Second-order conditions
From Glossary
This descends from classical optimization, using the second-order term in Taylor's expansion. For unconstrained optimization, the second-order necessary condition (for ) 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 and the mathematical program is in standard form, for a local maximum):
- Second-order necessary conditions. There exist Lagrange multipliers, (u,v), such that and for which: (1) and (2) is negative semi-definite on the tangent plane.
- Second-order sufficient conditions. The above necessary conditions hold but with (2) replaced by (2') H_x[L(x*,u,v)] is negative definite on the tangent plane.