Second-order cone programming

A second-order cone program (SOCP) is a convex optimization problem of the form


 * minimize $$\ f^T x \ $$
 * subject to
 * $$\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m$$
 * $$Fx = g \ $$

where the problem parameters are $$f \in \mathbb{R}^n, \ A_i \in \mathbb{R}^{{n_i}\times n}, \ b_i \in \mathbb{R}^{n_i}, \ c_i \in \mathbb{R}^n, \ d_i \in \mathbb{R}, \ F \in \mathbb{R}^{p\times n}$$, and $$g \in \mathbb{R}^p$$. $$x\in\mathbb{R}^n$$ is the optimization variable. $$\lVert x \rVert_2 $$ is the Euclidean norm and $$^T$$ indicates transpose. The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function $$(A x + b, c^T x + d)$$ to lie in the second-order cone in $$\mathbb{R}^{n_i + 1}$$.

SOCPs can be solved by interior point methods and in general, can be solved more efficiently than semidefinite programming (SDP) problems. Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics. Applications in quantitative finance include portfolio optimization; some market impact constraints, because they are not linear, cannot be solved by quadratic programming but can be formulated as SOCP problems.

Second-order cone
The standard or unit second-order cone of dimension $$n+1$$ is defined as

$$\mathcal{C}_{n+1}=\left\{ \begin{bmatrix} x \\ t \end{bmatrix} \Bigg| x \in \mathbb{R}^n, t\in \mathbb{R}, \|x\|_2\leq t \right\}$$.

The second-order cone is also known by quadratic cone or ice-cream cone or Lorentz cone. The standard second-order cone in $$\mathbb{R}^3$$ is $$\left\{(x,y,z) \Big| \sqrt{x^2 + y^2} \leq z \right\}$$.

The set of points satisfying a second-order cone constraint is the inverse image of the unit second-order cone under an affine mapping:

$$\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i \Leftrightarrow \begin{bmatrix} A_i \\ c_i^T \end{bmatrix} x + \begin{bmatrix} b_i \\ d_i \end{bmatrix} \in \mathcal{C}_{n_i+1}$$

and hence is convex.

The second-order cone can be embedded in the cone of the positive semidefinite matrices since

$$||x||\leq t \Leftrightarrow \begin{bmatrix} tI & x \\ x^T & t \end{bmatrix} \succcurlyeq 0,$$

i.e., a second-order cone constraint is equivalent to a linear matrix inequality (Here $$M\succcurlyeq 0 $$ means $$M $$ is semidefinite matrix). Similarly, we also have,

$$\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i \Leftrightarrow \begin{bmatrix} (c_i^T x+d_i)I & A_i x+b_i \\ (A_i x + b_i)^T & c_i^T x + d_i \end{bmatrix} \succcurlyeq 0$$.

Relation with other optimization problems
When $$A_i = 0$$ for $$i = 1,\dots,m$$, the SOCP reduces to a linear program. When $$c_i = 0 $$ for $$i = 1,\dots,m$$, the SOCP is equivalent to a convex quadratically constrained linear program.

Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint. Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program. The converse, however, is not valid: there are positive semidefinite cones that do not admit any second-order cone representation. In fact, while any closed convex semialgebraic set in the plane can be written as a feasible region of a SOCP, it is known that there exist convex semialgebraic sets that are not representable by SDPs, that is, there exist convex semialgebraic sets that can not be written as a feasible region of a SDP.

Quadratic constraint
Consider a convex quadratic constraint of the form


 * $$ x^T A x + b^T x + c \leq 0. $$

This is equivalent to the SOCP constraint


 * $$ \lVert A^{1/2} x + \frac{1}{2}A^{-1/2}b \rVert \leq  \left(\frac{1}{4}b^T A^{-1} b - c \right)^{\frac{1}{2}} $$

Stochastic linear programming
Consider a stochastic linear program in inequality form


 * minimize $$\ c^T x \ $$
 * subject to
 * $$\mathbb{P}(a_i^Tx \leq b_i) \geq p, \quad i = 1,\dots,m $$

where the parameters $$a_i \ $$ are independent Gaussian random vectors with mean $$\bar{a}_i$$ and covariance $$\Sigma_i \ $$ and $$p\geq0.5$$. This problem can be expressed as the SOCP


 * minimize $$\ c^T x \ $$
 * subject to
 * $$\bar{a}_i^T x + \Phi^{-1}(p) \lVert \Sigma_i^{1/2} x \rVert_2 \leq b_i, \quad i = 1,\dots,m $$

where $$\Phi^{-1}(\cdot) \ $$ is the inverse normal cumulative distribution function.

Stochastic second-order cone programming
We refer to second-order cone programs as deterministic second-order cone programs since data defining them are deterministic. Stochastic second-order cone programs are a class of optimization problems that are defined to handle uncertainty in data defining deterministic second-order cone programs.

Other examples
Other modeling examples are available at the MOSEK modeling cookbook.