Wat is Linear, quadratic, second-order cone and semidefinite programmes?

A linear programme minimises a linear function subject to linear constraints; a quadratic programme has a convex quadratic objective and linear constraints; a second-order cone programme (SOCP) adds constraints $\lVert A_ix + b_i\rVert_2 \le c_i^\top x + d_i$; a semidefinite programme constrains a symmetric matrix depending affinely on $x$ to be positive semidefinite.