Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones
An Euclidean Jordan algebra (EJA) is a triple \((V,\circ ,\left\langle \cdot ,\cdot \right\rangle ),\) where \(V\) is a finite-dimensional real vector space, \(\circ :V\times V\rightarrow V\) is a symmetric bilinear form (product) satisfying the Jordan Axiom \[ x\circ (x^{2}\circ y)=x^{2}\circ (x\circ y)\quad \forall x,y\in V, \] and \(\left\langle \cdot ,\cdot \right\rangle :V\times V\rightarrow V\) is a positive definite symmetric bilinear form (inner product)\ satisfying the associative axiom \[ \left\langle x\circ z,y\right\rangle =\left\langle x,z\circ y\right\rangle \quad \forall x,y,z\in V. \] It is known that a cone \(K\) is symmetric if and only if there exists some EJA \((V,\circ ,\left\langle \cdot ,\cdot \right\rangle )\) such that \( K=\left\{ x^{2},x\in V\right\} \) (the cone of squares of \(V\)). The monotone linear complementarity problems over a symmetric cone problem (SCLCP) considered by the authors consists of, given a symmetric cone \(K\) with associated EJA \((V,\circ ,\left\langle \cdot ,\cdot \right\rangle )\) and an affine subspace \(L\) of \(V\times V\) satisfying the\ monotone property, i.e., \[ \left\langle x_{1}-x_{2},y_{1}-y_{2}\right\rangle \geq 0\quad \forall \left( x_{1},y_{1}\right) ,\left( x_{2},y_{2}\right) \in L, \] find a pair \((x,s)\in K\times K\) such that \[ (x,s)\in L\text{ and }\left\langle x,s\right\rangle =0. \] The optimality conditions of several important optimization problems can be written in the form of SCLCP. For example, the solution of any Linear Optimization problem over symmetric cones can be formulated as SCLCP. The paper proposes an interior-point method for monotone linear complementarity problems over symmetric cones (SCLCP) that is based on barrier functions which are defined by a large class of univariate functions, called eligible kernel functions. Computational experience is provided.
Euclidean Jordan algebras and symmetric cones
scientific article; zbMATH DE number 5980386