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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 ]
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.
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.
The sum of these values is an upper bound because the soft constraints cannot assume a higher value. It is exact because the maximal values of soft constraints may derive from different evaluations: a soft constraint may be maximal for x = a {\displaystyle x=a} while another constraint is maximal for x = b {\displaystyle x=b} .
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 ...
An example of MUSCL type state parabolic-reconstruction. It is possible to extend the idea of linear-extrapolation to higher order reconstruction, and an example is shown in the diagram opposite. However, for this case the left and right states are estimated by interpolation of a second-order, upwind biased, difference equation.
In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given. The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, [ 1 ] Haupt et al. [ 2 ] and from Rody Oldenhuis software. [ 3 ]
In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint.