Introduction

Consider the following mathematical programming problem with inequality constraints: 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:msub> <m:mi>g</m:mi> <m:mi>i</m:mi> </m:msub> <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:mtext>&#x2003;</m:mtext><m:mi>i</m:mi><m:mo>=</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mo>&#x2026;</m:mo><m:mo>,</m:mo><m:mi>m</m:mi><m:mo>,</m:mo></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\min \;f\left( x \right)\;subject\;to\;x \in \;X,{g_i}\left( x \right) \leqslant 0\quad i = 1, \ldots ,m,$$ where f and g _i , i = 1,…, m are real-valued functions defined on a metric space X. Let g(x) = (g₁ (x),…, g _m (x)). We consider g as a map defined on X and mapping into ℝm. Denote the problem (1.1.1) as P(f, g).