# SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

## TL;DR

This paper introduces SOS-convex semi-algebraic functions, extending SOS-convex polynomials, and demonstrates that certain nonsmooth convex optimization problems can be efficiently solved via semi-definite programming, with applications to robust optimization.

## Contribution

The paper defines SOS-convex semi-algebraic functions and shows they enable solving a broad class of nonsmooth convex problems through SDP, extending existing relaxation results.

## Key findings

- Optimal solutions of SOS-convex semi-algebraic programs can be obtained via SDP.
- The approach applies to robust optimization problems with spectrahedron data uncertainty.
- Extended SDP relaxation results for broader classes of uncertainty sets.

## Abstract

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with $\ell_1$-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semi-algebraic programs can be found by solving a single semi-definite programming problem (SDP). We achieve the results by using tools from semi-algebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we outline how the derived results can be applied to show that robust SOS-convex optimization problems under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers the open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a robust solution from the semi-definite programming relaxation in this broader setting.