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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 ]
Examples of include the positive orthant + = {:}, positive semidefinite matrices +, and the second-order cone {(,): ‖ ‖}. Often f {\displaystyle f\ } is a linear function, in which case the conic optimization problem reduces to a linear program , a semidefinite program , and a second order cone program , respectively.
In LP, the objective and constraint functions are all linear. Quadratic programming are the next-simplest. In QP, the constraints are all linear, but the objective may be a convex quadratic function. Second order cone programming are more general. Semidefinite programming are more general. Conic optimization are even more general - see figure ...
Trapezoidal rule — second-order implicit method; Runge–Kutta methods — one of the two main classes of methods for initial-value problems Midpoint method — a second-order method with two stages; Heun's method — either a second-order method with two stages, or a third-order method with three stages
Second-order cone programming; Semidefinite embedding; Sequential linear-quadratic programming; Sequential minimal optimization; Sequential quadratic programming; Simplex algorithm; Simulated annealing; Simultaneous perturbation stochastic approximation; Social cognitive optimization; Space allocation problem; Space mapping; Special ordered set
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A commercial optimization solver for linear programming, non-linear programming, mixed integer linear programming, convex quadratic programming, convex quadratically constrained quadratic programming, second-order cone programming and their mixed integer counterparts. AMPL: CPLEX: Popular solver with an API for several programming languages.