A new efficient eigenvalue bounding method for convexity detection with applications in global optimization and control

We introduce a new method for the calculation of bounds on the eigenvalues of Hessian matrices of twice continuously differentiable functions. Eigenvalue bounds of Hessian matrices arise in a number of notoriously difficult tasks in computational chemical engineering. For example, Hessian matrix eigenvalue bounds are used in global nonlinear optimization, global convexity/concavity analysis in convex optimization, and global positive/negative invariance analysis in nonlinear control. We stress that the improvements in computational complexity to be stated below are only possible, because the desired Hessian matrix eigenvalue bounds are calculated without ever calculating the Hessian matrix itself. To the author’s knowledge the proposed method is the first Hessianmatrix-free approach to bounding Hessian matrix eigenvalues. We start with a more precise problem statement in the next section, summarize the methodological advances in a subsequent section, and turn to applications in the last section.