6.5 Conic Exponential Optimization

The structure of a typical conic optimization problem is

\[\begin{split}\begin{array}{lccccl} \mbox{minimize} & & & c^T x+c^f & & \\ \mbox{subject to} & l^c & \leq & A x & \leq & u^c, \\ & l^x & \leq & x & \leq & u^x, \\ & & & Fx+g & \in & \D, \end{array}\end{split}\]

(see Sec. 11 (Problem Formulation and Solutions) for detailed formulations). We recommend Sec. 6.2 (From Linear to Conic Optimization) for a tutorial on how problems of that form are represented in MOSEK and what data structures are relevant. Here we discuss how to set-up problems with the primal/dual exponential cones.

MOSEK supports two exponential cones, namely:

• Primal exponential cone:

  \[\EXP = \left\lbrace x \in \real^3: x_0 \geq x_1 \exp(x_2/x_1),\ x_0,x_1\geq 0 \right\rbrace.\]

• Dual exponential cone:

  \[\EXP^* = \left\lbrace s \in \real^3: s_0 \geq -s_2 e^{-1} \exp(s_1/s_2),\ s_2\leq 0,s_0\geq 0 \right\rbrace.\]

For example, consider the following constraint:

\[(x_4, x_0, x_2) \in \EXP\]

which describes a convex cone in \(\real^3\) given by the inequalities:

\[x_4 \geq x_0\exp(x_2/x_0),\ x_0,x_4\geq 0.\]

For other types of cones supported by MOSEK, see Sec. 13.6 (Supported domains) and the other tutorials in this chapter. Different cone types can appear together in one optimization problem.

6.5.1 Example CEO1

Consider the following basic conic exponential problem which involves some linear constraints and an exponential inequality:

(6.12)\[\begin{split}\begin{array} {lrcl} \mbox{minimize} & x_0 + x_1 & & \\ \mbox{subject to} & x_0+x_1+x_2 & = & 1, \\ & x_0 & \geq & x_1\exp(x_2/x_1), \\ & x_0, x_1 & \geq & 0. \end{array}\end{split}\]

The affine conic form of (6.12) is:

(6.13)\[\begin{split}\begin{array} {lrcl} \mbox{minimize} & x_0 + x_1 & & \\ \mbox{subject to} & x_0+x_1+x_2 & = & 1, \\ & Ix & \in & \EXP, \\ & x & \in & \real^3. \end{array}\end{split}\]

where \(I\) is the \(3\times 3\) identity matrix.

Setting up the linear part

The linear parts (constraints, variables, objective) are set up using exactly the same methods as for linear problems, and we refer to Sec. 6.1 (Linear Optimization) for all the details. The same applies to technical aspects such as defining an optimization task, retrieving the solution and so on.

Setting up the conic constraints

To define the conic constraints, we start by setting prob$F equal to the identity matrix in (6.13). The prob$g vector is set to zero. Lastly, the conic domain is specified as columns in a list-typed matrix called cones, with rows for each associated detail. In example (6.13) we have one conic constraint:

    prob$F <- rbind(c(1,0,0),c(0,1,0),c(0,0,1))
    prob$g <- c(0, 0, 0)
    prob$cones <- matrix(list("PEXP", 3, NULL), nrow=3, ncol=1)
    rownames(prob$cones) <- c("type","dim", "conepar")

The first row in prob$cones selects the "type" of conic domain, in this case the exponential cone "MSK_DOMAIN_PRIMAL_EXP_CONE" (note that PEXP, PRIMAL_EXP_CONE are valid aliases for this conic domain). The second row is used to provide the dimension ("dim") of the conic domain, which in this case has to be 3. The third row sets the parameters ("conepar") for parametric conic domains, but because the exponential cone is not parameterized we set this value as NULL.

Source code

Listing 6.6 Source code solving problem (6.12). Click here to download.

library("Rmosek")

ceo1 <- function()
{
    # Specify the non-conic part of the problem.
    prob <- list(sense="min")
    prob$c  <- c(1, 1, 0)
    prob$A  <- Matrix(c(1, 1, 1), nrow=1, sparse=TRUE)
    prob$bc <- rbind(blc=1, 
                     buc=1)
    prob$bx <- rbind(blx=rep(-Inf,3), 
                     bux=rep( Inf,3))
    
    # Specify the affine conic constraints.
    prob$F <- rbind(c(1,0,0),c(0,1,0),c(0,0,1))
    prob$g <- c(0, 0, 0)
    prob$cones <- matrix(list("PEXP", 3, NULL), nrow=3, ncol=1)
    rownames(prob$cones) <- c("type","dim", "conepar")

    # Solve the problem
    r <- mosek(prob)

    # Return the solution
    stopifnot(identical(r$response$code, 0))
    r$sol
}

ceo1()