Extended Lorentz cones and mixed complementarity problems

Abstract

In this paper we extend the notion of a Lorentz cone in a Euclidean space as follows: we divide the index set corresponding to the coordinates of points in two disjoint classes. By definition a point belongs to an extended Lorentz cone associated with this division, if the coordinates corresponding to one class are at least as large as the norm of the vector formed by the coordinates corresponding to the other class. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., order preserving) with respect to the partial order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular, a Cartesian product between an Euclidean space and any closed convex set in another Euclidean space is such a set. We use this property to find solutions of general mixed complementarity problems recursively.

Appendix: How large is the family of \(K\)-isotone mappings?

The remaining sections can be read without this one, which is entirely for the purpose of convincing the reader that the family of $K$-isotone mappings which occur in the condition “$I-F$ is $K$-isotone” of Proposition 2 and the corresponding condition in Theorem 3 is very wide.

Let $K,S\subset {\mathbb {R}}^m$ be pointed closed convex cones such that $K\subset S$. The function $f:{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ is called $K$-monotone if $x\le _K y$ implies $f(x)\le f(y)$. Both the $K$-monotone functions and the $K$-isotone mappings form a cone. If $f_1,\dots ,f_\ell :{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ are $K$-monotone and $w^1,\dots ,w^\ell \in K$, then it is easy to see that the mapping $F:{\mathbb {R}}^m\rightarrow {\mathbb {R}}^m$ defined by

$$\begin{aligned} F(x)=f_1(x)w^1+\dots +f_\ell (x)w^\ell \end{aligned}$$

(26)

is $K$-isotone. It is obvious that any $S$-monotone function is also $K$-monotone. Hence, if $f_1,\dots ,f_\ell :{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ are $S$-monotone, then the mapping $F$ defined by (26) is $K$-isotone. The pointed closed convex cone $S$ is called simplicial if there exists linearly independent vectors $u^1,\dots ,u^m\in {\mathbb {R}}^m$ such that

$$\begin{aligned} S={{\mathrm{cone}}}\{u^1,\dots ,u^m\}:=\{\lambda _1u^1+\dots +\lambda _mu^m:\lambda _1,\dots ,\lambda _m\ge 0\}. \end{aligned}$$

(27)

The vectors $u^1,\dots ,u^m$ are called the generators of $S$ and we say that $S$ is generated by $u^1,\dots ,u^m$. It can be shown that the dual $S^*$ of a simplicial cone $S$ is simplicial. Moreover, if $U:=(u^1,\dots ,u^m)$ (that is an $m\times m$ matrix with columns $u^1,\dots ,u^m$) and $(U^\top )^{-1}=(v^1,\dots ,v^m)$, then $S^*={{\mathrm{cone}}}\{v^1,\dots ,v^m\}$ [29]. Let $\{e^1,e^2,\dots ,e^m\}$ be the set of standard unit vectors in ${\mathbb {R}}^m$. The cone ${\mathbb {R}}^m_+=\{\lambda _1e^1+\dots +\lambda _me^m:\lambda _1,\dots ,\lambda _m\ge 0\}$ is called the nonnegative orthant. Let $S$ be the simplicial cone defined by (27). If $f:{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ is ${\mathbb {R}}^m_+$-monotone, then $\hat{f}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ defined by $\hat{f}(x_1u^1+\dots +x_mu^m)=f(x_1e^1+\dots +x_me^m)$ is $S$-monotone. If $g_1,\dots ,g_m:{\mathbb {R}}\rightarrow {\mathbb {R}}$ are monotone increasing, then obviously $g:{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} g(x_1u^1+\dots +x_mu^m)=g_1(x_1)+\dots +g_m(x_m) \end{aligned}$$

(28)

is $S$-monotone. Moreover, if $f:{\mathbb {R}}^m\rightarrow {\mathbb {R}}$ is $S$-monotone and $\psi :{\mathbb {R}}\rightarrow {\mathbb {R}}$ is monotone increasing, then it is straightforward to see that $\psi \circ f$ is also $S$-monotone. Hence, if all mappings $f_i$ in (26) are formed by using a combination of (28), the previous property and the conicity of the $S$-monotone functions, then the mapping $F$ defined by (26) is $K$-isotone for any pointed closed convex cone $K$ contained in $S$. For any such cone $K$ it is easy to construct a simplicial cone $S$ which contains $K$. From the definition of the dual of a cone it follows that ${\mathbb {R}}^m=\{0\}^*=(K\cap (-K))^*=K^*+(-K)^*=K^*-K^*$. Thus, the smallest linear subspace of ${\mathbb {R}}^m$ containing $K^*$ is ${\mathbb {R}}^m$ and hence the interior of $K^*$ is nonempty (see [26]). Therefore, there exist $m$ linearly independent vectors in $K^*$, that is, $K^*$ contains a simplicial cone $T$. Let $S$ be the dual of $T$. Then, obviously $K\subset S$.

The above constructions show that for any pointed closed convex cone the family of $K$-isotone mappings, used in Proposition 2 and Theorem 3 is very wide. Moreover, there may be many $K$-isotone mappings which are not of the above type. This topic is worth to be investigated in the future.