Conic program

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The standard form is

LaTeX: \min \{ c^T x : Ax=b, \; x \in K\},

where LaTeX: K is a cone (not necessarily convex). This is more general than it looks. Suppose LaTeX: S is any non-empty set, and we have the mathematical program:

LaTeX: \min \{ c^T x : x \in S\}.

Then, defining LaTeX: K = \{t(1,x): x \in S, \; t \ge 0\}, we have that the above problem is equivalent to the conic program:

LaTeX: \min \{ c^Tx : (x_0, x) \in K, \;  x_0 = 1 \}.

This problem class also includes the semi-definite program, as LaTeX: x could be a matrix.

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