Automatic robust convex programming

Abstract

This paper presents the robust optimization framework in the modelling language YALMIP, which carries out robust modelling and uncertainty elimination automatically and allows the user to concentrate on the high-level model. While introducing the software package, a brief summary of robust optimization is given, as well as some comments on modelling and tractability of complex convex uncertain optimization problems.

Keywords:

• robust optimization
• conic programming
• modelling software

Notes

Assuming the uncertainty set is bounded and has a strict interior.

This is not necessary, any polytope can be handled by expressing w as a convex combination of its vertices, if such vertex enumeration is available and tractable. The variable used in the convex combination lives, by definition, on a simplex.

This is not a restriction, since any polynomial can be rendered homogeneous on a simplex by multiplying monomial terms with suitable powers of .

To be precise, the theorem concerns a strict inequality of a homogeneous polynomial, and the restriction on the simplex can be relaxed to any set in the positive orthant excluding the origin.

For every row a _i , we have to maximize subject to . Since the dimension of w is n, there are 2n inequalities in the uncertainty constraint set. There will thus be 2n dual variables for each row. The dual variables are constrained by n equality constraints and 2n inequality constraints. Summing up and adding the original constraints and variables leads to the result.

Methods to impose strict inequalities in a sum-of-squares setting is a delicate issue beyond the scope of this example.