Second-order conditions

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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.

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.

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