What is a second-order cone constraint?

What is a second-order cone constraint?

Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones.

What is cone programming?

In this chapter we consider convex optimization problems of the form. The linear inequality is a generalized inequality with respect to a proper convex cone. It may include componentwise vector inequalities, second-order cone inequalities, and linear matrix inequalities.

What is quadratic cone?

The quadric cone is the simplest quadric ruled surface, i.e., it is a surface of degree 2 that contains infinitely many lines. In fact, the vertex of the cone is at the origin, and every line that connects the origin to a point of the surface lies on the cone.

What is linear programming in statistics?

Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

What is conic constraint?

A cone constraint specifies that the vector formed by a set of decision variables is constrained to lie within a closed convex pointed cone. The simplest example of such a cone is the non-negative orthant, the region where all variables are non-negative — the normal situation in an LP.

Is Semidefinite programming convex?

Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

Why are conics quadratic?

Conic sections are also known as quadratic relations because the equations which describe them are second order and not always functions. These conic sections are excellent mathematical models of the paths taken by planets, meteors, spacecrafts, light rays, and many other objects.

What is conic problem?

A conic optimization problem is a problem involving a constraint that the optimization variable be in some closed convex cone. Prominent examples are linear programs (LP), second order cone programs (SOCP), semidefinite problems (SDP), and copositive problems.

What is conic quadratic?

Conic quadratic optimization, also known as second-order cone optimization, is a straightforward generalization of linear optimization, in the sense that we optimize a linear function under linear (in)equalities with some variables belonging to one or more (rotated) quadratic cones.

What is exponential cone?

The exponential cone is defined as the set (yex/y≤z,y>0), see, e.g. Chandrasekara and Shah 2016 for a primer on exponential cone programming and the equivalent framework of relative entropy programming.

What is a Semidefinite programming problem?

Semidefinite programming is an extension of linear programming where the componentwise inequalities between vectors are replaced by matrix inequalities, or, equivalently the first orthant is replaced by the cone of positive semi-definite matrices. A semidefinite optimization problem has the form. (4.11)

What are the three degenerate conics?

THE THREE DEGENERATE CONICS ARE THE POINT, THE LINE, AND TWO INTERSECTING LINES.