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>