On Second-Order Cone Functions

We consider the second-order cone function (SOCF) <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>f</a:mi><a:mo>:</a:mo><a:msup><a:mrow><a:mi mathvariant="double-struck">R</a:mi></a:mrow><a:mrow><a:mi>n</a:mi></a:mrow></a:msup><a:mo>⟶</a:mo><a:mi mathvariant="double-struck">R</a:mi></a:math> defined by <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M2"><e:mi>f</e:mi><e:mfenced open="(" close=")" separators="|"><e:mrow><e:mi>x</e:mi></e:mrow></e:mfenced><e:mo>=</e:mo><e:msup><e:mrow><e:mi>c</e:mi></e:mrow><e:mrow><e:mi>T</e:mi></e:mrow></e:msup><e:mi>x</e:mi><e:mo>+</e:mo><e:mi>d</e:mi><e:mo>−</e:mo><e:mrow><e:mfenced open="‖" close="‖" separators="|"><e:mrow><e:mi>A</e:mi><e:mi>x</e:mi><e:mo>+</e:mo><e:mi>b</e:mi></e:mrow></e:mfenced></e:mrow></e:math> , with parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M3"><m:mi>c</m:mi><m:mo>∈</m:mo><m:msup><m:mrow><m:mi mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msup></m:math> , <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" id="M4"><p:mi>d</p:mi><p:mo mathvariant="double-struck">∈</p:mo><p:mi mathvariant="double-struck">R</p:mi></p:math> , <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M5"><t:mi>A</t:mi><t:mo>∈</t:mo><t:msup><t:mrow><t:mi mathvariant="double-struck">R</t:mi></t:mrow><t:mrow><t:mi>m</t:mi><t:mo>×</t:mo><t:mi>n</t:mi></t:mrow></t:msup></t:math> , and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M6"><w:mi>b</w:mi><w:mo>∈</w:mo><w:msup><w:mrow><w:mi mathvariant="double-struck">R</w:mi></w:mrow><w:mrow><w:mi>m</w:mi></w:mrow></w:msup></w:math> . Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" id="M7"><z:mi>f</z:mi></z:math> . The parameters <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" id="M8"><bb:mi>A</bb:mi></bb:math> and <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" id="M9"><db:mi>b</db:mi></db:math> are not uniquely determined. We show that every SOCF can be written in the form <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" id="M10"><fb:mi>f</fb:mi><fb:mfenced open="(" close=")" separators="|"><fb:mrow><fb:mi>x</fb:mi></fb:mrow></fb:mfenced><fb:mo>=</fb:mo><fb:msup><fb:mrow><fb:mi>c</fb:mi></fb:mrow><fb:mrow><fb:mi>T</fb:mi></fb:mrow></fb:msup><fb:mi>x</fb:mi><fb:mo>+</fb:mo><fb:mi>d</fb:mi><fb:mo>−</fb:mo><fb:msqrt><fb:mrow><fb:msup><fb:mrow><fb:mi>δ</fb:mi></fb:mrow><fb:mrow><fb:mn>2</fb:mn></fb:mrow></fb:msup><fb:mo>+</fb:mo><fb:msup><fb:mrow><fb:mfenced open="(" close=")" separators="|"><fb:mrow><fb:mi>x</fb:mi><fb:mo>−</fb:mo><fb:msub><fb:mrow><fb:mi>x</fb:mi></fb:mrow><fb:mi>∗</fb:mi></fb:msub></fb:mrow></fb:mfenced></fb:mrow><fb:mrow><fb:mi>T</fb:mi></fb:mrow></fb:msup><fb:mi>M</fb:mi><fb:mrow><fb:mfenced open="(" close=")" separators="|"><fb:mrow><fb:mi>x</fb:mi><fb:mo>−</fb:mo><fb:msub><fb:mrow><fb:mi>x</fb:mi></fb:mrow><fb:mi>∗</fb:mi></fb:msub></fb:mrow></fb:mfenced></fb:mrow></fb:mrow></fb:msqrt></fb:math> . We give necessary and sufficient conditions for the parameters <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" id="M11"><qb:mi>c</qb:mi></qb:math> , <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" id="M12"><sb:mi>d</sb:mi></sb:math> , <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" id="M13"><ub:mi>δ</ub:mi></ub:math> , <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" id="M14"><wb:mi>M</wb:mi><wb:mo>=</wb:mo><wb:msup><wb:mrow><wb:mi>A</wb:mi></wb:mrow><wb:mrow><wb:mi>T</wb:mi></wb:mrow></wb:msup><wb:mi>A</wb:mi></wb:math> , and <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" id="M15"><yb:msub><yb:mrow><yb:mi>x</yb:mi></yb:mrow><yb:mi>∗</yb:mi></yb:msub></yb:math> to be uniquely determined. We also give necessary and sufficient conditions for <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" id="M16"><ac:mi>f</ac:mi></ac:math> to be bounded above.