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The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function (+, +) to lie in the second-order cone in +. [ 1 ] SOCPs can be solved by interior point methods [ 2 ] and in general, can be solved more efficiently than semidefinite programming (SDP) problems. [ 3 ]
A hierarchy of convex optimization problems. (LP: linear programming, QP: quadratic programming, SOCP second-order cone program, SDP: semidefinite programming, CP: conic optimization.) Linear programming problems are the simplest convex programs. In LP, the objective and constraint functions are all linear. Quadratic programming are the next ...
The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the ...
In optimal control, the situation is more complicated because of the possibility of a singular solution.The generalized Legendre–Clebsch condition, [1] also known as convexity, [2] is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e.,
Similar to convex order, Laplace transform order is established by comparing the expectation of a function of the random variable where the function is from a special class: () = (). This makes the Laplace transform order an integral stochastic order with the generator set given by the function set defined above with α {\displaystyle ...
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize = ‖ ‖ = () = +.Here () = has the dimension 1x1 (the number of columns of ), so it is a scalar and equal to its own transpose, hence = and the quantity to minimize becomes
If the domain is just the real line, then () is just the second derivative ″ (), so the condition becomes ″ (). If m = 0 {\displaystyle m=0} then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that f ″ ( x ) ≥ 0 {\displaystyle f''(x)\geq 0} ), which implies the function is convex, and ...