Convexity and Concavity Detection in Computational Graphs: Tree Walks for Convexity Assessment

We examine symbolic tools associated with two modeling systems for mathematical programming, which can be used to automatically detect the presence or absence of convexity and concavity in the objective and constraint functions, as well as convexity of the feasible set in some cases. The coconut solver system Schichl (2004b) focuses on nonlinear global continuous optimization and possesses its own modeling language and data structures. The Dr.ampl meta-solver (Fourer and Orban, 2007) aims to analyze nonlinear differentiable optimization models and hooks into the ampl Solver Library (Gay, 2002). Our symbolic convexity analysis may be supplemented, when it returns inconclusive results, with a numerical phase that may detect non-convexity. We report numerical results using these tools on sets of test problems for both global and local optimization.