7. Optimization: the Kuhn-Tucker conditions for problems with inequality constraints 7.1 Optimization with inequality constraints: the Kuhn-Tucker conditions 7.2 Optimization with inequality constraints: the necessity of the Kuhn-Tucker conditions 7.3 Optimization with inequality constraints: the sufficiency of the Kuhn-Tucker conditions 7.4 Optimization with inequality constraints: nonnegativity conditions 7.5 Optimization: summary of conditions under which first-order conditions are necessary and sufficient