Lagrange-Type Functions

Consider the following problem P(f, g) 3.1.1<m:math display='block'> <m:mrow> <m:mi>min</m:mi><m:mtext>&#x2009;</m:mtext><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mtext>&#x2009;</m:mtext><m:mi>s</m:mi><m:mi>u</m:mi><m:mi>b</m:mi><m:mi>j</m:mi><m:mi>e</m:mi><m:mi>c</m:mi><m:mi>t</m:mi><m:mtext>&#x2009;</m:mtext><m:mi>t</m:mi><m:mi>o</m:mi><m:mtext>&#x2009;</m:mtext><m:mi>x</m:mi><m:mo>&#x2208;</m:mo><m:mtext>&#x2009;</m:mtext><m:mi>X</m:mi><m:mo>,</m:mo><m:mi>g</m:mi><m:mrow><m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo></m:mrow><m:mo>&#x2264;</m:mo><m:mn>0</m:mn><m:mo>,</m:mo></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\min \;f\left( x \right)\;subject\;to\;x \in \;X,g\left( x \right) \leqslant 0,$$ where X is a metric space, f is a real-valued function defined on X, and g maps X into ℝm, that is, g(x) = (g₁ (x),…, g _m (x)), where g _i are real-valued functions, defined on X.