We study a general equilibrium model formulated as a smooth system of equations coupled with complementarity conditions relative to the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$n$$</EquationSource> </InlineEquation>-dimensional Lorentz cone. For the purpose of analysis, as well as for the design of algorithms, we exploit the fact that the Lorentz cone is...</equationsource></inlineequation>

Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{a_i:i\in I\}$$</EquationSource> </InlineEquation> be a finite set in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\mathbb R ^n$$</EquationSource> </InlineEquation>. The illumination problem addressed in this work is about selecting an apex <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$z$$</EquationSource> </InlineEquation> in a prescribed set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$Z\subseteq \mathbb R ^n$$</EquationSource> </InlineEquation> and a unit vector <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$y\in \mathbb R ^n$$</EquationSource> </InlineEquation> so that the conic light beam <Equation ID="Equ55"> <EquationSource Format="TEX">$$\begin{aligned}...</equationsource></equation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation></equationsource></inlineequation>

Let <InlineEquation ID="IEq1270"> <EquationSource Format="TEX">$$\mathbb{M}_{m,n}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="double-struck">M</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math> </EquationSource> </InlineEquation> be the linear space of real matrices of dimension m × n. A variational problem that arises quite often in applications is that of minimizing a real-valued function f on some feasible set <InlineEquation ID="IEq128"> <EquationSource Format="TEX">$$\Upomega\subseteq \mathbb{M}_{m,n}.$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msub> <mi mathvariant="double-struck">M</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub>...</mo></mrow></math></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

We introduce an axiomatic formalism for the concept of the center of a set in a Euclidean space. Then we explain how to exploit possible symmetries and possible cyclicities in the set in order to localize its center. Special attention is paid to the determination of centers in cones of matrices....