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A second-order cone program (SOCP) is a convex optimization problem of the form minimize subject to ‖ + ‖ +, =, …, = where the ...
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.
On March 5, 2021, an edit titled "correct errors" removed an extremely useful formula. In particular, there used to be a formula for converting x T A T A x + b T x + c ≤ 0 {\displaystyle x^{T}A^{T}Ax+b^{T}x+c\leq 0} into an SOCP constraint, but it was replaced by a different one for x T A x + b T x + c ≤ 0 {\displaystyle x^{T}Ax+b^{T}x+c ...
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 ...
Second-order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs. Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is a generalization of linear and convex quadratic programming.
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There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.
A self-concordant function is a function satisfying a certain differential inequality, which makes it particularly easy for optimization using Newton's method [1]: Sub.6.2.4.2 A self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set.