$ is
invariant. The methods we propose solve upLCP whenever the following
assumptions are met.

###### Assumption 1

The matrix $M(\theta)$ is sufficient for all $\theta\in\Theta$.

###### Assumption 2

System (2) is feasible for all $\theta\in\Theta$.

We point any reader interested in background information on LCP, in particular
the definition of sufficient matrices, to the work of Cottle et al. [3].
Additionally, we note that Assumption 2 is not extremely restrictive. For
example, the claim of Assumption 2 is satisfied when upLCP results from the
reformulation of a biobjective linear program or biobjective quadratic program
that has been scalarized using the weighted sum approach (see [5], for
example) and $\Theta$ is taken to be $[0,1]$.

To our knowledge, the only other works in which solution procedures are
proposed for upLCP having the general form of System (2) are those of Väliaho
[12] and Chakraborty et al. [2]. Section 6 of the former work describes
various restrictions that can be placed on the structure of $M(\theta)$ in
order to guarantee finite execution of the procedures presented therein, but
all are more restrictive than our Assumption 1. Similarly, the methods
presented in the latter work require that $M(\theta)$ be a $P$-matrix for all
$\theta\in\Theta$, which is again more restrictive than our Assumption 1.
Solution methodology for the multiparametric counterpart to System (2), i.e.,
the case in which $\theta\in\hbox{$\mathbb{R}$}^{k}$ for $k>1$, is presented
in [1]. However, the techniques presented therein are not directly applicable
for scalar $\theta$. This motivates our current work, the remainder of which
is organized as follows. The proposed algorithm for solving upLCP is presented
in Section 2. We demonstrate the use of our proposed methodology on a
motivating example in Section 3. In Section 4 we discuss some classes of
optimization problems that can be reformulated as upLCP and can therefore be
solved using the methods presented in this work. Section 5 contains a brief
description of our implementation and the results of a small number of
computational tests. Finally, we give concluding remarks in Section 6.

##  2 Proposed Algorithm

We begin this section with a brief set of definitions. Given an instance of
upLCP, as presented in System (2), we define the matrix
$G(\theta):=\left[\begin{matrix}I&-M(\theta)\end{matrix}\right]$ and the index
set $\mathcal{E}:=\\{1,\dots,2h\\}$. Then a set
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}\subset\mathcal{E}$
is called a _basis_ if
$|\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}|=h$.
Moreover, a basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$ is
_complementary_ if
$\left|\\{i,i+h\\}\cap\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}\right|=1$
for each $i\in\\{1,\dots,h\\}$. Then, given a complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$, its
associated _invariancy interval_ is the set

$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}:=\left\\{\theta\in\Theta:G(\theta)^{-1}_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}q(\theta)\geq
0\right\\},$ (3)

where
$G(\theta)^{-1}_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
denotes the matrix comprised of the columns of $G(\theta)^{-1}$ whose indices
are in $\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$. We
note here that for a given complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$,
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
may be the union of disjoint intervals in $\Theta$. To see this, recognize
that: (i) for any complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$, the system
$G(\theta)^{-1}_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}q(\theta)\geq
0$ can be equivalently written as
$\dfrac{Adj\left(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}\right)}{det\left(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}\right)}q(\theta)\geq
0$, where $Adj(\cdot)$ and $det(\cdot)$ represent the matrix adjoint and
determinant, respectively; and (ii) both the adjoint and determinant of a
given matrix can be represented as polynomials of the elements of the matrix.
Hence, for each complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$,
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
is a possibly nonconvex subset of $\Theta$ defined by a set of rational
inequalities in $\theta$. Fortunately, this structure can be somewhat
improved. By Lemma 2.1 of [1] we know that, under Assumption 1, the sign of
$det\left(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}\right)$
is invariant over $\Theta$ for any complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$. Thus,
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
can be represented as

$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}:=\left\\{\theta\in\Theta:s_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}Adj(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}})q(\theta)\geq
0\right\\},$ (4)

where $s_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
represents the sign of
$det\left(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}\right)$
over $\Theta$. Note that in Equation 4,
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
is given by a system of inequalities that are polynomial in $\theta$. We now
provide Algorithm 1 in which we present a technique for solving upLCP.

Algorithm 1 Solve_upLCP($\Theta$) – Partition the parameter space $\Theta$.
Input: The set $\Theta$ as defined in Equation (1) for an instance of upLCP.
Output: A set $\mathcal{P}$ specifying a partition of $\Theta$. Each
$\mathcal{I}\in\mathcal{P}$ is a tuple of the form
$(\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*},\alpha^{*},\beta^{*})$
where $\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}$
represents a complementary basis and
$\alpha^{*},\beta^{*}\in\hbox{$\mathbb{R}$}$ specify an interval
$[\alpha^{*},\beta^{*}]\subseteq\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
such that $w(\theta)$ and $z(\theta)$ satisfy System (2) for all
$\theta\in[\alpha^{*},\beta^{*}]$.

1:Let $\mathcal{S}=\\{\Theta\\}$ and $\mathcal{P}=\emptyset$.

2:while $\mathcal{S}\neq\emptyset$ do select
$[\alpha^{\prime},\beta^{\prime}]$ from $\mathcal{S}$.

3: Set $\theta^{*}=\frac{\alpha^{\prime}+\beta^{\prime}}{2}$ and compute a
complemetary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}$ such
that
$\theta^{*}\in\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$.

4: Set
$\alpha^{*},\beta^{*}=\textsc{Get\\_Extremes}(\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*},\theta^{*},\alpha^{\prime},\beta^{\prime})$
and add
$(\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*},\alpha^{*},\beta^{*})$
to $\mathcal{P}$.

5: if $\alpha^{\prime}<\alpha^{*}$ then add $[\alpha^{\prime},\alpha^{*}]$ to
$\mathcal{S}$.

6: if $\beta^{\prime}>\beta^{*}$ then add $[\beta^{*},\beta^{\prime}]$ to
$\mathcal{S}$.

7:Return $\mathcal{P}$.

The majority of the work done in Algorithm 1 is the processing of a set
$\mathcal{S}$ of intervals using the while loop contained in lines 2–6. On
line 3, the midpoint $\theta^{*}$ of the current interval is computed and a
complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}$ is
sought for which $\theta^{*}$ is contained within
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$.
Such a complementary basis can be found by fixing $\theta$ to $\theta^{*}$ and
solving the resulting non-parametric LCP using, for example, the criss-cross
method of den Hertog et al. [4]. We note that the criss-cross method is a
particularly good choice in this case as it is guaranteed to solve the
resulting non-parametric LCP under Assumption 1. On line 4, the subroutine
Get_Extremes is used to compute $\alpha^{*},\beta^{*}\in\hbox{$\mathbb{R}$}$
that define a subinterval $[\alpha^{*},\beta^{*}]$ of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$
that contains $\theta^{*}$ and should be included in the final partition of
$\Theta$. Finally, lines 5–6 identify connected subintervals of
$[\alpha^{\prime},\beta^{\prime}]$ over which solutions to System (2) have yet
to be computed, if any exist, and those subintervals are added to the set
$\mathcal{S}$.

We pause now to discuss the subroutine Get_Extremes in more detail. There are
many ways to implement such a routine, but the strategies employed in our
implementation are presented in Algorithm 2.

Algorithm 2
Get_Extremes($\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*},\theta^{*},\alpha^{\prime},\beta^{\prime}$)
– Compute extreme values of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$.
Input: A complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}$,
$\theta^{*}\in\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$,
and $\alpha^{\prime},\beta^{\prime}\in\hbox{$\mathbb{R}$}$.
Output: $\alpha^{*},\beta^{*}\in\hbox{$\mathbb{R}$}$ that specify an interval
$[\alpha^{*},\beta^{*}]\subseteq\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$
such that $\theta^{*}\in[\alpha^{*},\beta^{*}]$ and $w(\theta)$ and
$z(\theta)$ satisfy System (2) for all $\theta\in[\alpha^{*},\beta^{*}]$.

1:Set $\alpha^{*}=\alpha^{\prime}$ and $\beta^{*}=\beta^{\prime}$.

2:for
$i\in\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}$ do

3: if
$degree\left(\left(Adj(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}})\right)_{i\bullet}q(\theta)\right)>0$
then

4: Let
$\mathcal{R}=\textsc{Get\\_Real\\_Roots}\left(\left(Adj(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}})\right)_{i\bullet}q(\theta)\right)$.

5: for $r\in\mathcal{R}$ do

6: if $multiplicity(r)$ is odd then

7: if $r\in(\alpha^{*},\theta^{*})$ then set $\alpha^{*}=r$.

8: else if $r\in(\theta^{*},\beta^{*})$ then set $\beta^{*}=r$.

9: else if $r==\alpha^{*}$ OR $r==\beta^{*}$ then

10: if
$\left.\frac{d}{d\theta}\left(s_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}\left(Adj(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}})\right)_{i\bullet}q(\theta)\right)\right|_{\theta=\theta^{*}}>0$
then set $\alpha^{*}=r$.

11: else set $\beta^{*}=r$.

12:Return $\alpha^{*},\beta^{*}$.

On a high-level, the work done in Algorithm 2 seeks to find the largest
connected interval that is a subset of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$
and contains $\theta^{*}$. To do this, we examine the roots of the nonconstant
polynomial functions that serve as the boundaries of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$
(lines 2–4). We note that any polynomial root finder capable of determining
root multiplicity can be used for the subroutine Get_Real_Roots (see, for
example, [10]). As roots of even multiplicity cannot be restrictive, we need
only consider roots of odd multiplicity (lines 5–6). Since we have
$\theta^{*}\in\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$,
we know that if no root occurs directly at $\theta^{*}$, then the endpoints of
the interval of interest are: (i) the root closest to $\theta^{*}$ on the
left, and (ii) the root closest to $\theta^{*}$ on the right (lines 7–8). If a
root does occur directly at $\theta^{*}$, however, then $\theta^{*}$ serves as
one of the endpoints of the interval of interest, and we must determine the
direction in which the function associated with that root increases in order
to determine which endpoint it is (lines 9–11). For this purpose, we use the
sign of the derivative of the associated function, evaluated at $\theta^{*}$.
Since the defining constraints of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$
are given as greater-than-or-equal-to constraints, we know that if the
aforementioned sign is positive, $\theta^{*}$ is the left endpoint of the
interval of interest. Otherwise, it is the right endpoint. We note that it is
possible that two functions defining boundaries of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{*}}$
each have roots at $\theta^{*}$ and, moreover, the derivatives of the two
functions, evaluated at $\theta^{*}$, may have opposite signs. In this case,
the interval of interest reduces to a singleton at $\theta^{*}$ and need not
be included in the partition of $\Theta$. Hence, such intervals can be
rejected upon discovery or removed from the partition in a post-processing
phase.

We finish this section by stating a small number of theoretical results
related to the methods proposed in Algorithms 1–2. For the sake of space, we
do not include proofs for these results – though, the proofs are not
challenging – and, as such, we state them as observations.

###### Observation 1

For a given complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$, the final
partition $\mathcal{P}$ of $\Theta$ may contain more than one connected
subinterval of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$.

###### Observation 2

Under Assumption 1, for a given complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}$, the number
of connected subintervals of
$\mathcal{II}_{\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}}$
present in the final partition $\mathcal{P}$ of $\Theta$ can be at most $n-h$,
where $n$ is the number of unique roots of odd multiplicity of the functions
$Adj(G(\theta)_{\bullet\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}})q(\theta)$
and $h=|\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}|$.

###### Observation 3

Under Assumptions 1–2, Algorithm 1 will complete in a finite number of
iterations.

##  3 Motivating Example

At the start of Chapter 1 of [8], the authors demonstrate that an instance of
LCP with $M=\left[\begin{array}[]{rr}2&-1\\\ 1&3\\\ \end{array}\right]$ will
have relatively nice properties. We use this as a starting point and consider
the following instance of mpLCP:

$\begin{array}[]{c}w-\left[\begin{array}[]{cc}2&-1+\theta_{2}\\\
1-\theta_{1}&3\\\ \end{array}\right]z=\left[\begin{array}[]{c}1-\theta_{1}\\\
-2+3\theta_{2}\\\ \end{array}\right]\\\\[2.84526pt] w^{\top}z=0\\\\[2.84526pt]
w,z\geq 0\end{array}$ (5)

We assume here that $(\theta_{1},\theta_{2})\in[-2,2]^{2}$ and note that it is
straightforward to verify that
$M(\theta)=\left[\begin{array}[]{cc}2&-1+\theta_{2}\\\ 1-\theta_{1}&3\\\
\end{array}\right]$ is sufficient for all
$(\theta_{1},\theta_{2})\in[-2,2]^{2}$ using Theorem 4.3 of [13]. We note that
beginning with a mpLCP rather than a upLCP allows us to visualize some of the
challenges associated with upLCP. We compute the parametric solution to System
(5) using the techniques proposed in [1]. The final partition of $\Theta$ is
depicted in Figure 1 and the complementary bases associated with each region,
as well as the parametric solutions associated with each basis, are provided
in Table 1.

Figure 1: Regions over which the computed parametric solutions are valid. Table 1: Parametric solutions to System (5). Region | Basic Variables and
---|---
| Associated Values
I | $w_{1}=1-\theta_{1}$
| $w_{2}=3\theta_{2}-2$
II | $w_{1}=-\theta_{1}-\theta_{2}^{2}+\frac{5}{3}\theta_{2}+\frac{1}{3}$
| $z_{2}=\frac{2}{3}-\theta_{2}$
III | $z_{1}=\frac{3\theta_{1}+3\theta_{2}^{2}-5\theta_{2}-1}{\theta_{1}\theta_{2}-\theta_{1}-\theta_{2}+7}$
|
$z_{2}=\frac{\theta_{1}^{2}-2\theta_{1}-6\theta_{2}+5}{\theta_{1}\theta_{2}-\theta_{1}-\theta_{2}+7}$
IV | $z_{1}=\frac{1}{2}\theta_{1}-\frac{1}{2}$
| $w_{2}=-\frac{1}{2}\theta_{1}^{2}+\theta_{1}+3\theta_{2}-\frac{5}{2}$

We note that, in an abuse of notation, we identify a basis in Table 1 by its
associated basic variables, rather than the elements of $\mathcal{E}$ that it
contains. Additionally, variables having zero value are omitted from Table 1.
The same convention for identifying complementary bases and omitting variables
with zero value is used throughout the remainder of this work.

We now reduce System (5) to an instance of upLCP by fixing
$\theta_{2}=\frac{1}{2}\theta_{1}$ (depicted in Figure 1 as a dotted black
line) and letting $\Theta=[-2,2]$. In observing Figure 1, we note that this
reduction will result in a realization of the scenario outlined in Observation
1. Namely, our final partition of $\Theta$ will contain more than one interval
associated with a single complementary basis. We believe that situations like
this have been one of the primary barriers to the development of methods for
solving upLCP as specified by System (2). We now demonstrate how the
techniques outlined in Algorithms 1–2 can be used to solve this instance of
upLCP. We begin by setting $\mathcal{S}=[-2,2]$ and then iterate through the
while loop on line 2 of Algorithm 1. Note that in the following discussion we
process elements from the set $\mathcal{S}$ using a last-in-first-out
strategy.

Iteration 1:

> Here $[\alpha^{\prime},\beta^{\prime}]=[-2,2]$ and we set $\theta^{*}=0$.
> Using the criss-cross method, we find that an optimal basis at $\theta^{*}$
> is
> $\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{1}=\\{w_{1}=-\frac{1}{4}\theta_{1}^{2}-\frac{1}{6}\theta_{1}+\frac{1}{3},z_{2}=-\frac{1}{2}\theta_{1}+\frac{2}{3}\\}$.
> We now enter Algorithm 2 via the routine Get_Extremes. The real roots
> associated with the function representing $w_{1}$ are approximately $-1.535$
> and $0.869$, whereas the function associated with $z_{2}$ has only one real
> root, approximately $1.333$. Of these roots, $-1.535$ is closest to
> $\theta^{*}$ on the left and $0.869$ is closest on the right. As such, these
> values are returned from Get_Extremes as $\alpha^{*}$ and $\beta^{*}$,
> respectively. On lines 5–6 we now add $[-2,-1.535]$ and $[0.869,2]$ to
> $\mathcal{S}$ and proceed to Iteration 2.

Iteration 2:

> For this and subsequent iterations, we omit details that are analogous to
> those of Iteration 1. We note that here we have
> $[\alpha^{\prime},\beta^{\prime}]=[0.869,2]$ and thus $\theta^{*}=1.4345$.
> An optimal basis at this point is
> $\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{2}=\\{z_{1}=\frac{1}{2}\theta_{1}-\frac{1}{2},w_{2}=-\frac{1}{2}\theta_{1}^{2}+\frac{5}{2}\theta_{1}-\frac{5}{2}\\}$
> and respective sets of approximate real roots for these functions are
> $\\{1\\}$ and $\\{1.382,3.618\\}$. Hence, we set $\alpha^{*}=1.382$ and
> leave $\beta^{*}$ as $2$. The only new interval added to $\mathcal{S}$ is
> then $[0.869,1.382]$.

Iteration 3:

> We now have $[\alpha^{\prime},\beta^{\prime}]=[0.869,1.382]$ and
> $\theta^{*}=1.1255$. An optimal basis here is
> $\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{3}=\\{z_{1}=\frac{-3\theta_{1}^{2}-2\theta_{1}+4}{-2\theta_{1}^{2}+6\theta_{1}-28},z_{2}=\frac{-2\theta_{1}^{2}+10\theta_{1}-10}{-\theta_{1}^{2}+3\theta_{1}-14}\\}$
> and respective sets of approximate real roots for these functions are
> $\\{0.869,-1.535\\}$ and $\\{1.382,3.618\\}$. As a result, $\alpha^{*}$ and
> $\beta^{*}$ are left as $0.869$ and $1.382$, respectively, and no intervals
> are added to $\mathcal{S}$.

Iteration 4:

> Finally, we have $[\alpha^{\prime},\beta^{\prime}]=[-2,-1.535]$ and
> $\theta^{*}=-1.7675$. An optimal basis at this point is
> $\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{3}$. Thus,
> using the previously computed sets of approximate real roots, we see that
> $\alpha^{*}$ and $\beta^{*}$ are left as $-2$ and $-1.535$, respectively,
> and no intervals are added to $\mathcal{S}$.

We have now completed the execution of Algorithm 1. The final partition of
$\Theta$ consists of the four intervals $[-1.535,0.869]$, $[1.382,2]$,
$[0.869,1.382]$, and $[-2,-1.535]$. The parametric solutions that are valid
over these intervals are those given in the respective descriptions of
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{1},\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{2},\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{3}$
and (again)
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}_{3}$ above.
It is straightforward to verify that this solution matches the one given in
Table 1 when $\theta_{2}$ is fixed to $\frac{1}{2}\theta_{1}$. Recognize,
though, that the representation of $w_{2}$ in the solution associated with
Region I in Table 1 reduces to $w_{2}=\frac{3}{2}\theta_{1}-2$. Hence, as
there exists no $\theta_{1}\in[-2,2]$ for which both $w_{1}=1-\theta_{1}$ and
$w_{2}=\frac{3}{2}\theta_{1}-2$ are non-negative, the complementary basis
$\mathbin{\ThisStyle{\scalebox{1.15}{$\SavedStyle\mathpzc{B}$}}}^{\prime}=\\{w_{1},w_{2}\\}$
is not associated with any interval in the final partition of $\Theta$ when
solving the reduced instance of upLCP.

##  4 Applicability to Other Classes of Problems

It is well known that LCP arises naturally as the system resulting from
applying the Karush-Kuhn-Tucker (KKT) optimality conditions to quadratic
programs (QP). Moreover, as a solution to the KKT system is guaranteed to give
an optimal solution for any convex QP, the methods presented herein are
directly applicable to the uni-parametric form of QP given by

$\displaystyle\min_{x}\quad$
$\displaystyle\frac{1}{2}x^{\top}Q(\theta)x+c(\theta)^{\top}x$ s.t.
$\displaystyle A(\theta)x\leq b(\theta)$ (6) $\displaystyle x\geq 0$
$\displaystyle\theta\in\Theta$

whenever $Q(\theta)$ is convex for all $\theta\in\Theta$, with $\Theta$
defined as in Equation (1). We note that this implies that the methods
presented herein are also applicable to: (i) biobjective problems having
linear and/or convex quadratic objectives and linear constraints (via
weighted-sum scalarization), and (ii) uni-parametric linear programs (by
replacing $Q(\theta)$ with the zero matrix in Problem (6)). To our knowledge,
the only other works that propose solution strategies for Problem (6) under
similar assumptions to those that are implied by our Assumption 1 are those of
Ritter [9], Väliaho [14], and Jonker et al. [7].

##  5 Computational Results

Using the Python programming language, we develop an implementation of the
methods described in Algorithms 1–2 in which the processing of set
$\mathcal{S}$ on line 2 of Algorithm 1 is performed in parallel, when desired.
Interested readers may obtain the code from
https://github.com/Nadelgren/upLCP_solver. The code is written so that the
user may specify any one of three types of problems: (i) upLCP, (ii) upQP, or
(iii) upLP. Problems given in the form of upQP or upLP are converted to upLCP
prior to utilization of the methods proposed herein, but the computed solution
is ultimately provided in the context of the originally presented problem. We
note that solutions for problems in the form of upQP or upLP contain not only
parametric values for the problem’s original decision variables, but also for
slack variables for each constraint and dual variables for each constraint and
non-negativity restriction.

As suggested in Section 2, our implementation utilizes the criss-cross method
of den Hertog et al. [4] to compute complementary bases on line 3 of Algorithm
1. Additionally, we employ the computer algebra system Pari/GP 2.14.0 [11] via
the Python library CyPari2 for all symbolic algebra needed to compute and
process (e.g., calculate roots, take derivatives, etc.) the polynomial
functions that define each invariancy interval.

We now present numerical results for two classes of problems. Instances of the
first class are obtained from biobjective QPs having convex objectives that
are scalarized using the weighted-sum approach and then reformulated as upLCP.
We refer to these instances as boQP instances. Instances of the second class
are obtained using some of the techniques outlined by Illés and Morapitiye
[6]. We refer to these instances as sufLCP instances. All instances, together
with a detailed description of the specific techniques use for their
generation, are provided along with the code at the above referenced url.

All tests were conducted on a machine running Linux Mint 20.0 and that had a
1.2 GHz Intel i3-1005G1 CPU with 12GB of RAM. When solving each instance of
upLCP, we permitted the processing of set $\mathcal{S}$ to be executed in
parallel, using four threads. In all, we generated five boQP instances for
each value of $h\in\\{50,75,100,125\\}$ and five sufLCP instances for each
value of $h\in\\{50,75,100,125,150,175\\}$. The CPU time required to solve
each boQP instance, as well as the number of invariancy intervals present in
the final partition for each of instance, is presented in Table 2. Analogous
data is provided for sufLCP instances in Table 3.

Table 2: Results for boQP Instances Instance |  | CPU | # of | Instance |  | CPU | # of
---|---|---|---|---|---|---|---
Size ($h$) | # | Time (s) | Intervals |  Size ($h$) | # | Time (s) | Intervals
50 | 1 | 18.61 | 15 | 100 | 1 | 915.22 | 33
| 2 | 24.90 | 13 |  | 2 | 1041.95 | 38
| 3 | 51.05 | 30 |  | 3 | 1377.16 | 39
| 4 | 18.17 | 22 |  | 4 | 806.51 | 27
| 5 | 8.07 | 11 |  | 5 | 1543.52 | 44
75 | 1 | 66.69 | 17 | 125 | 1 | 2813.07 | 38
| 2 | 185.42 | 27 |  | 2 | 4463.14 | 51
| 3 | 327.48 | 23 |  | 3 | 2137.46 | 45
| 4 | 266.87 | 26 |  | 4 | 705.88 | 5
| 5 | 222.91 | 24 |  | 5 | 1429.14 | 31
Table 3: Results for sufLCP Instances Instance |  | CPU | # of | Instance |  | CPU | # of
---|---|---|---|---|---|---|---
Size ($h$) | # | Time (s) | Intervals |  Size ($h$) | # | Time (s) | Intervals
50 | 1 | 1.42 | 15 | 125 | 1 | 496.97 | 37
| 2 | 3.34 | 11 |  | 2 | 303.12 | 25
| 3 | 5.73 | 17 |  | 3 | 64.05 | 44
| 4 | 2.81 | 10 |  | 4 | 169.82 | 29
| 5 | 3.94 | 6 |  | 5 | 283.03 | 35
75 | 1 | 19.88 | 20 | 150 | 1 | 369.89 | 42
| 2 | 27.35 | 22 |  | 2 | 814.69 | 48
| 3 | 39.17 | 52 |  | 3 | 96.95 | 34
| 4 | 6.23 | 9 |  | 4 | 597.42 | 61
| 5 | 26.41 | 115 |  | 5 | 442.07 | 37
100 | 1 | 80.94 | 23 | 175 | 1 | 1947.01 | 47
| 2 | 15.74 | 16 |  | 2 | 2066.37 | 42
| 3 | 250.15 | 37 |  | 3 | 1083.17 | 42
| 4 | 22.27 | 14 |  | 4 | 2228.88 | 54
| 5 | 41.50 | 18 |  | 5 | 2062.75 | 40

As we expect, we see that for both classes of problems the required CPU time
and number of intervals present in the final solution increase as problem size
increases.

##  6 Conclusion

We have presented a new method for solving upLCP, upQP, and upLP that is
capable of solving some classes of problems that were not able to be solved by
any known uni-parametric methodology from the literature. Moreover, we have
demonstrated empirically that our proposed technique can be used to solve
relatively large instances in reasonable time.

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