6.6 Geometric Programming

Geometric programs (GP) are a particular class of optimization problems which can be expressed in special polynomial form as positive sums of generalized monomials. More precisely, a geometric problem in canonical form is

(6.17)$$\begin{array}{r}\begin{array}{lll}\text{minimize}& f_0(x)& \\ \text{subject to}& f_i(x)\le 1,& i=1,\dots ,m,\\ & x_j>0,& j=1,\dots ,n,\end{array}\end{array}$$

where each $f_0,\dots ,f_m$ is a posynomial, that is a function of the form

$$f(x)=\sum _k{c}_k{x}_1^{{\alpha }_{k1}}{x}_2^{{\alpha }_{k2}}\cdots x_n^{{\alpha }_{kn}}$$

with arbitrary real ${\alpha }_{ki}$ and $c_k>0$. The standard way to formulate GPs in convex form is to introduce a variable substitution

$$x_i=\exp (y_i).$$

Under this substitution all constraints in a GP can be reduced to the form

(6.18)$$\log (\sum _k\exp (a_k^{T}y+b_k))\le 0$$

involving a log-sum-exp bound. Moreover, constraints involving only a single monomial in $x$ can be even more simply written as a linear inequality:

$$a_k^{T}y+b_k\le 0$$

We refer to the MOSEK Modeling Cookbook and to [BKVH07] for more details on this reformulation. A geometric problem formulated in convex form can be entered into MOSEK with the help of exponential cones.

6.6.1 Example GP1

The following problem comes from [BKVH07]. Consider maximizing the volume of a $h\times w\times d$ box subject to upper bounds on the area of the floor and of the walls and bounds on the ratios $h/w$ and $d/w$:

(6.19)$$\begin{array}{r}\begin{array}{rrl}\text{maximize}& hwd& \\ \text{subject to}& 2(hw+hd)& \le A_{\mathrm{wall}},\\ & wd& \le A_{\mathrm{floor}},\\ & \alpha & \le h/w\le \beta ,\\ & \gamma & \le d/w\le \delta .\end{array}\end{array}$$

The decision variables in the problem are $h,w,d$. We make a substitution

$$h=\exp (x),w=\exp (y),d=\exp (z)$$

after which (6.19) becomes

(6.20)$$\begin{array}{r}\begin{array}{rl}\text{maximize}& x+y+z\\ \text{subject to}& \log (\exp (x+y+\log (2/A_{\mathrm{wall}}))+\exp (x+z+\log (2/A_{\mathrm{wall}})))\le 0,\\ & y+z\le \log (A_{\mathrm{floor}}),\\ & \log (\alpha )\le x-y\le \log (\beta ),\\ & \log (\gamma )\le z-y\le \log (\delta ).\end{array}\end{array}$$

Next, we demonstrate how to implement a log-sum-exp constraint (6.18). It can be written as:

(6.21)$$\begin{array}{r}\begin{array}{l}{u}_k\ge \exp (a_k^{T}y+b_k),(\mathrm{equiv}.(u_k,1,a_k^{T}y+b_k)\in K_{\exp }),\\ \sum _k{u}_k=1.\end{array}\end{array}$$

This presentation requires one extra variable $u_k$ for each monomial appearing in the original posynomial constraint. In this case the affine conic constraints (ACC, see Sec. 6.2 (From Linear to Conic Optimization)) take the form:

$$\begin{array}{r}\left[\begin{array}{ccccc}0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 1& 0& 1& 0& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ u_1\\ u_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\\ \log (2/A_{\mathrm{wall}})\\ 0\\ 1\\ \log (2/A_{\mathrm{wall}})\end{array}\right]\in K_{\exp }\times K_{\exp }.\end{array}$$

As a matter of demonstration we will also add the constraint

$$u_1+u_2-1=0$$

as an affine conic constraint. It means that to define the all the ACCs we need to produce the following affine expressions (AFE) and store them:

$$u_1,u_2,x+y+\log (2/A_{\mathrm{wall}}),x+z+\log (2/A_{\mathrm{wall}}),1.0,u_1+u_2-1.0.$$

We implement it by adding all the affine expressions (AFE) and then picking the ones required for each ACC:

Listing 6.7 Implementation of log-sum-exp as in (6.21). Click here to download.

    {
        let afei = task.get_num_afe()?;
        let u1 = task.get_num_var()?;
        let u2 = u1+1;

        let afeidx = &[0, 1, 2, 2, 3, 3, 5, 5];
        let varidx = &[u1, u2, x, y, x, z, u1, u2];
        let fval   = &[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0];
        let gfull  = &[0.0, 0.0, (2.0/Aw).ln(), (2.0/Aw).ln(), 1.0, -1.0];

        task.append_vars(2)?;
        task.append_afes(6)?;

        task.put_var_bound_slice_const(u1, u1+2, Boundkey::FR, -INF, INF)?;

        // Affine expressions appearing in affine conic constraints
        // in this order:
        // u1, u2, x+y+log(2/Awall), x+z+log(2/Awall), 1.0, u1+u2-1.0
        task.put_afe_f_entry_list(afeidx, varidx, fval)?;
        task.put_afe_g_slice(afei, afei+6, gfull)?;
        {
            let dom = task.append_primal_exp_cone_domain()?;

            // (u1, 1, x+y+log(2/Awall)) \in EXP
            task.append_acc(dom, &[0, 4, 2], &[0.0,0.0,0.0])?;

            // (u2, 1, x+z+log(2/Awall)) \in EXP
            task.append_acc(dom, &[1, 4, 3], &[0.0,0.0,0.0])?;
        }
        {
            let dom = task.append_rzero_domain(1)?;
            // The constraint u1+u2-1 \in \ZERO is added also as an ACC
            task.append_acc(dom, &[5], &[0.0])?;
        }
    }

We can now use this function to assemble all constraints in the model. The linear part of the problem is entered as in Sec. 6.1 (Linear Optimization).

Listing 6.8 Source code solving problem (6.20). Click here to download.

#[allow(non_snake_case)]
fn max_volume_box(Aw : f64,
                  Af : f64,
                  alpha : f64,
                  beta : f64,
                  gamma : f64,
                  delta : f64) -> Result<Vec<f64>,String>
{
    let numvar = 3i32;  // Variables in original problem
    /* Create the optimization task. */
    let mut task = match Task::new() {
        Some(e) => e,
        None => return Err("Failed to create task".to_string()),
        }.with_callbacks();

    // Directs the log task stream to the user specified
    // method task_msg_obj.stream
    task.put_stream_callback(Streamtype::LOG, |msg| print!("{}",msg))?;

    // Add variables and constraints
    task.append_vars(numvar)?;

    let x = 0i32;
    let y = 1i32;
    let z = 2i32;

    // Objective is the sum of three first variables
    task.put_obj_sense(Objsense::MAXIMIZE)?;
    task.put_c_slice(0, numvar, &[1.0,1.0,1.0])?;

    task.put_var_bound_slice_const(0, numvar, Boundkey::FR, -INF, INF)?;

    task.append_cons(3)?;
    // s0+s1 < 1 <=> log(s0+s1) < 0
    task.put_aij_list(&[0,0,1,1,2,2],
                      &[y, z, x, y, z, y],
                      &[1.0, 1.0, 1.0, -1.0, 1.0, -1.0])?;

    task.put_con_bound(0,Boundkey::UP,-INF,Af.ln())?;
    task.put_con_bound(1,Boundkey::RA,alpha.ln(),beta.ln())?;
    task.put_con_bound(2,Boundkey::RA,gamma.ln(),delta.ln())?;


    {
        let afei = task.get_num_afe()?;
        let u1 = task.get_num_var()?;
        let u2 = u1+1;

        let afeidx = &[0, 1, 2, 2, 3, 3, 5, 5];
        let varidx = &[u1, u2, x, y, x, z, u1, u2];
        let fval   = &[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0];
        let gfull  = &[0.0, 0.0, (2.0/Aw).ln(), (2.0/Aw).ln(), 1.0, -1.0];

        task.append_vars(2)?;
        task.append_afes(6)?;

        task.put_var_bound_slice_const(u1, u1+2, Boundkey::FR, -INF, INF)?;

        // Affine expressions appearing in affine conic constraints
        // in this order:
        // u1, u2, x+y+log(2/Awall), x+z+log(2/Awall), 1.0, u1+u2-1.0
        task.put_afe_f_entry_list(afeidx, varidx, fval)?;
        task.put_afe_g_slice(afei, afei+6, gfull)?;
        {
            let dom = task.append_primal_exp_cone_domain()?;

            // (u1, 1, x+y+log(2/Awall)) \in EXP
            task.append_acc(dom, &[0, 4, 2], &[0.0,0.0,0.0])?;

            // (u2, 1, x+z+log(2/Awall)) \in EXP
            task.append_acc(dom, &[1, 4, 3], &[0.0,0.0,0.0])?;
        }
        {
            let dom = task.append_rzero_domain(1)?;
            // The constraint u1+u2-1 \in \ZERO is added also as an ACC
            task.append_acc(dom, &[5], &[0.0])?;
        }
    }

    let _trm = task.optimize()?;
    task.write_data("gp1.ptf")?;
    let mut xyz = vec![0.0; 3];
    task.get_xx_slice(Soltype::ITR, 0i32, numvar, xyz.as_mut_slice())?;

    // task.write_data("gp1.ptf")?;

    Ok(xyz.iter().map(|v| v.exp()).collect())
}

Given sample data we obtain the solution $h,w,d$ as follows:

Listing 6.9 Sample data for problem (6.19). Click here to download.

#[allow(non_snake_case)]
fn main() -> Result<(),String> {
    // maximize     h*w*d
    // subjecto to  2*(h*w + h*d) <= Awall
    //              w*d <= Afloor
    //              alpha <= h/w <= beta
    //              gamma <= d/w <= delta
    //
    // Variable substitutions:  h = exp(x), w = exp(y), d = exp(z).
    //
    // maximize     x+y+z
    // subject      log( exp(x+y+log(2/Awall)) + exp(x+z+log(2/Awall)) ) <= 0
    //                              y+z <= log(Afloor)
    //              log( alpha ) <= x-y <= log( beta )
    //              log( gamma ) <= z-y <= log( delta )
    //
    // Finally, the model we will implement:
    //
    // maximize   x+y+z
    // subject to s0 > exp(x+y+log(2/Awall); (s0,1,x+y+log(2/Awall)) in PEXP
    //            s1 > exp(x+z+log(2/Awall); (s1,1,x+z+log(2/Awall)) in PEXP
    //            s0+s1 < 1
    //
    //            y+z < log Afloor
    //
    //            x-y in [log alpha; log beta]
    //            z-y in [log gamma; log delta]
    //

    // (x,y,z) in pexp : x0 > x1 * exp(x2/x1)

    let Aw    = 200.0;
    let Af    = 50.0;
    let alpha = 2.0;
    let beta  = 10.0;
    let gamma = 2.0;
    let delta = 10.0;

    let hwd = max_volume_box(Aw, Af, alpha, beta, gamma, delta)?;

    println!("h={:.4} w={:.4} d={:.4}\n", hwd[0], hwd[1], hwd[2]);

    Ok(())
}