 # Second-order conditions

This descends from classical optimization, using the second-order term in Taylor's expansion. For unconstrained optimization, the second-order necessary condition (for $LaTeX: \textstyle f \in C^2$ ) is that the hessian is negative semi-definite (for a max). Second-order sufficient conditions are the first-order conditions plus the hessian be negative definite. For constrained optimization, the second-order conditions are similar, using projection for a regular mathematical program and the Lagrange Multiplier Rule. They are as follows (all functions are in $LaTeX: \textstyle C^2c$ and the mathematical program is in standard form, for $LaTeX: \textstyle x^*$ a local maximum):
Second-order necessary conditions. There exist Lagrange multipliers, (u,v), such that $LaTeX: \textstyle u \ge 0$ and $LaTeX: \textstyle ug(x^*)=0$ for which: (1) $LaTeX: \textstyle \nabla x [ \mbox{L}(x^*,u,v)]=0,$ and (2) $LaTeX: \textstyle \mbox{H}_x [\mbox{L}(x^*,u,v)]$ is negative semi-definite on the tangent plane.