Extremal problems involving isotropic sets and functions on spaces of rectangular matrices

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>.</mo> </mrow> </math> </EquationSource> </InlineEquation> Matrix optimization problems of such a degree of generality are not always easy to deal with, especially if the decision variable is a high-dimensional rectangular matrix. Sometimes, it is possible to reduce the size and complexity of the matrix optimization problem in the presence of symmetry assumptions (isotropy, orthogonal invariance, etc.). This work establishes a localization result for the solutions to a class of extremal problems involving isotropic sets and functions. Copyright Sociedad de Estadística e Investigación Operativa 2014