6.4 Power Cone Optimization

Conic optimization is a generalization of linear optimization, allowing constraints of the type

\[x^t \in \K_t,\]

where \(x^t\) is a subset of the problem variables and \(\K_t\) is a convex cone. Since the set \(\real^n\) of real numbers is also a convex cone, we can simply write a compound conic constraint \(x\in \K\) where \(\K=\K_1\times\cdots\times\K_l\) is a product of smaller cones and \(x\) is the full problem variable.

MOSEK can solve conic optimization problems of the form

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

where the domain restriction, \(x \in \K\), implies that all variables are partitioned into convex cones

\[x = (x^0, x^1, \ldots , x^{p-1}),\quad \mbox{with } x^t \in \K_t \subseteq \real^{n_t}.\]

In this tutorial we describe how to use the power cone. The primal power cone of dimension \(n\) with parameter \(0<\alpha<1\) is defined as:

\[\POW_n^{\alpha,1-\alpha} = \left\lbrace x \in \real^n: x_0^\alpha x_1^{1-\alpha}\geq\sqrt{\sum_{i=2}^{n-1}x_i^2},\ x_0,x_1\geq 0 \right\rbrace.\]

In particular, the most important special case is the three-dimensional power cone family:

\[\POW_3^{\alpha,1-\alpha} = \left\lbrace x \in \real^3: x_0^\alpha x_1^{1-\alpha}\geq |x_2|,\ x_0,x_1\geq 0 \right\rbrace.\]

For example, the conic constraint \((x,y,z)\in\POW_3^{0.25,0.75}\) is equivalent to \(x^{0.25}y^{0.75}\geq |z|\), or simply \(xy^3\geq z^4\) with \(x,y\geq 0\).

MOSEK also supports the dual power cone:

\[\left(\POW_n^{\alpha,1-\alpha}\right)^{*} = \left\lbrace x \in \real^n: \left(\frac{x_0}{\alpha}\right)^\alpha \left(\frac{x_1}{1-\alpha}\right)^{1-\alpha}\geq\sqrt{\sum_{i=2}^{n-1}x_i^2},\ x_0,x_1\geq 0 \right\rbrace.\]

For other types of cones supported by MOSEK see Sec. 6.3 (Conic Quadratic Optimization), Sec. 6.5 (Conic Exponential Optimization), Sec. 6.6 (Semidefinite Optimization). Different cone types can appear together in one optimization problem.

Furthermore, each variable may belong to one cone at most. The constraint \(x_i - x_j = 0\) would however allow \(x_i\) and \(x_j\) to belong to different cones with same effect.

6.4.1 Example POW1

Consider the following optimization problem which involves powers of variables:

(6.6)\[\begin{split}\begin{array} {lrcl} \mbox{maximize} & x^{0.2}y^{0.8} + z^{0.4} - x & & \\ \mbox{subject to} & x+y+\frac12 z & = & 2, \\ & x,y,z & \geq & 0. \end{array}\end{split}\]

With \((x,y,z)=(x_0,x_1,x_2)\) we convert it into conic form using auxiliary variables as bounds for the power expressions:

(6.7)\[\begin{split}\begin{array} {lrcl} \mbox{maximize} & x_3 + x_4 - x_0 & & \\ \mbox{subject to} & x_0+x_1+\frac12 x_2 & = & 2, \\ & (x_0,x_1,x_3) & \in & \POW_3^{0.2,0.8}, \\ & (x_2,x_5,x_4) & \in & \POW_3^{0.4,0.6}, \\ & x_5 & = & 1. \end{array}\end{split}\]

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

The conic constraints are specified as columns in a list-typed matrix called cones, with rows for each associated detail. In example (6.7) we have two conic constraints:

    NUMCONES <- 2
    prob$cones <- matrix(list(), nrow=3, ncol=NUMCONES)
    rownames(prob$cones) <- c("type","sub","conepar")

    prob$cones[,1] <- list("PPOW", c(1, 2, 4), c(0.2, 0.8))
    prob$cones[,2] <- list("PPOW", c(3, 6, 5), c(0.4, 0.6))

The first row selects the type of cone, i.e., the power cone "MSK_CT_PPOW", noting that the prefix MSK_CT_ is optional. The second row selects the vector of variables constrained to the cone, identified by index (counting from one in the R language). The third row selects the cone parameters. The existence of this third row is ignored by nonparametric cones (compare, e.g., to Sec. 6.3.1 (Example CQO1)) and any parameter value can be assigned to non-parametric cones when mixed with parametric ones.

The code below produces the answer of (6.6) which is

[ 0.06389298  0.78308564  2.30604283 ]

Source code

Listing 6.4 Source code solving problem (6.6). Click here to download.
library("Rmosek")

pow1 <- function()
{
    # Specify the non-conic part of the problem.
    prob <- list(sense="max")
    prob$c  <- c(-1, 0, 0, 1, 1, 0)
    prob$A  <- Matrix(c(1, 1, 0.5, 0, 0, 0), nrow=1, sparse=TRUE)
    prob$bc <- rbind(blc=2, 
                     buc=2)
    prob$bx <- rbind(blx=c(rep(-Inf,5), 1), 
                     bux=c(rep( Inf,5), 1))
    
    # Specify the cones.
    NUMCONES <- 2
    prob$cones <- matrix(list(), nrow=3, ncol=NUMCONES)
    rownames(prob$cones) <- c("type","sub","conepar")

    prob$cones[,1] <- list("PPOW", c(1, 2, 4), c(0.2, 0.8))
    prob$cones[,2] <- list("PPOW", c(3, 6, 5), c(0.4, 0.6))

    #
    # Non-parametric cones (such as "QUAD") are ignorant to the existence of 
    # the third row "conepar" and can be assigned any value (such as NaN).
    #

    # Solve the problem
    r <- mosek(prob)

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

pow1()