7.1 Separable Convex (SCopt) Interface

The Rmosek Package provides a way to add simple non-linear functions composed from a limited set of non-linear terms. Non-linear terms can be mixed with quadratic terms in objective and constraints. We consider problems which can be formulated as:

\[\begin{split}\begin{array} {lcccccc} \mbox{minimize} & & & z_0(x) + c^T x & & & \\ \mbox{subject to} & l^c_i & \leq & z_i(x) + a_i^T x & \leq & u^c_i & i=1\ldots m\\ & l^x & \leq & x & \leq & u^x, & \end{array}\end{split}\]

where \(x\in\real^n\) and each \(z_i : \real^n\rightarrow\real\) is separable, that is can be written as a sum

\[z_i(x) = \sum^{n}_{j=1} z_{i,j}(x_j).\]

The interface implements a limited set of functions which can appear as \(z_{i,j}\). They are:

Table 1 Functions supported by the SCopt interface.
Separable function Operator name Name
\(f x \ln (x)\) ent Entropy function
\(f e^{g x + h}\) exp Exponential function
\(f \ln (g x + h)\) log Logarithm
\(f (x+h)^g\) pow Power function

where \(f,g,h\in\real\) are constants. This formulation does not guarantee convexity. For MOSEK to be able to solve the problem, the following requirements must be met:

  • If the objective is minimized, the sum of non-linear terms must be convex, otherwise it must be concave.
  • Any constraint bounded below must be concave, and any constraint bounded above must be convex.
  • Each separable term must be twice differentiable within the bounds of the variable it is applied to.

Some simple rules can be followed to ensure that the problem satisfies MOSEK’s convexity and differentiability requirements. First of all, for any variable \(x_i\) used in a separable term, the variable bounds must define a range within which the function is twice differentiable. These bounds are defined in Table 2.

Table 2 Safe bounds for functions in the SCopt interface.
Separable function Operator name Safe \(x\) bounds
\(f x \ln (x)\) ent \(0 < x\).
\(f e^{g x + h}\) exp \(-\infty <x <\infty\).
\(f \ln (g x + h)\) log If \(g > 0\): \(-h/g < x\).
If \(g < 0\): \(x < -h/g\).
\(f (x+h)^g\) pow If \(g > 0\) and integer: \(-\infty <x <\infty\).
If \(g < 0\) and integer: either \(-h <x\) or \(x <-h\).
Otherwise: \(-h <x\).

To ensure convexity, we require that each \(z_i(x)\) is either a sum of convex terms or a sum of concave terms. Table 3 lists convexity conditions for the relevant ranges for \(f>0\) — changing the sign of \(f\) switches concavity/convexity.

Table 3 Convexity conditions for functions in the SCopt interface.
Separable function Operator name Convexity conditions
\(f x \ln (x)\) ent Convex within safe bounds.
\(f e^{g x + h}\) exp Convex for all \(x\).
\(f \ln (g x + h)\) log Concave within safe bounds.
\(f (x+h)^g\) pow If \(g\) is even integer: convex within safe bounds.

If \(g\) is odd integer:

  • concave if \((-\infty,-h)\),
  • convex if \((-h,\infty)\)

If \(0<g<1\): concave within safe bounds.

Otherwise: convex within safe bounds.

A problem involving linear combinations of variables (such as \(\ln(x_1+x_2)\)), can be converted to a separable problem using slack variables and additional equality constraints.

7.1.1 Example

Consider the following separable convex problem:

(1)\[\begin{split}\begin{array}{ll} \mbox{minimize} & \exp (x_2) - \ln(x_1) \\ \mbox{subject to} & x_2 \ln(x_2) \leq 0\\ & x_1^{1/2} - x_2 \geq 0\\ & \half\leq x_1, x_2 \leq 1. \end{array}\end{split}\]

Note that all nonlinear functions are well defined for \(x\) values satisfying the variable bounds strictly. This assures that function evaluation errors will not occur during the optimization process because MOSEK.

From this example the input arguments for a separable convex program follow easily. The linear part of the objective function and constraints, as well as the constraint and variable bounds, should all be specified as for linear programs (see Sec. 6.2 (Linear Optimization).). The only addition to this is the list called opro containing the list-typed operator matrices (for objective) and (for constraints).

##                type  j     f    g    h
opro[,1] <- list("LOG", 1,  -1.0, 1.0, 0.0)
opro[,2] <- list("EXP", 2,   1.0, 1.0, 0.0)

The operator matrices have a column for each operator and a row for each descriptive element. The opro matrix have five rows called \(\{type, j, f, g, h\}\), while the oprc matrix have six rows called \(\{type, i, j, f, g, h\}\). Row should specify the operator type in a string, being either, exponential EXP, logarithm LOG or power POW. Row (not in opro) should specify the index of the constraint to which the non-linear term should be added. Row should specify the variable index of the operator. Rows , and should specify the coefficients of the operator.

##                type  i     j    f    g    h
oprc[,1] <- list("ENT", 1,    2, 1.0, 0.0, 0.0)
oprc[,2] <- list("POW", 2,    1, 1.0, 0.5, 0.0)

Note that the definition of the entropy operator, was the only operator defined without \(g\) and \(h\). Thus, for entropy operators, these two unused rows in the operator matrix can be set to either zero or any empty definition (NULL, NA or NaN).