Critical Values of Multilinear Forms under Conic Constraints

Abstract

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Phi:\Pi_{i=1}^r E_i\to \mathbb{R}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mo>:</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal">Π</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>r</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be a multilinear form on the Cartesian product of finitely many Euclidean vector spaces. We suppose that each factor <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">E_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is equipped with its own closed convex cone <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">K_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We analyze the concept of critical point and critical value of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\Phi</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> when each argument of this function is restricted to a normalization constraint and a conic constraint. Our study encompasses the theory of cone-constrained singular values of bilinear and trilinear forms. </jats:p>