12.5 Affine Conic Constraints

Optimization Toolbox for MATLAB allows conic problems to be specified in another format, namely with affine conic constraints. A conic problem can thus be specified as:

(12.26)\[\begin{split}\begin{array} {lccccll} \mbox{minimize} & & & \sum_{j=1}^n c_j x_j + \sum_{j=1}^p \left\langle \barC_j, \barX_j \right\rangle + c^f & & & \\ \mbox{subject to} & l_i^c & \leq & \sum_{j=1}^n a_{ij} x_j + \sum_{j=1}^p \left\langle \barA_{ij}, \barX_j\right\rangle & \leq & u_i^c, & i = 1, \ldots, m,\\ & l_j^x & \leq & x_j & \leq & u_j^x, & j = 1, \ldots, n,\\ & & & Fx+g \in \K, & & & \\ & & & \barX_j \in \PSD^{r_j}, & & & j = 1, \ldots, p \end{array}\end{split}\]

where all the data has the same meaning as in Sec. 12.2 (Conic Optimization) and Sec. 12.3 (Semidefinite Optimization) and \(F\in\real^{k\times n}\), \(g\in\real^k\) specifiy the affine conic constraint.

Duality

The dual of problem (12.26) is

(12.27)\[\begin{split}\begin{array} {lcl} \mbox{maximize} & (l^c)^T s_l^c - (u^c)^T s_u^c + (l^x)^T s_l^x - (u^x)^T s_u^x - \dot{y}^T g + c^f &\\ \mbox{subject to} &\\ & A^T y + s_l^x - s_u^x + F^T \dot{y} = c, & \\ & -y + s_l^c - s_u^c = 0, &\\ & \barC_j - \sum_{i=0}^{m-1} y_i \barA_{ij} = \barS_j, & j=0,\ldots,p-1 \\ & s_l^c,s_u^c,s_l^x ,s_u^x \geq 0, & \\ & \dot{y} \in \K^*, & \\ & \barS_j \in \PSD^{r_j}, & j=0,\ldots,p-1. \end{array}\end{split}\]

Duality and infeasibility cerfificates behave analogously as in Sec. 12.2 (Conic Optimization).

Remark

A problem of this form is internally converted into the problem:

(12.28)\[\begin{split}\begin{array} {lccccll} \mbox{minimize} & & & \sum_{j=1}^n c_j x_j + \sum_{j=1}^p \left\langle \barC_j, \barX_j \right\rangle + c^f & & & \\ \mbox{subject to} & l_i^c & \leq & \sum_{j=1}^n a_{ij} x_j + \sum_{j=1}^p \left\langle \barA_{ij}, \barX_j\right\rangle & \leq & u_i^c, & i = 1, \ldots, m,\\ & g_i & \leq & z_i - \sum_{j=1}^n f_{ij} x_j & \leq & g_i & i = 1, \ldots, k,\\ & l_j^x & \leq & x_j & \leq & u_j^x, & j = 1, \ldots, n,\\ & & & z \in \K, & & & \\ & & & \barX_j \in \PSD^{r_j}, & & & j = 1, \ldots, p \end{array}\end{split}\]

which conforms with the format in Sec. 12.2 (Conic Optimization) and Sec. 12.3 (Semidefinite Optimization). The reformulated problem has \(n+k\) variables, \(m+k\) linear constraints and in total \(k\) variables in cones. The new columns and rows are appended at the end of the original ones. If a problem with affine conic constraints is saved to a file then this reformulation will be written.