15.11 Nonlinear interfaces (obsolete)ΒΆ

Important

This is a legacy document for users familiar with SCopt, DGopt, EXPopt, mskenopt, mskscopt and mskgpopt from previous versions of MOSEK. These interfaces have now been removed. We assume familiarity with documentation included in version 8. All problems expressible with this interface can (and should) be reformulated using the exponential cone and power cones.

New users should formulate problems involving powers, logarithms and exponentials directly in conic form.

Conversion tutorial

We recommend converting all nonlinear problems using SCopt, DGopt, EXPopt, mskenopt, mskscopt and mskgpopt into conic form. Depending on the values of \(f,g,h\) either the epigraph or hypograph of a SCopt function if convex, and a bounding variable can be introduced following the basic rules below. We assume all variables are within safe bounds where the SCopt operators are defined and convex. We also assume \(f>0\).

A more comprehensive modeling guide for these types of problems can be found in the MOSEK Modeling Cookbook.

Powers

Consider \(f(x+h)^g\). This can be reformulated using the power cone.

  • If \(g>1\) then \(t\geq f(x+h)^g\) is equivalent to \((t/f)^{1/g}\geq |x+h|\), that is \((t/f,1,x+h)\in \POW_3^{1/g,1-1/g}\).
  • If \(0<g<1\) then \(|t|\leq f(x+h)^g\) is equivalent to \((x+h, 1, t/f)\in \POW_3^{g,1-g}\).
  • If \(g<0\) then \(t\geq f(x+h)^g\) is equivalent to \((t/f)(x+h)^{-g}\geq 1\), that is \((t/f,x+h,1)\in \POW_3^{1/(1-g),-g/(1-g)}\).

Logarithm

The bound \(t\leq f\log(gx+h)\) is equivalent to \((gx+h, 1, t/f)\in\EXP\).

Entropy

The bound \(t\geq fx\log{x}\) is equivalent to \((1, x, -t/f)\in\EXP\).

Exponential

The bound \(t\geq f\exp(gx+h)\) is equivalent to \((t/f, 1, gx+h)\in\EXP\).

Exponential optimization (EXPopt), Geometric programming (mskgpopt)

For a basic tutorial in geometric programming (GP) see Sec. 6.8 (Geometric Programming).

An exponential optimization problem in standard form consists of constraints of the type:

\[t\geq \log\left(\sum_i\exp(a_i^Tx+b_i)\right).\]

This log-sum-exp bound is equivalent to

\[\sum_i \exp(a_i^Tx+b_i-t) \leq 1\]

and requires bounding each exponential function as explained above.

Dual geometric optimization (DGopt)

The objective function of a dual geometric problem involves maximizing expressions of the form

\[x\log{\frac{c}{x}} \quad \mathrm{and} \quad x_i\log{\frac{e^Tx}{x_i}},\]

which can be achieved using bounds \(t\leq x\log{\frac{y}{x}}\), that is \((t,x,y)\in\EXP\).